\usepackage or this document’s own macros. Corpus-wide, \bm occurs 327 times, \Perp is defined by no package and spans 10 documents, and 11,088 of 11,624 undefined occurrences are the source document’s own macros. A row that looks wrong here may be correct, and one that looks right here may not compile. The LaTeX report renders through the document’s own preamble and is the surface to judge from.4597 rows
The page, the confidence and the picture of an inline formula are its HOST LINE's --- a formula has none of its own. A line's confidence is not a formula's.
| Identifier | Page | Conf. | LaTeX source | Rendered | Image |
|---|---|---|---|---|---|
| lyche-numerical-linear-algebra_FO0001 | 7 | 1.000 | \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO0002 | 7 | 1.000 | \boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO0003 | 7 | 0.993 | \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO0004 | 7 | 0.993 | \boldsymbol{A} \boldsymbol{x}=\boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO0005 | 7 | 1.000 | \boldsymbol{b}-\boldsymbol{A} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO0006 | 7 | 1.000 | \lambda | | |
| lyche-numerical-linear-algebra_FO0007 | 7 | 0.997 | \boldsymbol{A} \boldsymbol{x}=\lambda \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO0008 | 21 | 1.000 | \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO0009 | 21 | 1.000 | \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO0010 | 179 | 0.987 | \boldsymbol{A}^{*} \boldsymbol{A}, \boldsymbol{A} \boldsymbol{A}^{*} | | |
| lyche-numerical-linear-algebra_FO0011 | 26 | 1.000 | p | | |
| lyche-numerical-linear-algebra_FO0012 | 267 | 1.000 | \omega | | |
| lyche-numerical-linear-algebra_FO0013 | 335 | 1.000 | \lambda_{m} | | |
| lyche-numerical-linear-algebra_FO0014 | 43 | 1.000 | T | | |
| lyche-numerical-linear-algebra_FO0020 | 49 | 1.000 | f(x)=x^{4} | | |
| lyche-numerical-linear-algebra_FO0024 | 120 | 1.000 | \boldsymbol{v}_{1} | | |
| lyche-numerical-linear-algebra_FO0026 | 27 | 1.000 | c | | |
| lyche-numerical-linear-algebra_FO0029 | 28 | 1.000 | \mathcal{S} | | |
| lyche-numerical-linear-algebra_FO0030 | 28 | 1.000 | \mathcal{T} | | |
| lyche-numerical-linear-algebra_FO0033 | 251 | 1.000 | n=100 | | |
| lyche-numerical-linear-algebra_FO0038 | 277 | 1.000 | n=2500 | | |
| lyche-numerical-linear-algebra_FO0039 | 292 | 0.996 | Q(x, y) | | |
| lyche-numerical-linear-algebra_FO0040 | 291 | 1.000 | Q | | |
| lyche-numerical-linear-algebra_FO0041 | 301 | 1.000 | \boldsymbol{x}-\boldsymbol{x}_{0} | | |
| lyche-numerical-linear-algebra_FO0042 | 301 | 1.000 | \mathbb{W}_{k} | | |
| lyche-numerical-linear-algebra_FO0044 | 51 | 1.000 | \mu_{1} | | |
| lyche-numerical-linear-algebra_FO0047 | 328 | 0.989 | R_{i} | | |
| lyche-numerical-linear-algebra_FO0048 | 268 | 1.000 | k_{n} | | |
| lyche-numerical-linear-algebra_FO0049 | 21 | 1.000 | n | | |
| lyche-numerical-linear-algebra_FO0050 | 268 | 1.000 | 10^{-8} | | |
| lyche-numerical-linear-algebra_FO0053 | 97 | 1.000 | K | | |
| lyche-numerical-linear-algebra_FO0056 | 21 | 1.000 | \mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C} | | |
| lyche-numerical-linear-algebra_FO0057 | 21 | 1.000 | v:=e | | |
| lyche-numerical-linear-algebra_FO0058 | 21 | 1.000 | v | | |
| lyche-numerical-linear-algebra_FO0059 | 21 | 1.000 | e | | |
| lyche-numerical-linear-algebra_FO0060 | 21 | 1.000 | \boldsymbol{x} \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO0061 | 21 | 1.000 | x_{i} \in \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO0062 | 21 | 1.000 | i=1, \ldots, n | | |
| lyche-numerical-linear-algebra_FO0063 | 21 | 1.000 | \boldsymbol{x}^{T} | | |
| lyche-numerical-linear-algebra_FO0064 | 22 | 1.000 | \mathbb{R}^{m \times n} | | |
| lyche-numerical-linear-algebra_FO0065 | 22 | 1.000 | m | | |
| lyche-numerical-linear-algebra_FO0066 | 22 | 0.998 | i | | |
| lyche-numerical-linear-algebra_FO0067 | 22 | 0.998 | j | | |
| lyche-numerical-linear-algebra_FO0068 | 22 | 0.998 | a_{i, j}, a_{i j} | | |
| lyche-numerical-linear-algebra_FO0069 | 22 | 1.000 | \boldsymbol{A}(i, j) | | |
| lyche-numerical-linear-algebra_FO0070 | 22 | 1.000 | (\boldsymbol{A})_{i, j} | | |
| lyche-numerical-linear-algebra_FO0071 | 22 | 1.000 | \boldsymbol{a}_{: j} | | |
| lyche-numerical-linear-algebra_FO0072 | 22 | 1.000 | \boldsymbol{a}_{i:}^{T} | | |
| lyche-numerical-linear-algebra_FO0073 | 22 | 1.000 | \boldsymbol{a}_{j} | | |
| lyche-numerical-linear-algebra_FO0074 | 22 | 1.000 | \boldsymbol{a}_{i}^{T} | | |
| lyche-numerical-linear-algebra_FO0075 | 22 | 1.000 | m=1 | | |
| lyche-numerical-linear-algebra_FO0076 | 22 | 1.000 | n=1 | | |
| lyche-numerical-linear-algebra_FO0077 | 22 | 1.000 | m=n | | |
| lyche-numerical-linear-algebra_FO0078 | 22 | 0.999 | \boldsymbol{A}, \boldsymbol{B}, \boldsymbol{C}, \cdots | | |
| lyche-numerical-linear-algebra_FO0079 | 22 | 1.000 | \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z}, \cdots | | |
| lyche-numerical-linear-algebra_FO0080 | 22 | 1.000 | x=a+i b | | |
| lyche-numerical-linear-algebra_FO0081 | 22 | 1.000 | a, b | | |
| lyche-numerical-linear-algebra_FO0082 | 22 | 1.000 | i^{2}=-1 | | |
| lyche-numerical-linear-algebra_FO0083 | 22 | 1.000 | \mathbb{C} | | |
| lyche-numerical-linear-algebra_FO0084 | 22 | 1.000 | a=\operatorname{Re} x | | |
| lyche-numerical-linear-algebra_FO0085 | 22 | 1.000 | b=\operatorname{Im} x | | |
| lyche-numerical-linear-algebra_FO0086 | 22 | 1.000 | x | | |
| lyche-numerical-linear-algebra_FO0087 | 22 | 1.000 | \bar{x}:=a-i b | | |
| lyche-numerical-linear-algebra_FO0088 | 22 | 1.000 | |x|:=\sqrt{\bar{x} x}=\sqrt{a^{2}+b^{2}} | | |
| lyche-numerical-linear-algebra_FO0089 | 23 | 1.000 | e^{x+y}=e^{x} e^{y} | | |
| lyche-numerical-linear-algebra_FO0090 | 23 | 1.000 | x, y \in \mathbb{C} | | |
| lyche-numerical-linear-algebra_FO0091 | 23 | 1.000 | \boldsymbol{A} \in \mathbb{C}^{m \times n} | | |
| lyche-numerical-linear-algebra_FO0092 | 23 | 1.000 | \boldsymbol{x} \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO0093 | 23 | 1.000 | \boldsymbol{x}^{*}:=\overline{\boldsymbol{x}}^{T}=\left[\bar{x}_{1}, \ldots, \bar{x}_{n}\right] | | |
| lyche-numerical-linear-algebra_FO0094 | 23 | 0.809 | x^{*}=x^{T} | | |
| lyche-numerical-linear-algebra_FO0095 | 23 | 1.000 | \boldsymbol{x}, \boldsymbol{y} \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO0096 | 23 | 1.000 | a \in \mathbb{C} | | |
| lyche-numerical-linear-algebra_FO0097 | 23 | 1.000 | \boldsymbol{C}:=\boldsymbol{A}+\boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO0098 | 23 | 1.000 | \boldsymbol{A}, \boldsymbol{B}, \boldsymbol{C} | | |
| lyche-numerical-linear-algebra_FO0099 | 23 | 1.000 | c_{i j}:=a_{i j}+b_{i j} | | |
| lyche-numerical-linear-algebra_FO0100 | 23 | 1.000 | i, j | | |
| lyche-numerical-linear-algebra_FO0101 | 23 | 1.000 | \boldsymbol{C}:=\alpha \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO0102 | 23 | 1.000 | c_{i j}:=\alpha a_{i j} | | |
| lyche-numerical-linear-algebra_FO0103 | 23 | 1.000 | \boldsymbol{C}:=\boldsymbol{A} \boldsymbol{B}, \boldsymbol{C}=\boldsymbol{A} \cdot \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO0104 | 23 | 1.000 | \boldsymbol{C}=\boldsymbol{A} * \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO0105 | 23 | 1.000 | \boldsymbol{A} \in | | |
| lyche-numerical-linear-algebra_FO0106 | 23 | 1.000 | \mathbb{C}^{m \times p}, \boldsymbol{B} \in \mathbb{C}^{p \times n}, \boldsymbol{C} \in \mathbb{C}^{m \times n} | | |
| lyche-numerical-linear-algebra_FO0107 | 23 | 1.000 | c_{i j}:=\sum_{k=1}^{p} a_{i k} b_{k j} | | |
| lyche-numerical-linear-algebra_FO0108 | 23 | 1.000 | i=1, \ldots, m | | |
| lyche-numerical-linear-algebra_FO0109 | 23 | 1.000 | j=1, \ldots, n | | |
| lyche-numerical-linear-algebra_FO0110 | 23 | 1.000 | \boldsymbol{C}:=\boldsymbol{A} \times \boldsymbol{B}, \boldsymbol{D}:=\boldsymbol{A} / \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO0111 | 23 | 1.000 | \boldsymbol{E}:= | | |
| lyche-numerical-linear-algebra_FO0112 | 23 | 1.000 | \boldsymbol{A} \wedge r | | |
| lyche-numerical-linear-algebra_FO0113 | 23 | 1.000 | c_{i j}:=a_{i j} b_{i j}, d_{i j}:=a_{i j} / b_{i j} | | |
| lyche-numerical-linear-algebra_FO0114 | 23 | 1.000 | e_{i j}:=a_{i j}^{r} | | |
| lyche-numerical-linear-algebra_FO0115 | 23 | 1.000 | r | | |
| lyche-numerical-linear-algebra_FO0116 | 23 | 1.000 | \boldsymbol{A} / \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO0117 | 23 | 1.000 | \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO0118 | 23 | 1.000 | \boldsymbol{C}=\boldsymbol{A} \times \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO0119 | 23 | 1.000 | \boldsymbol{A} \in \mathbb{R}^{m \times n} | | |
| lyche-numerical-linear-algebra_FO0120 | 23 | 1.000 | \boldsymbol{A}^{T} | | |
| lyche-numerical-linear-algebra_FO0121 | 23 | 1.000 | \boldsymbol{A}^{*} | | |
| lyche-numerical-linear-algebra_FO0122 | 23 | 1.000 | n \times m | | |
| lyche-numerical-linear-algebra_FO0123 | 23 | 1.000 | a_{i j}^{T}:=a_{j i} | | |
| lyche-numerical-linear-algebra_FO0124 | 23 | 1.000 | a_{i j}^{*}:=\bar{a}_{j i} | | |
| lyche-numerical-linear-algebra_FO0125 | 23 | 1.000 | n, p | | |
| lyche-numerical-linear-algebra_FO0126 | 23 | 1.000 | (\boldsymbol{A} \boldsymbol{B})^{T}=\boldsymbol{B}^{T} \boldsymbol{A}^{T} | | |
| lyche-numerical-linear-algebra_FO0127 | 23 | 1.000 | (\boldsymbol{A} \boldsymbol{B})^{*}=\boldsymbol{B}^{*} \boldsymbol{A}^{*} | | |
| lyche-numerical-linear-algebra_FO0128 | 23 | 1.000 | \boldsymbol{A} \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO0129 | 23 | 1.000 | \boldsymbol{A}^{T}=\boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO0130 | 23 | 1.000 | \boldsymbol{A}^{*}=\boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO0131 | 23 | 1.000 | \boldsymbol{I}_{n}=\boldsymbol{I}:=\left[\delta_{i j}\right]_{i, j=1}^{n} | | |
| lyche-numerical-linear-algebra_FO0132 | 23 | 1.000 | \boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO0133 | 23 | 1.000 | \boldsymbol{e}_{1}, \boldsymbol{e}_{2}, \ldots, \boldsymbol{e}_{n} | | |
| lyche-numerical-linear-algebra_FO0134 | 24 | 1.000 | \boldsymbol{A} \in \mathbb{R}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO0135 | 24 | 1.000 | a_{i j}=0 | | |
| lyche-numerical-linear-algebra_FO0136 | 24 | 1.000 | i \neq j | | |
| lyche-numerical-linear-algebra_FO0137 | 24 | 1.000 | i>j | | |
| lyche-numerical-linear-algebra_FO0138 | 24 | 1.000 | i<j | | |
| lyche-numerical-linear-algebra_FO0139 | 24 | 1.000 | i>j+1 | | |
| lyche-numerical-linear-algebra_FO0140 | 24 | 1.000 | i<j+1 | | |
| lyche-numerical-linear-algebra_FO0141 | 24 | 1.000 | |i-j|>1 | | |
| lyche-numerical-linear-algebra_FO0142 | 24 | 0.999 | d | | |
| lyche-numerical-linear-algebra_FO0143 | 24 | 0.999 | |i-j|>d | | |
| lyche-numerical-linear-algebra_FO0144 | 24 | 1.000 | n \times n | | |
| lyche-numerical-linear-algebra_FO0145 | 24 | 1.000 | b_{i i}:=d_{i} | | |
| lyche-numerical-linear-algebra_FO0146 | 24 | 1.000 | i=1, \ldots, n, b_{i+1, i}:=a_{i}, b_{i, i+1}:=c_{i} | | |
| lyche-numerical-linear-algebra_FO0147 | 24 | 1.000 | i=1, \ldots, n-1 | | |
| lyche-numerical-linear-algebra_FO0148 | 24 | 1.000 | b_{i j}:=0 | | |
| lyche-numerical-linear-algebra_FO0149 | 24 | 1.000 | 1 \leq i_{1}<i_{2}<\cdots<i_{r} \leq m, 1 \leq j_{1}<j_{2}<\cdots< | | |
| lyche-numerical-linear-algebra_FO0150 | 24 | 1.000 | j_{c} \leq n | | |
| lyche-numerical-linear-algebra_FO0151 | 24 | 1.000 | \boldsymbol{A}(\boldsymbol{i}, \boldsymbol{j}) \in \mathbb{C}^{r \times c} | | |
| lyche-numerical-linear-algebra_FO0152 | 24 | 1.000 | \boldsymbol{i}:=\left[i_{1}, \ldots, i_{r}\right] | | |
| lyche-numerical-linear-algebra_FO0153 | 24 | 1.000 | \boldsymbol{j}:=\left[j_{1}, \ldots, j_{c}\right] | | |
| lyche-numerical-linear-algebra_FO0154 | 25 | 1.000 | \mathbb{R}^{2} | | |
| lyche-numerical-linear-algebra_FO0155 | 25 | 1.000 | \mathbb{R}^{3} | | |
| lyche-numerical-linear-algebra_FO0156 | 25 | 1.000 | \mathcal{V} | | |
| lyche-numerical-linear-algebra_FO0157 | 25 | 1.000 | +: \mathcal{V} \times \mathcal{V} \longrightarrow \mathcal{V} | | |
| lyche-numerical-linear-algebra_FO0158 | 25 | 0.992 | \cdot: \mathbb{R} \times \mathcal{V} \longrightarrow \mathcal{V} | | |
| lyche-numerical-linear-algebra_FO0159 | 25 | 1.000 | \boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w} | | |
| lyche-numerical-linear-algebra_FO0160 | 25 | 1.000 | c, d | | |
| lyche-numerical-linear-algebra_FO0161 | 25 | 1.000 | \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO0162 | 25 | 1.000 | \boldsymbol{u}+\boldsymbol{v} | | |
| lyche-numerical-linear-algebra_FO0163 | 25 | 0.960 | \boldsymbol{u}+\boldsymbol{v}=\boldsymbol{v}+\boldsymbol{u} | | |
| lyche-numerical-linear-algebra_FO0164 | 25 | 0.994 | \boldsymbol{u}+(\boldsymbol{v}+\boldsymbol{w})=(\boldsymbol{u}+\boldsymbol{v})+\boldsymbol{w} | | |
| lyche-numerical-linear-algebra_FO0165 | 25 | 0.999 | \mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0166 | 25 | 0.999 | \boldsymbol{u}+\mathbf{0}=\boldsymbol{u} | | |
| lyche-numerical-linear-algebra_FO0167 | 25 | 1.000 | \boldsymbol{u} | | |
| lyche-numerical-linear-algebra_FO0168 | 25 | 1.000 | -\boldsymbol{u} | | |
| lyche-numerical-linear-algebra_FO0169 | 25 | 1.000 | \boldsymbol{u}+(-\boldsymbol{u})=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0170 | 25 | 1.000 | c \cdot \boldsymbol{u} | | |
| lyche-numerical-linear-algebra_FO0171 | 25 | 1.000 | c \cdot(\boldsymbol{u}+\boldsymbol{v})=c \cdot \boldsymbol{u}+c \cdot \boldsymbol{v} | | |
| lyche-numerical-linear-algebra_FO0172 | 25 | 1.000 | (c+d) \cdot \boldsymbol{u}=c \cdot \boldsymbol{u}+d \cdot \boldsymbol{u} | | |
| lyche-numerical-linear-algebra_FO0173 | 25 | 1.000 | c \cdot(d \cdot \boldsymbol{u})=(c d) \cdot \boldsymbol{u} | | |
| lyche-numerical-linear-algebra_FO0174 | 25 | 1.000 | 1 \cdot \boldsymbol{u}=\boldsymbol{u} | | |
| lyche-numerical-linear-algebra_FO0175 | 25 | 0.589 | c \boldsymbol{v} | | |
| lyche-numerical-linear-algebra_FO0176 | 25 | 0.589 | c \cdot \boldsymbol{v} | | |
| lyche-numerical-linear-algebra_FO0177 | 25 | 1.000 | \boldsymbol{u}-\boldsymbol{v}:=\boldsymbol{u}+(-\boldsymbol{v}) | | |
| lyche-numerical-linear-algebra_FO0178 | 25 | 0.974 | \boldsymbol{u} \in \mathcal{V} | | |
| lyche-numerical-linear-algebra_FO0179 | 25 | 1.000 | 0 \boldsymbol{u}=\mathbf{0}, c \mathbf{0}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0180 | 25 | 1.000 | -\boldsymbol{u}=(-1) \boldsymbol{u} | | |
| lyche-numerical-linear-algebra_FO0181 | 25 | 1.000 | n \in \mathbb{N} | | |
| lyche-numerical-linear-algebra_FO0182 | 25 | 1.000 | \mathcal{D} | | |
| lyche-numerical-linear-algebra_FO0183 | 25 | 1.000 | d \in \mathbb{N} | | |
| lyche-numerical-linear-algebra_FO0184 | 25 | 1.000 | \boldsymbol{f}, \boldsymbol{g}: \mathcal{D} \rightarrow \mathbb{R}^{d} | | |
| lyche-numerical-linear-algebra_FO0185 | 25 | 1.000 | \boldsymbol{f}, \boldsymbol{g} | | |
| lyche-numerical-linear-algebra_FO0186 | 25 | 1.000 | \boldsymbol{f}(t)=\boldsymbol{g}(t) | | |
| lyche-numerical-linear-algebra_FO0187 | 25 | 1.000 | t \in \mathcal{D} | | |
| lyche-numerical-linear-algebra_FO0188 | 25 | 0.998 | \boldsymbol{f}(t)=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0189 | 25 | 1.000 | \boldsymbol{f} | | |
| lyche-numerical-linear-algebra_FO0190 | 25 | 1.000 | -\boldsymbol{f}=(-1) \boldsymbol{f} | | |
| lyche-numerical-linear-algebra_FO0191 | 25 | 1.000 | d>1 | | |
| lyche-numerical-linear-algebra_FO0192 | 25 | 0.997 | n \geq 0 | | |
| lyche-numerical-linear-algebra_FO0193 | 25 | 0.997 | \Pi_{n} | | |
| lyche-numerical-linear-algebra_FO0194 | 25 | 1.000 | p: \mathbb{R} \rightarrow \mathbb{R}, p: \mathbb{R} \rightarrow \mathbb{C} | | |
| lyche-numerical-linear-algebra_FO0195 | 25 | 1.000 | p: \mathbb{C} \rightarrow \mathbb{C} | | |
| lyche-numerical-linear-algebra_FO0196 | 26 | 1.000 | a_{0}, \ldots, a_{n} | | |
| lyche-numerical-linear-algebra_FO0197 | 26 | 1.000 | 0 \leq k \leq n | | |
| lyche-numerical-linear-algebra_FO0198 | 26 | 1.000 | p(t)=a_{0}+\cdots+a_{k} t^{k} | | |
| lyche-numerical-linear-algebra_FO0199 | 26 | 1.000 | a_{k} \neq 0 | | |
| lyche-numerical-linear-algebra_FO0200 | 26 | 1.000 | n \geq 1 | | |
| lyche-numerical-linear-algebra_FO0201 | 26 | 1.000 | \mathcal{X}:=\left\{\boldsymbol{x}_{1}, \ldots, \boldsymbol{x}_{n}\right\} | | |
| lyche-numerical-linear-algebra_FO0202 | 26 | 1.000 | c_{1}, \ldots, c_{n} | | |
| lyche-numerical-linear-algebra_FO0203 | 26 | 1.000 | c_{1} \boldsymbol{x}_{1}+\cdots+c_{n} \boldsymbol{x}_{n} | | |
| lyche-numerical-linear-algebra_FO0204 | 26 | 1.000 | \boldsymbol{x}_{1}, \ldots, \boldsymbol{x}_{n} | | |
| lyche-numerical-linear-algebra_FO0205 | 26 | 1.000 | c_{j} \boldsymbol{x}_{j} \neq \mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0206 | 26 | 0.999 | \mathcal{X} | | |
| lyche-numerical-linear-algebra_FO0207 | 26 | 0.999 | \operatorname{span}(\mathcal{X}) | | |
| lyche-numerical-linear-algebra_FO0208 | 26 | 1.000 | \left\{\boldsymbol{x}_{1}, \ldots, \boldsymbol{x}_{n}\right\} | | |
| lyche-numerical-linear-algebra_FO0209 | 26 | 1.000 | \mathcal{V}=\operatorname{span}\left(\left\{\boldsymbol{x}_{1}, \ldots, \boldsymbol{x}_{n}\right\}\right) | | |
| lyche-numerical-linear-algebra_FO0210 | 26 | 1.000 | \boldsymbol{x}=\left[x_{1}, \ldots, x_{m}\right]^{T} | | |
| lyche-numerical-linear-algebra_FO0211 | 26 | 1.000 | \mathbb{C}^{m} | | |
| lyche-numerical-linear-algebra_FO0212 | 26 | 0.991 | \boldsymbol{x}=x_{1} \boldsymbol{e}_{1}+x_{2} \boldsymbol{e}_{2}+\cdots+x_{m} \boldsymbol{e}_{m} | | |
| lyche-numerical-linear-algebra_FO0213 | 26 | 0.991 | \mathbb{C}^{m}=\operatorname{span}\left(\left\{\boldsymbol{e}_{1}, \ldots, \boldsymbol{e}_{m}\right\}\right) | | |
| lyche-numerical-linear-algebra_FO0214 | 26 | 1.000 | \mathbb{R}^{m} | | |
| lyche-numerical-linear-algebra_FO0215 | 26 | 1.000 | \Pi=\cup_{n} \Pi_{n} | | |
| lyche-numerical-linear-algebra_FO0216 | 26 | 1.000 | \Pi | | |
| lyche-numerical-linear-algebra_FO0217 | 26 | 1.000 | \Pi= | | |
| lyche-numerical-linear-algebra_FO0218 | 26 | 0.986 | \operatorname{span}\left(\left\{p_{1}, \ldots, p_{m}\right\}\right) | | |
| lyche-numerical-linear-algebra_FO0219 | 26 | 0.986 | p_{1}, \ldots, p_{m} | | |
| lyche-numerical-linear-algebra_FO0220 | 26 | 1.000 | p_{j} | | |
| lyche-numerical-linear-algebra_FO0221 | 26 | 1.000 | j=1, \ldots, m | | |
| lyche-numerical-linear-algebra_FO0222 | 26 | 0.958 | \mathcal{X}=\left\{\boldsymbol{x}_{1}, \ldots, \boldsymbol{x}_{n}\right\} | | |
| lyche-numerical-linear-algebra_FO0223 | 26 | 1.000 | \boldsymbol{x} \in \operatorname{span}(\mathcal{X}) | | |
| lyche-numerical-linear-algebra_FO0224 | 26 | 1.000 | \boldsymbol{x}=c_{1} \boldsymbol{x}_{1}+\cdots+ | | |
| lyche-numerical-linear-algebra_FO0225 | 26 | 0.995 | c_{n} \boldsymbol{x}_{n} | | |
| lyche-numerical-linear-algebra_FO0226 | 26 | 0.995 | \boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{n} | | |
| lyche-numerical-linear-algebra_FO0227 | 26 | 1.000 | \boldsymbol{w}_{1}, \ldots, \boldsymbol{w}_{k} | | |
| lyche-numerical-linear-algebra_FO0228 | 26 | 1.000 | k \leq n | | |
| lyche-numerical-linear-algebra_FO0229 | 27 | 1.000 | k>n | | |
| lyche-numerical-linear-algebra_FO0230 | 27 | 1.000 | \boldsymbol{w}_{1} | | |
| lyche-numerical-linear-algebra_FO0231 | 27 | 1.000 | \mathcal{X}_{0}:=\left\{\boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{n}\right\} | | |
| lyche-numerical-linear-algebra_FO0232 | 27 | 1.000 | \boldsymbol{w}_{1}=c_{1} \boldsymbol{v}_{1}+\cdots+c_{n} \boldsymbol{v}_{n} | | |
| lyche-numerical-linear-algebra_FO0233 | 27 | 1.000 | \boldsymbol{w}_{1} \neq \mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0234 | 27 | 1.000 | c_{i_{1}} | | |
| lyche-numerical-linear-algebra_FO0235 | 27 | 1.000 | \boldsymbol{v}_{i_{1}} | | |
| lyche-numerical-linear-algebra_FO0236 | 27 | 1.000 | \boldsymbol{v} | | |
| lyche-numerical-linear-algebra_FO0237 | 27 | 1.000 | \mathcal{X}_{1}:= | | |
| lyche-numerical-linear-algebra_FO0238 | 27 | 1.000 | \left\{\boldsymbol{w}_{1}, \boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{i_{1}-1}, \boldsymbol{v}_{i_{1}+1}, \ldots, \boldsymbol{v}_{n}\right\} | | |
| lyche-numerical-linear-algebra_FO0239 | 27 | 1.000 | \boldsymbol{w}_{2} | | |
| lyche-numerical-linear-algebra_FO0240 | 27 | 1.000 | \mathcal{X}_{1} | | |
| lyche-numerical-linear-algebra_FO0241 | 27 | 1.000 | \boldsymbol{w}_{2}=d_{i_{1}} \boldsymbol{w}_{1}+\sum_{j \neq i_{1}} d_{j} \boldsymbol{v}_{j} | | |
| lyche-numerical-linear-algebra_FO0242 | 27 | 0.900 | d_{i_{2}} \neq 0 | | |
| lyche-numerical-linear-algebra_FO0243 | 27 | 0.900 | i_{2} | | |
| lyche-numerical-linear-algebra_FO0244 | 27 | 0.900 | i_{2} \neq i_{1} | | |
| lyche-numerical-linear-algebra_FO0245 | 27 | 0.900 | \boldsymbol{w}_{2}=d_{1} \boldsymbol{w}_{1} | | |
| lyche-numerical-linear-algebra_FO0246 | 27 | 1.000 | \boldsymbol{w} | | |
| lyche-numerical-linear-algebra_FO0247 | 27 | 1.000 | \mathcal{X}_{2} | | |
| lyche-numerical-linear-algebra_FO0248 | 27 | 0.998 | \boldsymbol{v}_{i_{2}} | | |
| lyche-numerical-linear-algebra_FO0249 | 27 | 1.000 | n-2 | | |
| lyche-numerical-linear-algebra_FO0250 | 27 | 1.000 | \mathcal{X}_{n} | | |
| lyche-numerical-linear-algebra_FO0251 | 27 | 1.000 | \boldsymbol{w}_{1}, \ldots, \boldsymbol{w}_{n} | | |
| lyche-numerical-linear-algebra_FO0252 | 27 | 1.000 | \boldsymbol{w}_{k} | | |
| lyche-numerical-linear-algebra_FO0254 | 27 | 1.000 | \left\{\boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{n}\right\} | | |
| lyche-numerical-linear-algebra_FO0255 | 27 | 1.000 | \operatorname{span}\left\{\boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{n}\right\}=\mathcal{V} | | |
| lyche-numerical-linear-algebra_FO0256 | 27 | 0.999 | \left\{\boldsymbol{v}_{i_{1}}, \ldots, \boldsymbol{v}_{i_{k}}\right\} | | |
| lyche-numerical-linear-algebra_FO0257 | 27 | 1.000 | \mathcal{V}=\operatorname{span}\left\{\boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{n}\right\} | | |
| lyche-numerical-linear-algebra_FO0258 | 27 | 1.000 | \operatorname{dim} \mathcal{V} | | |
| lyche-numerical-linear-algebra_FO0259 | 27 | 1.000 | \mathcal{X}=\left\{\boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{n}\right\} | | |
| lyche-numerical-linear-algebra_FO0260 | 27 | 1.000 | \mathcal{Y}=\left\{\boldsymbol{w}_{1}, \ldots, \boldsymbol{w}_{k}\right\} | | |
| lyche-numerical-linear-algebra_FO0261 | 27 | 1.000 | \mathcal{Y} | | |
| lyche-numerical-linear-algebra_FO0262 | 27 | 1.000 | n \leq k | | |
| lyche-numerical-linear-algebra_FO0263 | 27 | 1.000 | n=k | | |
| lyche-numerical-linear-algebra_FO0264 | 27 | 1.000 | \left\{\boldsymbol{e}_{1}, \ldots, \boldsymbol{e}_{n}\right\} | | |
| lyche-numerical-linear-algebra_FO0265 | 27 | 0.996 | \left\{\boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{k}\right\} | | |
| lyche-numerical-linear-algebra_FO0266 | 28 | 1.000 | \boldsymbol{v}_{k+1} | | |
| lyche-numerical-linear-algebra_FO0267 | 28 | 1.000 | \boldsymbol{u}, \boldsymbol{v} \in \mathcal{S} | | |
| lyche-numerical-linear-algebra_FO0268 | 28 | 1.000 | c \boldsymbol{u} | | |
| lyche-numerical-linear-algebra_FO0269 | 28 | 1.000 | \boldsymbol{u} \in \mathcal{S} | | |
| lyche-numerical-linear-algebra_FO0270 | 28 | 1.000 | V 1-V 5 | | |
| lyche-numerical-linear-algebra_FO0271 | 28 | 1.000 | S 1-S 5 | | |
| lyche-numerical-linear-algebra_FO0272 | 28 | 1.000 | \{\mathbf{0}\} | | |
| lyche-numerical-linear-algebra_FO0273 | 28 | 1.000 | \mathcal{X}=\left\{\boldsymbol{x}_{1}, \ldots, \boldsymbol{x}_{n}\right\} \subseteq \mathcal{V} | | |
| lyche-numerical-linear-algebra_FO0274 | 28 | 1.000 | \mathcal{S} \oplus \mathcal{T} | | |
| lyche-numerical-linear-algebra_FO0275 | 28 | 1.000 | \mathcal{S} \cap \mathcal{T}=\{\mathbf{0}\} | | |
| lyche-numerical-linear-algebra_FO0276 | 29 | 1.000 | \left\{\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{p}\right\} | | |
| lyche-numerical-linear-algebra_FO0277 | 29 | 1.000 | \mathcal{S} \cap \mathcal{T} | | |
| lyche-numerical-linear-algebra_FO0278 | 29 | 1.000 | \left\{\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{p}\right\}=\emptyset | | |
| lyche-numerical-linear-algebra_FO0279 | 29 | 1.000 | \left\{\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{p}, \boldsymbol{s}_{1}, \ldots, \boldsymbol{s}_{q}\right\} | | |
| lyche-numerical-linear-algebra_FO0280 | 29 | 1.000 | \left\{\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{p}, \boldsymbol{t}_{1}, \ldots, \boldsymbol{t}_{r}\right\} | | |
| lyche-numerical-linear-algebra_FO0281 | 29 | 1.000 | \boldsymbol{x} \in \mathcal{S}+\mathcal{T} | | |
| lyche-numerical-linear-algebra_FO0282 | 29 | 1.000 | \mathcal{S}+\mathcal{T} | | |
| lyche-numerical-linear-algebra_FO0283 | 29 | 1.000 | \boldsymbol{u}+\boldsymbol{s}+\boldsymbol{t}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0284 | 29 | 1.000 | \boldsymbol{u}:=\sum_{j=1}^{p} \alpha_{j} \boldsymbol{u}_{j}, \boldsymbol{s}:=\sum_{j=1}^{q} \rho_{j} \boldsymbol{s}_{j} | | |
| lyche-numerical-linear-algebra_FO0285 | 29 | 1.000 | \boldsymbol{t}:=\sum_{j=1}^{r} \sigma_{j} \boldsymbol{t}_{j} | | |
| lyche-numerical-linear-algebra_FO0286 | 29 | 1.000 | \boldsymbol{s}=-(\boldsymbol{u}+\boldsymbol{t}) | | |
| lyche-numerical-linear-algebra_FO0287 | 29 | 1.000 | \boldsymbol{s} \in | | |
| lyche-numerical-linear-algebra_FO0288 | 29 | 1.000 | \boldsymbol{s} | | |
| lyche-numerical-linear-algebra_FO0289 | 29 | 1.000 | \boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{p} | | |
| lyche-numerical-linear-algebra_FO0290 | 29 | 1.000 | \boldsymbol{s}:= | | |
| lyche-numerical-linear-algebra_FO0291 | 29 | 1.000 | \sum_{j=1}^{p} \beta_{j} \boldsymbol{u}_{j} | | |
| lyche-numerical-linear-algebra_FO0292 | 29 | 1.000 | \mathbf{0}=\sum_{j=1}^{p} \beta_{j} \boldsymbol{u}_{j}-\sum_{j=1}^{q} \rho_{j} \boldsymbol{s}_{j} | | |
| lyche-numerical-linear-algebra_FO0293 | 29 | 1.000 | \beta_{1}=\cdots=\beta_{p}=\rho_{1}=\cdots=\rho_{q}=0 | | |
| lyche-numerical-linear-algebra_FO0294 | 29 | 1.000 | \boldsymbol{s}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0295 | 29 | 1.000 | \boldsymbol{u}+\boldsymbol{t}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0296 | 29 | 1.000 | \alpha_{1}=\cdots=\alpha_{p}=\sigma_{1}=\cdots=\sigma_{r}=0 | | |
| lyche-numerical-linear-algebra_FO0297 | 29 | 1.000 | \left\{\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{p}, \boldsymbol{s}_{1}, \ldots, \boldsymbol{s}_{q}, \boldsymbol{t}_{1}, \ldots, \boldsymbol{t}_{r}\right\} | | |
| lyche-numerical-linear-algebra_FO0298 | 29 | 1.000 | \operatorname{dim}\{\mathbf{0}\}=0 | | |
| lyche-numerical-linear-algebra_FO0299 | 29 | 1.000 | \left\{\boldsymbol{s}_{1}, \ldots, \boldsymbol{s}_{n}\right\} | | |
| lyche-numerical-linear-algebra_FO0300 | 29 | 1.000 | \left\{\boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{m}\right\} | | |
| lyche-numerical-linear-algebra_FO0301 | 29 | 1.000 | \boldsymbol{s}_{j} | | |
| lyche-numerical-linear-algebra_FO0302 | 29 | 1.000 | \boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{m} | | |
| lyche-numerical-linear-algebra_FO0303 | 30 | 1.000 | \boldsymbol{x} \in \mathcal{S} | | |
| lyche-numerical-linear-algebra_FO0304 | 30 | 1.000 | \boldsymbol{x}=\sum_{j=1}^{n} c_{j} \boldsymbol{s}_{j}=\sum_{i=1}^{m} b_{i} \boldsymbol{v}_{i} | | |
| lyche-numerical-linear-algebra_FO0305 | 30 | 1.000 | \boldsymbol{b}:= | | |
| lyche-numerical-linear-algebra_FO0306 | 30 | 1.000 | \left[b_{1}, \ldots, \boldsymbol{b}_{m}\right]^{T}, \boldsymbol{c}:=\left[c_{1}, \ldots, c_{n}\right]^{T} | | |
| lyche-numerical-linear-algebra_FO0307 | 30 | 1.000 | \boldsymbol{b}=\boldsymbol{A} \boldsymbol{c} | | |
| lyche-numerical-linear-algebra_FO0308 | 30 | 1.000 | \boldsymbol{A}=\left[a_{i j}\right] \in \mathbb{C}^{m \times n} | | |
| lyche-numerical-linear-algebra_FO0309 | 30 | 0.999 | a_{i j} | | |
| lyche-numerical-linear-algebra_FO0310 | 30 | 0.999 | \boldsymbol{s}_{j} \in \mathcal{V} | | |
| lyche-numerical-linear-algebra_FO0311 | 30 | 0.980 | \left(c_{j}\right) | | |
| lyche-numerical-linear-algebra_FO0312 | 30 | 0.980 | \left(b_{i}\right) | | |
| lyche-numerical-linear-algebra_FO0313 | 30 | 1.000 | b_{i}=\sum_{j=1}^{n} a_{i j} c_{j} | | |
| lyche-numerical-linear-algebra_FO0314 | 30 | 1.000 | i= | | |
| lyche-numerical-linear-algebra_FO0315 | 30 | 1.000 | 1, \ldots, m | | |
| lyche-numerical-linear-algebra_FO0316 | 30 | 1.000 | \boldsymbol{b}:=\boldsymbol{A} \boldsymbol{c}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0317 | 30 | 1.000 | \boldsymbol{c}=\left[c_{1}, \ldots, c_{n}\right]^{T} | | |
| lyche-numerical-linear-algebra_FO0318 | 30 | 1.000 | \boldsymbol{x}:=\sum_{j=1}^{n} c_{j} \boldsymbol{s}_{j} | | |
| lyche-numerical-linear-algebra_FO0319 | 30 | 0.995 | \boldsymbol{x}=\sum_{i=1}^{m} b_{i} \boldsymbol{v}_{i} | | |
| lyche-numerical-linear-algebra_FO0320 | 30 | 0.995 | \boldsymbol{b}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0321 | 30 | 0.995 | \boldsymbol{x}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0322 | 30 | 1.000 | \boldsymbol{c}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0323 | 30 | 1.000 | \mathcal{V}=\mathbb{R}^{m} | | |
| lyche-numerical-linear-algebra_FO0324 | 30 | 1.000 | m \times n | | |
| lyche-numerical-linear-algebra_FO0325 | 30 | 1.000 | \boldsymbol{X}=\left[\boldsymbol{x}_{1}, \ldots, \boldsymbol{x}_{n}\right] | | |
| lyche-numerical-linear-algebra_FO0326 | 30 | 0.998 | \boldsymbol{X} \boldsymbol{c} | | |
| lyche-numerical-linear-algebra_FO0327 | 30 | 0.998 | \boldsymbol{c}= | | |
| lyche-numerical-linear-algebra_FO0328 | 30 | 1.000 | \left[c_{1}, \ldots, c_{n}\right]^{T} | | |
| lyche-numerical-linear-algebra_FO0329 | 30 | 1.000 | \boldsymbol{x}_{j} \in \mathcal{V}, j=1, \ldots, n | | |
| lyche-numerical-linear-algebra_FO0330 | 30 | 1.000 | \mathcal{R}(\boldsymbol{X}) | | |
| lyche-numerical-linear-algebra_FO0331 | 30 | 1.000 | \boldsymbol{X} | | |
| lyche-numerical-linear-algebra_FO0332 | 30 | 1.000 | \mathcal{R}\left(\boldsymbol{X}^{T}\right) | | |
| lyche-numerical-linear-algebra_FO0333 | 30 | 1.000 | \mathcal{N}(\boldsymbol{X}):=\left\{\boldsymbol{y} \in \mathbb{R}^{n}: \boldsymbol{X} \boldsymbol{y}=\mathbf{0}\right\} | | |
| lyche-numerical-linear-algebra_FO0334 | 30 | 1.000 | \mathcal{N}(\boldsymbol{X}) | | |
| lyche-numerical-linear-algebra_FO0335 | 30 | 1.000 | \operatorname{null}(\boldsymbol{X}) | | |
| lyche-numerical-linear-algebra_FO0336 | 31 | 0.922 | \boldsymbol{X} \in | | |
| lyche-numerical-linear-algebra_FO0337 | 31 | 1.000 | \mathbb{C}^{m \times n} | | |
| lyche-numerical-linear-algebra_FO0338 | 31 | 1.000 | \operatorname{rank}(\boldsymbol{X})=\operatorname{rank}\left(\boldsymbol{X}^{*}\right) | | |
| lyche-numerical-linear-algebra_FO0339 | 31 | 1.000 | \operatorname{rank}(\boldsymbol{X})+\operatorname{null}(\boldsymbol{X})=n | | |
| lyche-numerical-linear-algebra_FO0340 | 31 | 1.000 | \operatorname{rank}(\boldsymbol{X})+\operatorname{null}\left(\boldsymbol{X}^{*}\right)=m | | |
| lyche-numerical-linear-algebra_FO0341 | 31 | 1.000 | x_{j} | | |
| lyche-numerical-linear-algebra_FO0342 | 31 | 1.000 | b_{i} | | |
| lyche-numerical-linear-algebra_FO0343 | 32 | 0.992 | \boldsymbol{a}_{j}=\left[a_{1 j}, \ldots, \boldsymbol{a}_{m j}\right]^{T} \in \mathbb{C}^{m} | | |
| lyche-numerical-linear-algebra_FO0344 | 32 | 0.992 | \boldsymbol{b}=\left[b_{1}, \ldots, b_{m}\right]^{T} \in \mathbb{C}^{m} | | |
| lyche-numerical-linear-algebra_FO0345 | 32 | 1.000 | m<n, m=n | | |
| lyche-numerical-linear-algebra_FO0346 | 32 | 1.000 | m>n | | |
| lyche-numerical-linear-algebra_FO0347 | 32 | 1.000 | m<n | | |
| lyche-numerical-linear-algebra_FO0348 | 32 | 0.570 | \boldsymbol{A x}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0349 | 32 | 1.000 | \boldsymbol{A} \boldsymbol{x}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0350 | 32 | 1.000 | \mathbb{R}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO0351 | 32 | 0.658 | \boldsymbol{b} \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO0352 | 32 | 1.000 | \boldsymbol{B}=[\boldsymbol{A} \boldsymbol{b}] \in \mathbb{R}^{n \times(n+1)} | | |
| lyche-numerical-linear-algebra_FO0353 | 32 | 0.997 | \boldsymbol{z} \in \mathbb{R}^{n+1} | | |
| lyche-numerical-linear-algebra_FO0354 | 32 | 0.997 | \boldsymbol{B} \boldsymbol{z}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0355 | 32 | 0.896 | \boldsymbol{z}=\left[\begin{array}{c}\tilde{\boldsymbol{z}} \\ z_{n+1}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO0356 | 32 | 0.896 | \tilde{\boldsymbol{z}}=\left[z_{1}, \ldots, z_{n}\right]^{T} \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO0357 | 32 | 0.896 | z_{n+1} \in \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO0358 | 33 | 0.737 | z_{n+1}=0 | | |
| lyche-numerical-linear-algebra_FO0359 | 33 | 0.737 | \boldsymbol{A} \tilde{\boldsymbol{z}}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0360 | 33 | 0.737 | \tilde{\boldsymbol{z}} | | |
| lyche-numerical-linear-algebra_FO0361 | 33 | 0.997 | \boldsymbol{x}:=-\tilde{\boldsymbol{z}} / z_{n+1} | | |
| lyche-numerical-linear-algebra_FO0362 | 33 | 1.000 | \boldsymbol{A} \boldsymbol{y}=\boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO0363 | 33 | 1.000 | \boldsymbol{x}, \boldsymbol{y} \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO0364 | 33 | 1.000 | \boldsymbol{A}(\boldsymbol{x}-\boldsymbol{y})=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0365 | 33 | 1.000 | \boldsymbol{x}-\boldsymbol{y}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0366 | 33 | 1.000 | \boldsymbol{x}=\boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO0367 | 33 | 1.000 | m< | | |
| lyche-numerical-linear-algebra_FO0368 | 33 | 1.000 | \boldsymbol{b} \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO0369 | 33 | 1.000 | \boldsymbol{B} \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO0370 | 33 | 1.000 | \boldsymbol{A} \boldsymbol{B}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO0371 | 33 | 1.000 | \boldsymbol{C} \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO0372 | 33 | 1.000 | \boldsymbol{C} \boldsymbol{A}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO0373 | 33 | 1.000 | \boldsymbol{C} | | |
| lyche-numerical-linear-algebra_FO0374 | 33 | 1.000 | \boldsymbol{A}^{-1} | | |
| lyche-numerical-linear-algebra_FO0375 | 33 | 1.000 | \boldsymbol{A}^{-1} \boldsymbol{A}=\boldsymbol{A} \boldsymbol{A}^{-1}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO0376 | 33 | 0.998 | \boldsymbol{A}, \boldsymbol{B}, \boldsymbol{C} \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO0377 | 33 | 0.998 | \boldsymbol{A} \boldsymbol{B}= | | |
| lyche-numerical-linear-algebra_FO0378 | 33 | 1.000 | \boldsymbol{B} \boldsymbol{A}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO0379 | 33 | 1.000 | \boldsymbol{A}^{-1}=\boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO0380 | 33 | 1.000 | \boldsymbol{C} \boldsymbol{x}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0381 | 33 | 1.000 | \boldsymbol{A} \boldsymbol{B} \boldsymbol{x}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0382 | 33 | 1.000 | \boldsymbol{B} \boldsymbol{x}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0383 | 34 | 1.000 | \boldsymbol{C} \boldsymbol{x}=(\boldsymbol{A} \boldsymbol{B}) \boldsymbol{x}=\boldsymbol{A}(\boldsymbol{B} \boldsymbol{x})=\boldsymbol{A} \mathbf{0}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0384 | 34 | 1.000 | \tilde{\boldsymbol{x}} | | |
| lyche-numerical-linear-algebra_FO0385 | 34 | 1.000 | \boldsymbol{A} \tilde{\boldsymbol{x}}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0386 | 34 | 1.000 | \boldsymbol{B} \boldsymbol{x}=\tilde{\boldsymbol{x}} | | |
| lyche-numerical-linear-algebra_FO0387 | 34 | 1.000 | \boldsymbol{C} \boldsymbol{x}=(\boldsymbol{A} \boldsymbol{B}) \boldsymbol{x}=\boldsymbol{A}(\boldsymbol{B} \boldsymbol{x})=\boldsymbol{A} \tilde{\boldsymbol{x}}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0388 | 34 | 1.000 | \boldsymbol{A} \boldsymbol{b}_{i}=\boldsymbol{e}_{i} | | |
| lyche-numerical-linear-algebra_FO0389 | 34 | 1.000 | \boldsymbol{b}_{i} | | |
| lyche-numerical-linear-algebra_FO0390 | 34 | 1.000 | \boldsymbol{B}=\left[\boldsymbol{b}_{1}, \ldots, \boldsymbol{b}_{n}\right] | | |
| lyche-numerical-linear-algebra_FO0391 | 34 | 0.968 | \boldsymbol{A} \boldsymbol{B}=\left[\boldsymbol{A} \boldsymbol{b}_{1}, \ldots, \boldsymbol{A} \boldsymbol{b}_{n}\right]=\left[\boldsymbol{e}_{1}, \ldots, \boldsymbol{e}_{n}\right]=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO0392 | 34 | 1.000 | 1.8 \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO0393 | 34 | 1.000 | \boldsymbol{B} \boldsymbol{C}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO0394 | 34 | 1.000 | \boldsymbol{A} \boldsymbol{B}=\boldsymbol{B} \boldsymbol{C}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO0395 | 34 | 1.000 | \boldsymbol{A}=\boldsymbol{C} | | |
| lyche-numerical-linear-algebra_FO0396 | 34 | 0.995 | \boldsymbol{A}, \boldsymbol{B} \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO0397 | 34 | 1.000 | \left(\boldsymbol{A}^{-1}\right)^{-1}=\boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO0398 | 34 | 1.000 | \boldsymbol{C}=\boldsymbol{A} \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO0399 | 34 | 1.000 | \boldsymbol{C}^{-1}=\boldsymbol{B}^{-1} \boldsymbol{A}^{-1} | | |
| lyche-numerical-linear-algebra_FO0400 | 34 | 1.000 | \left(\boldsymbol{A}^{T}\right)^{-1}=\left(\boldsymbol{A}^{-1}\right)^{T}=: \boldsymbol{A}^{-T} | | |
| lyche-numerical-linear-algebra_FO0401 | 34 | 1.000 | \left(\boldsymbol{A}^{*}\right)^{-1}=\left(\boldsymbol{A}^{-1}\right)^{*}=: \boldsymbol{A}^{-*} | | |
| lyche-numerical-linear-algebra_FO0402 | 34 | 1.000 | c \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO0403 | 34 | 1.000 | (c \boldsymbol{A})^{-1}=\frac{1}{c} \boldsymbol{A}^{-1} | | |
| lyche-numerical-linear-algebra_FO0404 | 34 | 1.000 | \boldsymbol{A}^{-1} \boldsymbol{A}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO0405 | 34 | 1.000 | \left(\boldsymbol{B}^{-1} \boldsymbol{A}^{-1}\right)(\boldsymbol{A} \boldsymbol{B})=\boldsymbol{B}^{-1}\left(\boldsymbol{A}^{-1} \boldsymbol{A}\right) \boldsymbol{B}=\boldsymbol{B}^{-1} \boldsymbol{B}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO0406 | 34 | 1.000 | \boldsymbol{A} \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO0407 | 34 | 1.000 | \boldsymbol{I}=\boldsymbol{I}^{T}=\left(\boldsymbol{A}^{-1} \boldsymbol{A}\right)^{T}=\boldsymbol{A}^{T}\left(\boldsymbol{A}^{-1}\right)^{T} | | |
| lyche-numerical-linear-algebra_FO0408 | 34 | 1.000 | \left(\boldsymbol{A}^{-1}\right)^{T} | | |
| lyche-numerical-linear-algebra_FO0409 | 34 | 1.000 | \frac{1}{c} \boldsymbol{A}^{-1} | | |
| lyche-numerical-linear-algebra_FO0410 | 35 | 1.000 | n! | | |
| lyche-numerical-linear-algebra_FO0411 | 35 | 1.000 | \{1,2, \ldots, n\} | | |
| lyche-numerical-linear-algebra_FO0412 | 35 | 1.000 | \operatorname{sign}(\sigma) | | |
| lyche-numerical-linear-algebra_FO0413 | 35 | 1.000 | \sigma | | |
| lyche-numerical-linear-algebra_FO0414 | 35 | 1.000 | \epsilon | | |
| lyche-numerical-linear-algebra_FO0415 | 35 | 1.000 | \epsilon(i)= | | |
| lyche-numerical-linear-algebra_FO0416 | 35 | 1.000 | i, i=1,2 | | |
| lyche-numerical-linear-algebra_FO0417 | 35 | 1.000 | \sigma=\{2,1\} | | |
| lyche-numerical-linear-algebra_FO0418 | 35 | 1.000 | n=3 | | |
| lyche-numerical-linear-algebra_FO0419 | 35 | 0.719 | \{1,2,3\} | | |
| lyche-numerical-linear-algebra_FO0420 | 35 | 0.999 | \operatorname{sign}(\{1,2,3\})=\operatorname{sign}(\{2,3,1\})=\operatorname{sign}(\{3,1,2\})=1 | | |
| lyche-numerical-linear-algebra_FO0421 | 35 | 0.749 | \operatorname{sign}(\{2,1,3\})=\operatorname{sign}(\{3,2,1\})=\operatorname{sign}(\{1,3,2\})=-1 | | |
| lyche-numerical-linear-algebra_FO0422 | 35 | 1.000 | \operatorname{det}(\boldsymbol{A})= | | |
| lyche-numerical-linear-algebra_FO0423 | 36 | 1.000 | a_{11} a_{22} \cdots a_{n n} | | |
| lyche-numerical-linear-algebra_FO0424 | 36 | 1.000 | \operatorname{det}(\boldsymbol{I})=1 | | |
| lyche-numerical-linear-algebra_FO0425 | 36 | 1.000 | \operatorname{det}(\boldsymbol{B})=-\operatorname{det}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO0426 | 36 | 1.000 | \alpha, \operatorname{det}(\boldsymbol{B})=\alpha \operatorname{det}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO0427 | 36 | 1.000 | \boldsymbol{A}_{i, j} | | |
| lyche-numerical-linear-algebra_FO0428 | 36 | 1.000 | 1 \leq i, j \leq n | | |
| lyche-numerical-linear-algebra_FO0429 | 36 | 1.000 | \operatorname{det}\left(\boldsymbol{A}_{i j}\right) | | |
| lyche-numerical-linear-algebra_FO0430 | 36 | 0.969 | \left(x_{1}, y_{1}\right) | | |
| lyche-numerical-linear-algebra_FO0431 | 36 | 0.969 | \left(x_{2}, y_{2}\right) | | |
| lyche-numerical-linear-algebra_FO0432 | 36 | 1.000 | \operatorname{det}(\boldsymbol{A})=0 | | |
| lyche-numerical-linear-algebra_FO0433 | 37 | 1.000 | \boldsymbol{A}, \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO0434 | 37 | 0.984 | \operatorname{det}(\boldsymbol{A B})=\operatorname{det}(\boldsymbol{A}) \operatorname{det}(\boldsymbol{B}) | | |
| lyche-numerical-linear-algebra_FO0435 | 37 | 1.000 | \operatorname{det}\left(\boldsymbol{A}^{T}\right)=\operatorname{det}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO0436 | 37 | 1.000 | \operatorname{det}\left(\boldsymbol{A}^{*}\right)=\overline{\operatorname{det}(\boldsymbol{A})} | | |
| lyche-numerical-linear-algebra_FO0437 | 37 | 1.000 | \operatorname{det}(a \boldsymbol{A})=a^{n} \operatorname{det}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO0438 | 37 | 0.943 | \boldsymbol{A}=\left[\begin{array}{ll}\boldsymbol{C} & \boldsymbol{D} \\ \mathbf{0} & \boldsymbol{E}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO0439 | 37 | 0.943 | \boldsymbol{C}, \boldsymbol{E} | | |
| lyche-numerical-linear-algebra_FO0440 | 37 | 0.943 | \operatorname{det}(\boldsymbol{A})=\operatorname{det}(\boldsymbol{C}) \operatorname{det}(\boldsymbol{E}) | | |
| lyche-numerical-linear-algebra_FO0441 | 37 | 1.000 | \boldsymbol{x}= | | |
| lyche-numerical-linear-algebra_FO0442 | 37 | 0.982 | \left[x_{1}, x_{2}, \ldots, x_{n}\right]^{T} | | |
| lyche-numerical-linear-algebra_FO0443 | 37 | 1.000 | \boldsymbol{A}_{j}(\boldsymbol{b}) | | |
| lyche-numerical-linear-algebra_FO0444 | 37 | 0.999 | \operatorname{adj}(\boldsymbol{A}) \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO0445 | 37 | 0.999 | \operatorname{adj}(\boldsymbol{A})_{i, j}=(-1)^{i+j} \operatorname{det}\left(\boldsymbol{A}_{j, i}\right) | | |
| lyche-numerical-linear-algebra_FO0446 | 37 | 1.000 | \boldsymbol{A}_{j, i} | | |
| lyche-numerical-linear-algebra_FO0447 | 37 | 1.000 | \boldsymbol{A} \in \mathbb{C}^{m \times p}, \boldsymbol{B} \in \mathbb{C}^{p \times n} | | |
| lyche-numerical-linear-algebra_FO0448 | 37 | 1.000 | 1 \leq r \leq \min \{m, n, p\} | | |
| lyche-numerical-linear-algebra_FO0449 | 37 | 1.000 | \boldsymbol{i}=\left\{i_{1}, \ldots, i_{r}\right\} | | |
| lyche-numerical-linear-algebra_FO0450 | 37 | 1.000 | \boldsymbol{j}=\left\{j_{1}, \ldots, j_{r}\right\} | | |
| lyche-numerical-linear-algebra_FO0451 | 37 | 1.000 | 1 \leq i_{1}<i_{2}<\cdots<i_{r} \leq m | | |
| lyche-numerical-linear-algebra_FO0452 | 37 | 1.000 | 1 \leq j_{1}<j_{2}<\cdots<j_{r} \leq n | | |
| lyche-numerical-linear-algebra_FO0453 | 37 | 0.999 | \boldsymbol{k}=\left\{k_{1}, \ldots, k_{r}\right\} | | |
| lyche-numerical-linear-algebra_FO0454 | 37 | 0.999 | 1 \leq k_{1}<k_{2}<\cdots<k_{r} \leq p | | |
| lyche-numerical-linear-algebra_FO0455 | 38 | 0.937 | \lambda \in \mathbb{C} | | |
| lyche-numerical-linear-algebra_FO0456 | 38 | 0.937 | \lambda, \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO0457 | 38 | 1.000 | \sigma(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO0458 | 38 | 1.000 | \sigma(\boldsymbol{I})=\{1, \ldots, 1\}=\{1\} | | |
| lyche-numerical-linear-algebra_FO0459 | 38 | 1.000 | \lambda \in | | |
| lyche-numerical-linear-algebra_FO0460 | 38 | 0.995 | \sigma(\boldsymbol{A}) \Longleftrightarrow \operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})=0 | | |
| lyche-numerical-linear-algebra_FO0461 | 38 | 0.699 | (\lambda, \boldsymbol{x}) | | |
| lyche-numerical-linear-algebra_FO0462 | 38 | 1.000 | (\boldsymbol{A}-\lambda \boldsymbol{I}) \boldsymbol{x}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0463 | 38 | 1.000 | \boldsymbol{A}-\lambda \boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO0464 | 38 | 1.000 | \operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})=0 | | |
| lyche-numerical-linear-algebra_FO0465 | 38 | 1.000 | \operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I}) | | |
| lyche-numerical-linear-algebra_FO0466 | 38 | 1.000 | r(\lambda) | | |
| lyche-numerical-linear-algebra_FO0467 | 38 | 0.998 | n-2, \operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I}) | | |
| lyche-numerical-linear-algebra_FO0468 | 38 | 1.000 | \operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})=(-1)^{n} \operatorname{det}(\lambda \boldsymbol{I}-\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO0469 | 38 | 1.000 | \operatorname{det}(\lambda \boldsymbol{I}-\boldsymbol{A})=0 | | |
| lyche-numerical-linear-algebra_FO0470 | 39 | 0.958 | \pi_{\boldsymbol{A}}: \mathbb{C} \rightarrow | | |
| lyche-numerical-linear-algebra_FO0471 | 39 | 1.000 | \pi_{\boldsymbol{A}}(\lambda)=\operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I}) | | |
| lyche-numerical-linear-algebra_FO0472 | 39 | 0.994 | \lambda_{1}, \ldots, \lambda_{n} | | |
| lyche-numerical-linear-algebra_FO0473 | 39 | 1.000 | \boldsymbol{A} \overline{\boldsymbol{x}}=\bar{\lambda} \overline{\boldsymbol{x}} | | |
| lyche-numerical-linear-algebra_FO0474 | 39 | 0.945 | \pi_{\boldsymbol{A}} | | |
| lyche-numerical-linear-algebra_FO0475 | 39 | 1.000 | c_{n-1}=(-1)^{n-1} \operatorname{trace}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO0476 | 39 | 1.000 | c_{0}=\pi_{\boldsymbol{A}}(0)=\operatorname{det}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO0477 | 39 | 1.000 | d_{n-1}=(-1)^{n-1}\left(\lambda_{1}+\cdots+\lambda_{n}\right) | | |
| lyche-numerical-linear-algebra_FO0478 | 39 | 1.000 | d_{0}=\lambda_{1} \cdots \lambda_{n} | | |
| lyche-numerical-linear-algebra_FO0479 | 39 | 1.000 | c_{j}=d_{j} | | |
| lyche-numerical-linear-algebra_FO0480 | 39 | 1.000 | \boldsymbol{A}=\left[\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO0481 | 39 | 1.000 | \operatorname{trace}(\boldsymbol{A})=4, \operatorname{det}(\boldsymbol{A})=3 | | |
| lyche-numerical-linear-algebra_FO0482 | 39 | 1.000 | \pi_{\boldsymbol{A}}(\lambda)=\lambda^{2}-4 \lambda+3 | | |
| lyche-numerical-linear-algebra_FO0483 | 39 | 1.000 | \Longleftrightarrow \boldsymbol{A} \boldsymbol{x}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0484 | 39 | 1.000 | \boldsymbol{x} \neq 0 \Longleftrightarrow \boldsymbol{A} \boldsymbol{x}=\mathbf{0} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO0485 | 39 | 1.000 | \boldsymbol{x} \neq | | |
| lyche-numerical-linear-algebra_FO0486 | 39 | 0.981 | 0 \Longleftrightarrow | | |
| lyche-numerical-linear-algebra_FO0487 | 40 | 0.848 | \boldsymbol{A}=\left[\begin{array}{cc}\boldsymbol{B} & \boldsymbol{D} \\ 0 & \boldsymbol{C}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO0488 | 40 | 0.998 | \pi_{\boldsymbol{A}}=\pi_{\boldsymbol{B}} \cdot \pi_{\boldsymbol{C}} | | |
| lyche-numerical-linear-algebra_FO0489 | 40 | 1.000 | \boldsymbol{A}, \boldsymbol{B} \in \mathbb{R}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO0490 | 40 | 1.000 | 2 n \times 2 n | | |
| lyche-numerical-linear-algebra_FO0491 | 40 | 1.000 | \boldsymbol{A}, \ldots, \boldsymbol{Z} | | |
| lyche-numerical-linear-algebra_FO0492 | 40 | 1.000 | \boldsymbol{W}, \boldsymbol{X}, \boldsymbol{Y} | | |
| lyche-numerical-linear-algebra_FO0493 | 40 | 1.000 | \boldsymbol{Z} | | |
| lyche-numerical-linear-algebra_FO0494 | 40 | 1.000 | m \times m | | |
| lyche-numerical-linear-algebra_FO0495 | 40 | 1.000 | m=2^{k} | | |
| lyche-numerical-linear-algebra_FO0496 | 40 | 1.000 | k \geq 0 | | |
| lyche-numerical-linear-algebra_FO0497 | 40 | 1.000 | \mathcal{O}\left(m^{\log _{2}(7)}\right) | | |
| lyche-numerical-linear-algebra_FO0498 | 40 | 1.000 | \log _{2}(7) \approx 2.8<3 | | |
| lyche-numerical-linear-algebra_FO0499 | 41 | 1.000 | 2 \times 2 | | |
| lyche-numerical-linear-algebra_FO0500 | 41 | 1.000 | a, b, c, d | | |
| lyche-numerical-linear-algebra_FO0501 | 41 | 1.000 | a d-b c \neq 0 | | |
| lyche-numerical-linear-algebra_FO0502 | 41 | 1.000 | \boldsymbol{B}, \boldsymbol{C} \in | | |
| lyche-numerical-linear-algebra_FO0503 | 41 | 1.000 | \mathbb{R}^{n \times m} | | |
| lyche-numerical-linear-algebra_FO0504 | 41 | 1.000 | n, m \in \mathbb{N} | | |
| lyche-numerical-linear-algebra_FO0505 | 41 | 1.000 | \left(\boldsymbol{I}+\boldsymbol{C}^{T} \boldsymbol{A}^{-1} \boldsymbol{B}\right)^{-1} | | |
| lyche-numerical-linear-algebra_FO0506 | 41 | 1.000 | \boldsymbol{u}, \boldsymbol{v} \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO0507 | 41 | 1.000 | \boldsymbol{v}^{T} \boldsymbol{u} \neq 1 | | |
| lyche-numerical-linear-algebra_FO0508 | 41 | 1.000 | \boldsymbol{I}-\boldsymbol{u} \boldsymbol{v}^{T} | | |
| lyche-numerical-linear-algebra_FO0509 | 41 | 0.813 | \boldsymbol{I}-\tau \boldsymbol{u} \boldsymbol{v}^{T} | | |
| lyche-numerical-linear-algebra_FO0510 | 41 | 0.813 | \tau=1 /\left(\boldsymbol{v}^{T} \boldsymbol{u}-1\right) | | |
| lyche-numerical-linear-algebra_FO0511 | 41 | 1.000 | \boldsymbol{C}:=\boldsymbol{A}^{-1} | | |
| lyche-numerical-linear-algebra_FO0512 | 41 | 1.000 | \boldsymbol{a} \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO0513 | 41 | 1.000 | \overline{\boldsymbol{A}} | | |
| lyche-numerical-linear-algebra_FO0514 | 41 | 1.000 | \boldsymbol{a}^{T} | | |
| lyche-numerical-linear-algebra_FO0515 | 41 | 0.995 | \overline{\boldsymbol{A}}=\boldsymbol{A}-\boldsymbol{e}_{i}\left(\boldsymbol{e}_{i}^{T} \boldsymbol{A}-\boldsymbol{a}^{T}\right) | | |
| lyche-numerical-linear-algebra_FO0516 | 41 | 0.995 | \boldsymbol{e}_{i} | | |
| lyche-numerical-linear-algebra_FO0517 | 41 | 1.000 | \overline{\boldsymbol{C}}=\overline{\boldsymbol{A}}^{-1} | | |
| lyche-numerical-linear-algebra_FO0518 | 41 | 1.000 | \boldsymbol{a} | | |
| lyche-numerical-linear-algebra_FO0519 | 41 | 0.999 | \overline{\boldsymbol{C}} | | |
| lyche-numerical-linear-algebra_FO0520 | 41 | 0.999 | \lambda \neq 0 | | |
| lyche-numerical-linear-algebra_FO0521 | 41 | 1.000 | \boldsymbol{A}, \boldsymbol{B}, \boldsymbol{C}, \boldsymbol{E} \in | | |
| lyche-numerical-linear-algebra_FO0522 | 41 | 1.000 | \boldsymbol{B} \boldsymbol{C} | | |
| lyche-numerical-linear-algebra_FO0523 | 41 | 1.000 | \boldsymbol{L} \in \mathbb{R}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO0524 | 41 | 1.000 | \boldsymbol{A}=\boldsymbol{L}+\boldsymbol{L}^{T} | | |
| lyche-numerical-linear-algebra_FO0525 | 42 | 1.000 | \boldsymbol{E}^{T} \boldsymbol{A} \boldsymbol{E}=\boldsymbol{S}+\boldsymbol{S}^{T} | | |
| lyche-numerical-linear-algebra_FO0526 | 42 | 1.000 | \boldsymbol{S}=\boldsymbol{E}^{T} \boldsymbol{L} \boldsymbol{E} | | |
| lyche-numerical-linear-algebra_FO0527 | 42 | 1.000 | \boldsymbol{E}^{T} \boldsymbol{A} \boldsymbol{E} | | |
| lyche-numerical-linear-algebra_FO0528 | 42 | 1.000 | \left(x_{1}, y_{1}, z_{1}\right),\left(x_{2}, y_{2}, z_{2}\right),\left(x_{3}, y_{3}, z_{3}\right) | | |
| lyche-numerical-linear-algebra_FO0529 | 43 | 1.000 | P_{i}=\left(x_{i}, y_{i}\right), i=1,2,3 | | |
| lyche-numerical-linear-algebra_FO0530 | 43 | 0.816 | T \mathrm{is}^{2} | | |
| lyche-numerical-linear-algebra_FO0531 | 43 | 0.999 | \prod_{i>j}\left(x_{i}-x_{j}\right)=\prod_{i=2}^{n}\left(x_{i}-x_{1}\right)\left(x_{i}-x_{2}\right) \cdots\left(x_{i}-x_{i-1}\right) | | |
| lyche-numerical-linear-algebra_FO0532 | 43 | 1.000 | \boldsymbol{\alpha}=\left[\alpha_{1}, \ldots, \alpha_{n}\right]^{T}, \boldsymbol{\beta}= | | |
| lyche-numerical-linear-algebra_FO0533 | 43 | 1.000 | \left[\beta_{1}, \ldots, \beta_{n}\right]^{T} | | |
| lyche-numerical-linear-algebra_FO0534 | 43 | 0.996 | a_{i, j}=1 /\left(\alpha_{i}+\beta_{j}\right), i, j= | | |
| lyche-numerical-linear-algebra_FO0535 | 43 | 1.000 | 1,2, \ldots, n | | |
| lyche-numerical-linear-algebra_FO0536 | 43 | 0.997 | A(T)=A\left(A B P_{3} P_{1}\right)+A\left(P_{3} B C P_{2}\right)-A\left(P_{1} A C P_{2}\right) | | |
| lyche-numerical-linear-algebra_FO0537 | 43 | 0.997 | x_{n}^{k} | | |
| lyche-numerical-linear-algebra_FO0538 | 43 | 0.997 | k | | |
| lyche-numerical-linear-algebra_FO0539 | 43 | 0.997 | k+1 | | |
| lyche-numerical-linear-algebra_FO0540 | 43 | 0.997 | k=n-1, n-2, \ldots, 1 | | |
| lyche-numerical-linear-algebra_FO0541 | 44 | 1.000 | P=\prod_{i=1}^{n} \prod_{j=1}^{n} a_{i j} | | |
| lyche-numerical-linear-algebra_FO0542 | 44 | 1.000 | \boldsymbol{\gamma}=\left[\gamma_{1}, \ldots, \gamma_{n}\right]^{T} | | |
| lyche-numerical-linear-algebra_FO0543 | 44 | 1.000 | \prod_{j=1}^{n}\left(\alpha_{i}+\beta_{j}\right) | | |
| lyche-numerical-linear-algebra_FO0544 | 44 | 1.000 | i=1,2, \ldots, n | | |
| lyche-numerical-linear-algebra_FO0545 | 44 | 0.999 | n-1 | | |
| lyche-numerical-linear-algebra_FO0546 | 44 | 0.999 | \alpha_{r}+\beta_{s} | | |
| lyche-numerical-linear-algebra_FO0547 | 44 | 1.000 | \operatorname{det}(\boldsymbol{C}) | | |
| lyche-numerical-linear-algebra_FO0548 | 44 | 1.000 | n(n-1) | | |
| lyche-numerical-linear-algebra_FO0549 | 44 | 0.997 | \operatorname{det}(\boldsymbol{C})=q(\alpha, \beta) | | |
| lyche-numerical-linear-algebra_FO0550 | 44 | 0.997 | q | | |
| lyche-numerical-linear-algebra_FO0551 | 44 | 1.000 | \alpha_{i} | | |
| lyche-numerical-linear-algebra_FO0552 | 44 | 1.000 | \beta_{j} | | |
| lyche-numerical-linear-algebra_FO0553 | 44 | 1.000 | \operatorname{det}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO0554 | 44 | 1.000 | \alpha_{i}=\alpha_{j} | | |
| lyche-numerical-linear-algebra_FO0555 | 44 | 1.000 | \beta_{r}=\beta_{s} | | |
| lyche-numerical-linear-algebra_FO0556 | 44 | 1.000 | r \neq s | | |
| lyche-numerical-linear-algebra_FO0557 | 44 | 1.000 | q(\boldsymbol{\alpha}, \boldsymbol{\beta}) | | |
| lyche-numerical-linear-algebra_FO0558 | 44 | 1.000 | g(\boldsymbol{\alpha}) | | |
| lyche-numerical-linear-algebra_FO0559 | 44 | 1.000 | g(\boldsymbol{\beta}) | | |
| lyche-numerical-linear-algebra_FO0560 | 44 | 1.000 | \boldsymbol{\alpha} | | |
| lyche-numerical-linear-algebra_FO0561 | 44 | 1.000 | \boldsymbol{\beta} | | |
| lyche-numerical-linear-algebra_FO0562 | 44 | 1.000 | k=1 | | |
| lyche-numerical-linear-algebra_FO0563 | 44 | 1.000 | \beta_{i}+\alpha_{i}=0, i=1,2, \ldots, n | | |
| lyche-numerical-linear-algebra_FO0564 | 44 | 1.000 | \boldsymbol{A}^{-1}=\left(b_{j, k}\right) | | |
| lyche-numerical-linear-algebra_FO0565 | 44 | 1.000 | \boldsymbol{H}_{n}=\left(h_{i, j}\right) | | |
| lyche-numerical-linear-algebra_FO0566 | 44 | 1.000 | h_{i, j}=1 /(i+j-1) | | |
| lyche-numerical-linear-algebra_FO0567 | 44 | 1.000 | t_{i, j}^{n} | | |
| lyche-numerical-linear-algebra_FO0568 | 44 | 1.000 | \boldsymbol{T}_{n}=\boldsymbol{H}_{n}^{-1} | | |
| lyche-numerical-linear-algebra_FO0569 | 45 | 1.000 | \boldsymbol{L} | | |
| lyche-numerical-linear-algebra_FO0570 | 45 | 1.000 | \boldsymbol{U} | | |
| lyche-numerical-linear-algebra_FO0571 | 45 | 1.000 | \boldsymbol{A}= | | |
| lyche-numerical-linear-algebra_FO0572 | 45 | 1.000 | \boldsymbol{L} \boldsymbol{U} | | |
| lyche-numerical-linear-algebra_FO0573 | 45 | 1.000 | \boldsymbol{A}=\boldsymbol{L} \boldsymbol{D} \boldsymbol{U} | | |
| lyche-numerical-linear-algebra_FO0574 | 45 | 1.000 | \boldsymbol{D} | | |
| lyche-numerical-linear-algebra_FO0575 | 45 | 1.000 | \boldsymbol{A}=\boldsymbol{Q} \boldsymbol{R} | | |
| lyche-numerical-linear-algebra_FO0576 | 45 | 1.000 | \boldsymbol{Q} | | |
| lyche-numerical-linear-algebra_FO0577 | 45 | 1.000 | \boldsymbol{Q}^{*} \boldsymbol{Q}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO0578 | 45 | 1.000 | \boldsymbol{R} | | |
| lyche-numerical-linear-algebra_FO0579 | 46 | 0.996 | [a, b], n+1 \geq 2 | | |
| lyche-numerical-linear-algebra_FO0580 | 46 | 0.996 | [a, b] | | |
| lyche-numerical-linear-algebra_FO0581 | 46 | 1.000 | y | | |
| lyche-numerical-linear-algebra_FO0582 | 46 | 1.000 | \boldsymbol{y}:=\left[y_{1}, \ldots, y_{n+1}\right]^{T} \in \mathbb{R}^{n+1} | | |
| lyche-numerical-linear-algebra_FO0583 | 46 | 1.000 | g:[a, b] \rightarrow \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO0584 | 47 | 1.000 | a \leq x_{1}<x_{2}<\cdots<x_{n+1} \leq b | | |
| lyche-numerical-linear-algebra_FO0585 | 47 | 1.000 | n+1 | | |
| lyche-numerical-linear-algebra_FO0586 | 47 | 1.000 | g | | |
| lyche-numerical-linear-algebra_FO0587 | 47 | 1.000 | n=1, g | | |
| lyche-numerical-linear-algebra_FO0588 | 47 | 1.000 | f | | |
| lyche-numerical-linear-algebra_FO0589 | 47 | 1.000 | [a, b]=[-1,1] | | |
| lyche-numerical-linear-algebra_FO0590 | 47 | 1.000 | y_{i}=f\left(x_{i}\right), i=1, \ldots, 14 | | |
| lyche-numerical-linear-algebra_FO0591 | 48 | 1.000 | p_{i} | | |
| lyche-numerical-linear-algebra_FO0592 | 48 | 1.000 | \leq 1 | | |
| lyche-numerical-linear-algebra_FO0593 | 48 | 1.000 | p_{1} | | |
| lyche-numerical-linear-algebra_FO0594 | 48 | 0.656 | \leq 2 | | |
| lyche-numerical-linear-algebra_FO0595 | 48 | 0.656 | C^{2} | | |
| lyche-numerical-linear-algebra_FO0596 | 49 | 0.950 | D_{2} | | |
| lyche-numerical-linear-algebra_FO0597 | 49 | 0.950 | \boldsymbol{y} \in | | |
| lyche-numerical-linear-algebra_FO0598 | 49 | 1.000 | \mathbb{R}^{n+1} | | |
| lyche-numerical-linear-algebra_FO0599 | 49 | 1.000 | x_{1}, \ldots, x_{n+1} | | |
| lyche-numerical-linear-algebra_FO0600 | 49 | 1.000 | \mu_{1}, \mu_{n+1} | | |
| lyche-numerical-linear-algebra_FO0601 | 49 | 1.000 | g \in C^{2}[a, b] | | |
| lyche-numerical-linear-algebra_FO0602 | 49 | 1.000 | g\left(x_{i}\right)=y_{i}, \quad i=1,2, \ldots, n+1 | | |
| lyche-numerical-linear-algebra_FO0603 | 49 | 1.000 | \boldsymbol{D}_{\mathbf{2}} | | |
| lyche-numerical-linear-algebra_FO0604 | 49 | 1.000 | g^{\prime \prime}(a)=\mu_{1}, \quad g^{\prime \prime}(b)=\mu_{n+1} | | |
| lyche-numerical-linear-algebra_FO0605 | 49 | 1.000 | N | | |
| lyche-numerical-linear-algebra_FO0606 | 49 | 1.000 | \mu_{1}=\mu_{n+1}=0 | | |
| lyche-numerical-linear-algebra_FO0607 | 49 | 1.000 | n=2 | | |
| lyche-numerical-linear-algebra_FO0608 | 49 | 1.000 | f:[0,2] \rightarrow \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO0609 | 49 | 1.000 | [a, b]=[0,2], \boldsymbol{y}:=[0,1,16]^{T} | | |
| lyche-numerical-linear-algebra_FO0610 | 49 | 1.000 | \mu_{1}=g^{\prime \prime}(0)=0, \mu_{3}=g^{\prime \prime}(2)= | | |
| lyche-numerical-linear-algebra_FO0611 | 49 | 1.000 | x_{1}=0, x_{2}=1 | | |
| lyche-numerical-linear-algebra_FO0612 | 49 | 1.000 | x_{3}=2 | | |
| lyche-numerical-linear-algebra_FO0613 | 49 | 1.000 | p_{2} | | |
| lyche-numerical-linear-algebra_FO0614 | 49 | 1.000 | p_{1}(1)=p_{2}(1)=1, p_{1}^{\prime}(1)=p_{2}^{\prime}(1)=4, p_{1}^{\prime \prime}(1)=p_{2}^{\prime \prime}(1)= | | |
| lyche-numerical-linear-algebra_FO0615 | 49 | 1.000 | g \in C^{2}[0,2] | | |
| lyche-numerical-linear-algebra_FO0616 | 49 | 1.000 | g(0)=p_{1}(0)=0, g(1)=p_{2}(1)=1, g(2)= | | |
| lyche-numerical-linear-algebra_FO0617 | 49 | 1.000 | p_{2}(2)=16, g^{\prime \prime}(0)=p_{1}^{\prime \prime}(0)=0 | | |
| lyche-numerical-linear-algebra_FO0618 | 49 | 1.000 | g^{\prime \prime}(2)=p_{2}^{\prime \prime}(2)=48 | | |
| lyche-numerical-linear-algebra_FO0619 | 49 | 1.000 | p_{1}^{\prime \prime \prime}(x)=9 \neq | | |
| lyche-numerical-linear-algebra_FO0620 | 49 | 1.000 | 39=p_{2}^{\prime \prime \prime}(x) | | |
| lyche-numerical-linear-algebra_FO0621 | 49 | 1.000 | 3(n-1) C^{2} | | |
| lyche-numerical-linear-algebra_FO0622 | 49 | 1.000 | 4 n | | |
| lyche-numerical-linear-algebra_FO0623 | 49 | 1.000 | p_{1}, \ldots, p_{n} | | |
| lyche-numerical-linear-algebra_FO0625 | 50 | 1.000 | (0,2) | | |
| lyche-numerical-linear-algebra_FO0626 | 50 | 1.000 | a<b | | |
| lyche-numerical-linear-algebra_FO0627 | 50 | 1.000 | h=(b-a) / n | | |
| lyche-numerical-linear-algebra_FO0628 | 50 | 1.000 | n \geq 2, x_{i}=a+(i-1) h | | |
| lyche-numerical-linear-algebra_FO0629 | 50 | 1.000 | y_{i}, \mu_{i} | | |
| lyche-numerical-linear-algebra_FO0630 | 50 | 1.000 | i=1, \ldots, n+1 | | |
| lyche-numerical-linear-algebra_FO0631 | 51 | 1.000 | 1 \leq i \leq n | | |
| lyche-numerical-linear-algebra_FO0632 | 51 | 1.000 | p_{i}\left(x_{i}\right)=c_{i, 1}=y_{i} | | |
| lyche-numerical-linear-algebra_FO0633 | 51 | 1.000 | p_{i}^{\prime \prime}(x)=2 c_{i, 3}+6 c_{i, 4}\left(x-x_{i}\right) | | |
| lyche-numerical-linear-algebra_FO0634 | 51 | 1.000 | c_{i, 3} | | |
| lyche-numerical-linear-algebra_FO0635 | 51 | 1.000 | p_{i}^{\prime \prime}\left(x_{i}\right)= | | |
| lyche-numerical-linear-algebra_FO0636 | 51 | 1.000 | 2 c_{i, 3}=\mu_{i} | | |
| lyche-numerical-linear-algebra_FO0637 | 51 | 1.000 | c_{i, 4} | | |
| lyche-numerical-linear-algebra_FO0638 | 51 | 1.000 | p_{i}^{\prime \prime}\left(x_{i+1}\right)=\mu_{i}+6 h c_{i, 4}=\mu_{i+1} | | |
| lyche-numerical-linear-algebra_FO0639 | 51 | 1.000 | c_{i, 2} | | |
| lyche-numerical-linear-algebra_FO0640 | 51 | 1.000 | p_{i}\left(x_{i+1}\right)=y_{i}+c_{i, 2} h+\frac{\mu_{i}}{2} h^{2}+\frac{\mu_{i+1}-\mu_{i}}{6 h} h^{3}=y_{i+1} | | |
| lyche-numerical-linear-algebra_FO0641 | 51 | 1.000 | j=0,1,2,3 | | |
| lyche-numerical-linear-algebra_FO0642 | 51 | 1.000 | \left(x-x_{i}\right)^{j} | | |
| lyche-numerical-linear-algebra_FO0643 | 51 | 0.999 | \mu_{1}, \ldots | | |
| lyche-numerical-linear-algebra_FO0644 | 51 | 1.000 | \mu_{n+1} | | |
| lyche-numerical-linear-algebra_FO0645 | 51 | 0.973 | \mu_{1}, \ldots, \mu_{n+1} | | |
| lyche-numerical-linear-algebra_FO0646 | 51 | 0.973 | p_{i-1}\left(x_{i}\right)=p_{i}\left(x_{i}\right)=y_{i} | | |
| lyche-numerical-linear-algebra_FO0647 | 51 | 1.000 | p_{i-1}^{\prime \prime}\left(x_{i}\right)=p_{i}^{\prime \prime}\left(x_{i}\right)=\mu_{i} | | |
| lyche-numerical-linear-algebra_FO0648 | 51 | 1.000 | i=2, \ldots, n | | |
| lyche-numerical-linear-algebra_FO0649 | 51 | 1.000 | g \in C^{2} | | |
| lyche-numerical-linear-algebra_FO0650 | 51 | 1.000 | p_{i-1}^{\prime}\left(x_{i}\right)=p_{i}^{\prime}\left(x_{i}\right) | | |
| lyche-numerical-linear-algebra_FO0651 | 51 | 1.000 | g\left(x_{i}\right)=y_{i}, i=1, \ldots, n+1, g^{\prime \prime}(a)=p_{1}^{\prime \prime}\left(x_{1}\right)=\mu_{1} | | |
| lyche-numerical-linear-algebra_FO0652 | 51 | 1.000 | g^{\prime \prime}(b)= | | |
| lyche-numerical-linear-algebra_FO0653 | 51 | 1.000 | p_{n}^{\prime \prime}\left(x_{n+1}\right)=\mu_{n+1} | | |
| lyche-numerical-linear-algebra_FO0654 | 51 | 1.000 | \mu_{2}, \ldots, \mu_{n} | | |
| lyche-numerical-linear-algebra_FO0655 | 51 | 1.000 | n \geq 3 | | |
| lyche-numerical-linear-algebra_FO0656 | 52 | 1.000 | \boldsymbol{A}=\left[a_{i j}\right] \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO0657 | 52 | 1.000 | \left|x_{k}\right|=\max _{i}\left|x_{i}\right| | | |
| lyche-numerical-linear-algebra_FO0658 | 52 | 0.999 | \max _{1 \leq i \leq n}\left|x_{i}\right|=\left|x_{k}\right| \leq \frac{\left|b_{k}\right|}{\sigma_{k}} \leq \max _{1 \leq i \leq n}\left(\frac{\left|b_{i}\right|}{\sigma_{i}}\right) | | |
| lyche-numerical-linear-algebra_FO0659 | 52 | 0.999 | \max _{1 \leq i \leq n}\left|x_{i}\right| \leq 0 | | |
| lyche-numerical-linear-algebra_FO0660 | 52 | 1.000 | g_{1} | | |
| lyche-numerical-linear-algebra_FO0661 | 52 | 1.000 | g_{2} | | |
| lyche-numerical-linear-algebra_FO0662 | 52 | 1.000 | g:=g_{1}-g_{2} | | |
| lyche-numerical-linear-algebra_FO0663 | 52 | 1.000 | \boldsymbol{y}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0664 | 52 | 1.000 | \left[\mu_{2}, \ldots, \mu_{n}\right]^{T} | | |
| lyche-numerical-linear-algebra_FO0665 | 52 | 1.000 | \mu_{1}=\mu_{n+1} | | |
| lyche-numerical-linear-algebra_FO0666 | 52 | 1.000 | c_{i, j} | | |
| lyche-numerical-linear-algebra_FO0667 | 52 | 1.000 | g=0 | | |
| lyche-numerical-linear-algebra_FO0668 | 52 | 1.000 | g_{1}=g_{2} | | |
| lyche-numerical-linear-algebra_FO0669 | 52 | 1.000 | B | | |
| lyche-numerical-linear-algebra_FO0670 | 52 | 1.000 | [a, b]=[0,4] | | |
| lyche-numerical-linear-algebra_FO0671 | 52 | 1.000 | \boldsymbol{y}= | | |
| lyche-numerical-linear-algebra_FO0672 | 52 | 0.997 | \left[0, \frac{1}{6}, \frac{2}{3}, \frac{1}{6}, 0\right] | | |
| lyche-numerical-linear-algebra_FO0673 | 53 | 1.000 | \mu_{2}=\mu_{4}=1, \mu_{3}=-2 | | |
| lyche-numerical-linear-algebra_FO0674 | 53 | 1.000 | \{0,1,2,3,4\} | | |
| lyche-numerical-linear-algebra_FO0675 | 53 | 1.000 | (0,4) | | |
| lyche-numerical-linear-algebra_FO0676 | 53 | 1.000 | D^{2} | | |
| lyche-numerical-linear-algebra_FO0677 | 53 | 1.000 | \boldsymbol{A}=\operatorname{tridiag}\left(a_{i}, d_{i}, c_{i}\right) \in | | |
| lyche-numerical-linear-algebra_FO0678 | 54 | 1.000 | \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO0679 | 54 | 1.000 | \boldsymbol{A}=\boldsymbol{L} \boldsymbol{U} | | |
| lyche-numerical-linear-algebra_FO0680 | 54 | 1.000 | \boldsymbol{A} \boldsymbol{x}=\boldsymbol{L} \boldsymbol{U} \boldsymbol{x}=\boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO0681 | 54 | 1.000 | \boldsymbol{L} \boldsymbol{z}=\boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO0682 | 54 | 1.000 | \boldsymbol{U} \boldsymbol{x}=\boldsymbol{z} | | |
| lyche-numerical-linear-algebra_FO0683 | 54 | 1.000 | u_{1}, u_{2}, \ldots, u_{n-1} | | |
| lyche-numerical-linear-algebra_FO0684 | 54 | 0.999 | u_{n} \neq 0 | | |
| lyche-numerical-linear-algebra_FO0685 | 54 | 0.999 | \boldsymbol{z} | | |
| lyche-numerical-linear-algebra_FO0686 | 54 | 1.000 | \boldsymbol{l} \in \mathbb{C}^{n-1}, \boldsymbol{u} \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO0687 | 54 | 0.852 | \boldsymbol{a}, \boldsymbol{c} \in \mathbb{C}^{n-1}, \boldsymbol{d} \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO0688 | 55 | 1.000 | \boldsymbol{l}, \in \mathbb{C}^{n-1} | | |
| lyche-numerical-linear-algebra_FO0689 | 55 | 1.000 | \boldsymbol{u} \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO0690 | 55 | 1.000 | \boldsymbol{b} \in \mathbb{C}^{n, r} | | |
| lyche-numerical-linear-algebra_FO0691 | 55 | 1.000 | r \in \mathbb{N} | | |
| lyche-numerical-linear-algebra_FO0692 | 55 | 1.000 | u_{k} | | |
| lyche-numerical-linear-algebra_FO0693 | 55 | 1.000 | k=1, \ldots, n | | |
| lyche-numerical-linear-algebra_FO0694 | 55 | 1.000 | a_{0}:=c_{n}:=0 | | |
| lyche-numerical-linear-algebra_FO0695 | 55 | 1.000 | \left|u_{1}\right|=\left|d_{1}\right|=\sigma_{1}+\left|c_{1}\right| | | |
| lyche-numerical-linear-algebra_FO0696 | 55 | 1.000 | \left|u_{k}\right| \geq | | |
| lyche-numerical-linear-algebra_FO0697 | 55 | 0.999 | \sigma_{k}+\left|c_{k}\right| | | |
| lyche-numerical-linear-algebra_FO0698 | 55 | 0.999 | 1 \leq k \leq n-1 | | |
| lyche-numerical-linear-algebra_FO0699 | 55 | 0.999 | \left|c_{k}\right| /\left|u_{k}\right|<1 | | |
| lyche-numerical-linear-algebra_FO0700 | 56 | 1.000 | u_{1}=d_{1} | | |
| lyche-numerical-linear-algebra_FO0701 | 56 | 1.000 | l_{k} | | |
| lyche-numerical-linear-algebra_FO0702 | 56 | 1.000 | 3 n-3 | | |
| lyche-numerical-linear-algebra_FO0703 | 56 | 1.000 | r(5 n-4) | | |
| lyche-numerical-linear-algebra_FO0704 | 56 | 1.000 | O(n) | | |
| lyche-numerical-linear-algebra_FO0705 | 56 | 1.000 | 8 n-7 | | |
| lyche-numerical-linear-algebra_FO0706 | 56 | 1.000 | r=1 | | |
| lyche-numerical-linear-algebra_FO0707 | 56 | 0.939 | [0,1] | | |
| lyche-numerical-linear-algebra_FO0708 | 56 | 0.939 | u | | |
| lyche-numerical-linear-algebra_FO0709 | 56 | 1.000 | u(0)=u(1)= | | |
| lyche-numerical-linear-algebra_FO0710 | 56 | 1.000 | f(x)=1 | | |
| lyche-numerical-linear-algebra_FO0711 | 56 | 1.000 | u(x)=x(x-1) / 2 | | |
| lyche-numerical-linear-algebra_FO0712 | 56 | 1.000 | O\left(n^{3}\right) | | |
| lyche-numerical-linear-algebra_FO0713 | 57 | 1.000 | h:=1 /(m+1) | | |
| lyche-numerical-linear-algebra_FO0714 | 57 | 0.816 | x_{j}:=j h | | |
| lyche-numerical-linear-algebra_FO0715 | 57 | 0.816 | j=0,1, \ldots, m+1 | | |
| lyche-numerical-linear-algebra_FO0716 | 57 | 0.958 | v_{j} | | |
| lyche-numerical-linear-algebra_FO0717 | 57 | 0.958 | u\left(x_{j}\right) | | |
| lyche-numerical-linear-algebra_FO0718 | 57 | 1.000 | h^{2} | | |
| lyche-numerical-linear-algebra_FO0719 | 57 | 1.000 | \boldsymbol{T} | | |
| lyche-numerical-linear-algebra_FO0720 | 57 | 0.999 | \boldsymbol{T}=\operatorname{tridiag}\left(a_{i}, d_{i}, c_{i}\right) \in | | |
| lyche-numerical-linear-algebra_FO0721 | 57 | 1.000 | \mathbb{R}^{m \times m} | | |
| lyche-numerical-linear-algebra_FO0722 | 57 | 1.000 | a_{i}=c_{i}=-1 | | |
| lyche-numerical-linear-algebra_FO0723 | 57 | 1.000 | d_{i}=2 | | |
| lyche-numerical-linear-algebra_FO0724 | 58 | 1.000 | \boldsymbol{A}_{1} | | |
| lyche-numerical-linear-algebra_FO0725 | 58 | 1.000 | \boldsymbol{A}_{2} | | |
| lyche-numerical-linear-algebra_FO0726 | 58 | 1.000 | \boldsymbol{A}_{3} | | |
| lyche-numerical-linear-algebra_FO0727 | 58 | 1.000 | \operatorname{det}\left(\boldsymbol{A}_{3}\right)=4 \neq 0 | | |
| lyche-numerical-linear-algebra_FO0728 | 58 | 1.000 | \left|d_{1}\right|>\left|c_{1}\right| | | |
| lyche-numerical-linear-algebra_FO0729 | 58 | 0.574 | a_{i} \neq 0 | | |
| lyche-numerical-linear-algebra_FO0730 | 58 | 0.574 | i=1, \ldots, n-2 | | |
| lyche-numerical-linear-algebra_FO0731 | 58 | 0.574 | L U | | |
| lyche-numerical-linear-algebra_FO0732 | 58 | 1.000 | d_{n} \neq 0 | | |
| lyche-numerical-linear-algebra_FO0733 | 58 | 0.999 | k=1, \ldots, n-1 | | |
| lyche-numerical-linear-algebra_FO0734 | 58 | 0.991 | \left|u_{k}\right|>\left|c_{k}\right| | | |
| lyche-numerical-linear-algebra_FO0735 | 58 | 0.798 | \left|u_{1}\right|=\left|d_{1}\right|>\left|c_{1}\right| | | |
| lyche-numerical-linear-algebra_FO0736 | 58 | 0.798 | 1 \leq k \leq n-2 | | |
| lyche-numerical-linear-algebra_FO0737 | 58 | 1.000 | k=n-1 | | |
| lyche-numerical-linear-algebra_FO0738 | 58 | 1.000 | a_{n-1} \neq 0 | | |
| lyche-numerical-linear-algebra_FO0739 | 58 | 1.000 | \left|u_{k+1}\right|>\left|d_{k+1}\right|-\left|a_{k}\right| \geq\left|c_{k+1}\right| | | |
| lyche-numerical-linear-algebra_FO0740 | 58 | 1.000 | \left|u_{n}\right|>\left|d_{n}\right|-\left|a_{n-1}\right| \geq 0 | | |
| lyche-numerical-linear-algebra_FO0741 | 58 | 1.000 | a_{n-1}=0 | | |
| lyche-numerical-linear-algebra_FO0742 | 58 | 1.000 | \left|u_{n}\right| \geq\left|d_{n}\right|>0 | | |
| lyche-numerical-linear-algebra_FO0743 | 58 | 1.000 | \boldsymbol{T} \boldsymbol{v}=\boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO0744 | 58 | 1.000 | \boldsymbol{v}=\boldsymbol{T}^{-1} \boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO0745 | 58 | 1.000 | \boldsymbol{T}^{-1} | | |
| lyche-numerical-linear-algebra_FO0746 | 58 | 1.000 | \boldsymbol{T}^{-1} \boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO0747 | 58 | 1.000 | O\left(n^{2}\right) | | |
| lyche-numerical-linear-algebra_FO0748 | 59 | 1.000 | \boldsymbol{x} \in | | |
| lyche-numerical-linear-algebra_FO0749 | 59 | 1.000 | L | | |
| lyche-numerical-linear-algebra_FO0750 | 59 | 1.000 | x=0 | | |
| lyche-numerical-linear-algebra_FO0751 | 59 | 1.000 | x=L | | |
| lyche-numerical-linear-algebra_FO0752 | 59 | 1.000 | F | | |
| lyche-numerical-linear-algebra_FO0753 | 59 | 1.000 | (L, 0) | | |
| lyche-numerical-linear-algebra_FO0754 | 59 | 1.000 | y(x) | | |
| lyche-numerical-linear-algebra_FO0755 | 59 | 1.000 | R | | |
| lyche-numerical-linear-algebra_FO0756 | 59 | 1.000 | u:[0,1] \rightarrow \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO0757 | 59 | 1.000 | u(t):=y(t L) | | |
| lyche-numerical-linear-algebra_FO0758 | 59 | 1.000 | u^{\prime \prime}(t)= | | |
| lyche-numerical-linear-algebra_FO0759 | 59 | 1.000 | L^{2} y^{\prime \prime}(t L) | | |
| lyche-numerical-linear-algebra_FO0760 | 59 | 1.000 | u=0 | | |
| lyche-numerical-linear-algebra_FO0761 | 59 | 1.000 | F=0 | | |
| lyche-numerical-linear-algebra_FO0762 | 59 | 1.000 | K=0 | | |
| lyche-numerical-linear-algebra_FO0763 | 59 | 1.000 | u(t)=\sin (\pi t) | | |
| lyche-numerical-linear-algebra_FO0764 | 59 | 1.000 | u^{\prime \prime}(t)=-\pi^{2} u(t) | | |
| lyche-numerical-linear-algebra_FO0765 | 59 | 1.000 | K=\pi^{2} | | |
| lyche-numerical-linear-algebra_FO0766 | 59 | 1.000 | F=\frac{\pi^{2} R}{L^{2}} | | |
| lyche-numerical-linear-algebra_FO0767 | 59 | 1.000 | m \in \mathbb{N}, h:=1 /(m+ | | |
| lyche-numerical-linear-algebra_FO0768 | 60 | 1.000 | v_{j} \approx u(j h) | | |
| lyche-numerical-linear-algebra_FO0769 | 60 | 1.000 | j=0, \ldots, m+1 | | |
| lyche-numerical-linear-algebra_FO0770 | 60 | 1.000 | \lambda:=h^{2} K | | |
| lyche-numerical-linear-algebra_FO0771 | 60 | 1.000 | \lambda=4 \sin ^{2}(\pi h / 2) | | |
| lyche-numerical-linear-algebra_FO0772 | 60 | 1.000 | \lambda= | | |
| lyche-numerical-linear-algebra_FO0773 | 60 | 1.000 | h^{2} K=\frac{h^{2} F L^{2}}{R} | | |
| lyche-numerical-linear-algebra_FO0774 | 60 | 1.000 | h | | |
| lyche-numerical-linear-algebra_FO0775 | 60 | 1.000 | \frac{\pi^{2} R}{L^{2}} | | |
| lyche-numerical-linear-algebra_FO0776 | 60 | 0.997 | \boldsymbol{T}=\operatorname{tridiag}(-1,2,-1) | | |
| lyche-numerical-linear-algebra_FO0777 | 60 | 0.990 | a, d \in \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO0778 | 60 | 0.990 | \mathbf{1 D} | | |
| lyche-numerical-linear-algebra_FO0779 | 60 | 1.000 | |d|>2|a| | | |
| lyche-numerical-linear-algebra_FO0780 | 61 | 1.000 | m=3 | | |
| lyche-numerical-linear-algebra_FO0781 | 61 | 1.000 | \boldsymbol{T}_{1}=\left(t_{k j}\right)_{k, j}= | | |
| lyche-numerical-linear-algebra_FO0782 | 61 | 0.999 | \operatorname{tridiag}(a, d, a) \in \mathbb{R}^{m \times m} | | |
| lyche-numerical-linear-algebra_FO0783 | 61 | 0.999 | m \geq 2, a, d \in \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO0784 | 61 | 0.999 | h=1 /(m+1) | | |
| lyche-numerical-linear-algebra_FO0785 | 61 | 1.000 | \boldsymbol{T}_{1} \boldsymbol{s}_{j}=\lambda_{j} \boldsymbol{s}_{j} | | |
| lyche-numerical-linear-algebra_FO0786 | 61 | 1.000 | 1<k<m | | |
| lyche-numerical-linear-algebra_FO0787 | 61 | 1.000 | k=1, m | | |
| lyche-numerical-linear-algebra_FO0788 | 61 | 1.000 | j \pi h=j \pi /(m+1) \in(0, \pi) | | |
| lyche-numerical-linear-algebra_FO0789 | 61 | 0.988 | (0, \pi) | | |
| lyche-numerical-linear-algebra_FO0790 | 61 | 1.000 | \boldsymbol{T}_{1} | | |
| lyche-numerical-linear-algebra_FO0791 | 62 | 1.000 | i=\sqrt{-1} | | |
| lyche-numerical-linear-algebra_FO0792 | 62 | 1.000 | \boldsymbol{A}^{*}:=\overline{\boldsymbol{A}}^{T} | | |
| lyche-numerical-linear-algebra_FO0793 | 62 | 1.000 | \boldsymbol{x} \neq 0 | | |
| lyche-numerical-linear-algebra_FO0794 | 62 | 0.998 | \boldsymbol{x}^{*} | | |
| lyche-numerical-linear-algebra_FO0795 | 62 | 0.998 | \boldsymbol{x}^{*} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO0796 | 62 | 0.998 | \lambda=\frac{\boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{x}}{\boldsymbol{x}^{*} \boldsymbol{x}} | | |
| lyche-numerical-linear-algebra_FO0797 | 62 | 0.996 | \bar{\lambda}=\lambda^{*}=\frac{\left(\boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{x}\right)^{*}}{\left(\boldsymbol{x}^{*} \boldsymbol{x}\right)^{*}}=\frac{\boldsymbol{x}^{*} \boldsymbol{A}^{*} \boldsymbol{x}}{\boldsymbol{x}^{*} \boldsymbol{x}}=\frac{\boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{x}}{\boldsymbol{x}^{*} \boldsymbol{x}}=\lambda | | |
| lyche-numerical-linear-algebra_FO0798 | 62 | 0.725 | (\mu, \boldsymbol{y}) | | |
| lyche-numerical-linear-algebra_FO0799 | 62 | 0.725 | \mu \neq \lambda | | |
| lyche-numerical-linear-algebra_FO0800 | 62 | 1.000 | \boldsymbol{y}^{*} | | |
| lyche-numerical-linear-algebra_FO0801 | 62 | 1.000 | \mu | | |
| lyche-numerical-linear-algebra_FO0802 | 62 | 1.000 | \lambda \neq \mu | | |
| lyche-numerical-linear-algebra_FO0803 | 62 | 1.000 | \boldsymbol{y}^{*} \boldsymbol{x}=0 | | |
| lyche-numerical-linear-algebra_FO0804 | 62 | 1.000 | \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO0805 | 63 | 0.997 | \left[\begin{array}{ll}\boldsymbol{A}_{11} & \boldsymbol{A}_{12} \\ \boldsymbol{A}_{21} & \boldsymbol{A}_{22}\end{array}\right]=\left[\begin{array}{l|ll}1 & 2 & 3 \\ \hline 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO0806 | 63 | 0.949 | \left[\boldsymbol{a}_{: 1}, \boldsymbol{a}_{: 2}, \boldsymbol{a}_{: 3}\right]=\left[\begin{array}{c|c|c}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO0807 | 63 | 0.125 | \left[\begin{array}{c}\boldsymbol{a}_{1}^{T} \\ \boldsymbol{a}_{2:}^{T} \\ \boldsymbol{a}_{3:}^{T}\end{array}\right]=\left[\begin{array}{l}\frac{123}{45} 6 \\ \hline \frac{789}{788}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO0808 | 63 | 0.673 | \left[\boldsymbol{A}_{11}, \boldsymbol{A}_{12}\right]=\left[\begin{array}{c|cc}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO0809 | 63 | 0.902 | (i i) | | |
| lyche-numerical-linear-algebra_FO0810 | 63 | 0.902 | (i i i) \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO0811 | 63 | 1.000 | \boldsymbol{A} \in \mathbb{C}^{m \times p} | | |
| lyche-numerical-linear-algebra_FO0812 | 63 | 1.000 | \boldsymbol{B} \in \mathbb{C}^{p \times n} | | |
| lyche-numerical-linear-algebra_FO0813 | 63 | 1.000 | \boldsymbol{B}=\left[\boldsymbol{b}_{: 1}, \ldots, \boldsymbol{b}_{: n}\right] | | |
| lyche-numerical-linear-algebra_FO0814 | 63 | 1.000 | \boldsymbol{A} \boldsymbol{e}_{j} | | |
| lyche-numerical-linear-algebra_FO0815 | 63 | 1.000 | j=1, \ldots, p | | |
| lyche-numerical-linear-algebra_FO0816 | 63 | 1.000 | \boldsymbol{A}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO0817 | 63 | 1.000 | \boldsymbol{e}_{i}^{T} \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO0818 | 63 | 1.000 | \boldsymbol{A} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO0819 | 64 | 1.000 | \boldsymbol{B}=\left[\boldsymbol{B}_{1}, \boldsymbol{B}_{2}\right] | | |
| lyche-numerical-linear-algebra_FO0820 | 64 | 1.000 | \boldsymbol{B}_{1} \in \mathbb{C}^{p \times r} | | |
| lyche-numerical-linear-algebra_FO0821 | 64 | 1.000 | \boldsymbol{B}_{2} \in \mathbb{C}^{p \times(n-r)} | | |
| lyche-numerical-linear-algebra_FO0822 | 64 | 0.993 | \boldsymbol{A}=\left[\begin{array}{l}\boldsymbol{A}_{1} \\ \boldsymbol{A}_{2}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO0823 | 64 | 0.993 | \boldsymbol{A}_{1} \in \mathbb{C}^{k \times p} | | |
| lyche-numerical-linear-algebra_FO0824 | 64 | 0.993 | \boldsymbol{A}_{2} \in \mathbb{C}^{(m-k) \times p} | | |
| lyche-numerical-linear-algebra_FO0825 | 64 | 0.985 | \boldsymbol{A}=\left[\boldsymbol{A}_{1}, \boldsymbol{A}_{2}\right] | | |
| lyche-numerical-linear-algebra_FO0826 | 64 | 0.985 | \boldsymbol{B}=\left[\begin{array}{l}\boldsymbol{B}_{1} \\ \boldsymbol{B}_{2}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO0827 | 64 | 0.985 | \boldsymbol{A}_{1} \in \mathbb{C}^{m \times s}, \boldsymbol{A}_{2} \in \mathbb{C}^{m \times(p-s)}, \boldsymbol{B}_{1} \in | | |
| lyche-numerical-linear-algebra_FO0828 | 64 | 1.000 | \mathbb{C}^{s \times n} | | |
| lyche-numerical-linear-algebra_FO0829 | 64 | 1.000 | \boldsymbol{B}_{2} \in \mathbb{C}^{(p-s) \times n} | | |
| lyche-numerical-linear-algebra_FO0830 | 64 | 1.000 | \boldsymbol{A}=\left[\begin{array}{ll}\boldsymbol{A}_{11} & \boldsymbol{A}_{12} \\ \boldsymbol{A}_{21} & \boldsymbol{A}_{22}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO0831 | 64 | 1.000 | \boldsymbol{B}=\left[\begin{array}{ll}\boldsymbol{B}_{11} & \boldsymbol{B}_{12} \\ \boldsymbol{B}_{21} & \boldsymbol{B}_{22}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO0832 | 64 | 1.000 | \boldsymbol{A}_{11} | | |
| lyche-numerical-linear-algebra_FO0833 | 64 | 1.000 | \boldsymbol{A}_{21} | | |
| lyche-numerical-linear-algebra_FO0834 | 64 | 1.000 | \boldsymbol{B}_{11} | | |
| lyche-numerical-linear-algebra_FO0835 | 64 | 1.000 | \boldsymbol{B}_{12} | | |
| lyche-numerical-linear-algebra_FO0836 | 65 | 1.000 | \boldsymbol{A}_{i k} \boldsymbol{B}_{k j} | | |
| lyche-numerical-linear-algebra_FO0837 | 65 | 1.000 | \boldsymbol{A}, \boldsymbol{A}_{11} | | |
| lyche-numerical-linear-algebra_FO0838 | 65 | 1.000 | \boldsymbol{A}_{22} | | |
| lyche-numerical-linear-algebra_FO0839 | 65 | 1.000 | \boldsymbol{B}:=\boldsymbol{A}^{-1} | | |
| lyche-numerical-linear-algebra_FO0840 | 66 | 1.000 | \boldsymbol{B}_{21}= | | |
| lyche-numerical-linear-algebra_FO0841 | 66 | 0.577 | \mathbf{0 A}_{11}^{-1}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0842 | 66 | 1.000 | \boldsymbol{B}_{22} \boldsymbol{A}_{22}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO0843 | 66 | 1.000 | a_{i i} | | |
| lyche-numerical-linear-algebra_FO0844 | 66 | 1.000 | a_{i i}^{-1}, i=1, \ldots, n | | |
| lyche-numerical-linear-algebra_FO0845 | 66 | 0.785 | a_{11} | | |
| lyche-numerical-linear-algebra_FO0846 | 66 | 0.785 | a_{11} \neq \mathbf{0} | | |
| lyche-numerical-linear-algebra_FO0847 | 66 | 0.785 | \boldsymbol{A}^{-1}=\left[a_{11}^{-1}\right] | | |
| lyche-numerical-linear-algebra_FO0848 | 66 | 1.000 | \boldsymbol{A} \in \mathbb{C}^{(k+1) \times(k+1)} | | |
| lyche-numerical-linear-algebra_FO0849 | 66 | 1.000 | \boldsymbol{A}_{k} \in \mathbb{C}^{k \times k} | | |
| lyche-numerical-linear-algebra_FO0850 | 66 | 1.000 | 2.4 \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO0851 | 66 | 1.000 | \boldsymbol{A}_{k} | | |
| lyche-numerical-linear-algebra_FO0852 | 66 | 1.000 | \left(a_{k+1, k+1}\right) | | |
| lyche-numerical-linear-algebra_FO0853 | 66 | 1.000 | \boldsymbol{c} \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO0854 | 66 | 1.000 | a_{11}, \ldots, a_{k k} | | |
| lyche-numerical-linear-algebra_FO0855 | 66 | 1.000 | \boldsymbol{A}_{k}^{-1} | | |
| lyche-numerical-linear-algebra_FO0856 | 66 | 1.000 | a_{i i}^{-1}, i=1, \ldots, k | | |
| lyche-numerical-linear-algebra_FO0857 | 66 | 0.515 | \boldsymbol{C}=\boldsymbol{A B}=\left(c_{i j}\right) | | |
| lyche-numerical-linear-algebra_FO0858 | 66 | 1.000 | \boldsymbol{A}=\left(a_{i j}\right) | | |
| lyche-numerical-linear-algebra_FO0859 | 66 | 1.000 | \boldsymbol{B}=\left(b_{i j}\right) | | |
| lyche-numerical-linear-algebra_FO0860 | 66 | 1.000 | c_{i i}=a_{i i} b_{i i} | | |
| lyche-numerical-linear-algebra_FO0861 | 67 | 0.822 | \left\{\left(x-x_{i}\right)^{j}\right\}_{0 \leq j \leq n} | | |
| lyche-numerical-linear-algebra_FO0862 | 67 | 0.822 | n .^{3} | | |
| lyche-numerical-linear-algebra_FO0863 | 67 | 1.000 | D_{1} | | |
| lyche-numerical-linear-algebra_FO0864 | 67 | 1.000 | g^{\prime}(a)= | | |
| lyche-numerical-linear-algebra_FO0865 | 67 | 1.000 | s_{1} | | |
| lyche-numerical-linear-algebra_FO0866 | 67 | 1.000 | g^{\prime}(b)=s_{n+1} | | |
| lyche-numerical-linear-algebra_FO0867 | 67 | 1.000 | s_{n+1} | | |
| lyche-numerical-linear-algebra_FO0868 | 67 | 1.000 | \delta^{2} y_{i}:=y_{i+1}-2 y_{i}+y_{i-1}, i=2, \ldots, n | | |
| lyche-numerical-linear-algebra_FO0869 | 67 | 1.000 | g^{\prime}\left(x_{1}\right) | | |
| lyche-numerical-linear-algebra_FO0870 | 67 | 1.000 | g^{\prime}\left(x_{n+1}\right) | | |
| lyche-numerical-linear-algebra_FO0871 | 67 | 1.000 | x_{i} | | |
| lyche-numerical-linear-algebra_FO0872 | 68 | 0.999 | h \in C^{2}[a, b] | | |
| lyche-numerical-linear-algebra_FO0873 | 68 | 0.999 | h\left(x_{i}\right)=g\left(x_{i}\right), i=1, \ldots, n+1 | | |
| lyche-numerical-linear-algebra_FO0874 | 68 | 1.000 | \int_{a}^{b} g^{\prime \prime} e^{\prime \prime}=\left[g^{\prime \prime} e^{\prime}\right]_{a}^{b}-\int_{a}^{b} g^{\prime \prime \prime} e^{\prime} | | |
| lyche-numerical-linear-algebra_FO0875 | 68 | 1.000 | g^{\prime \prime} | | |
| lyche-numerical-linear-algebra_FO0876 | 68 | 1.000 | g^{\prime \prime}(b)=g^{\prime \prime}(a)=0 | | |
| lyche-numerical-linear-algebra_FO0877 | 68 | 1.000 | g^{\prime \prime \prime} | | |
| lyche-numerical-linear-algebra_FO0878 | 68 | 0.998 | v_{i} | | |
| lyche-numerical-linear-algebra_FO0879 | 68 | 0.998 | x_{i}, x_{i+1} | | |
| lyche-numerical-linear-algebra_FO0880 | 68 | 0.998 | e\left(x_{i}\right)=0 | | |
| lyche-numerical-linear-algebra_FO0881 | 68 | 1.000 | h=g+e | | |
| lyche-numerical-linear-algebra_FO0882 | 68 | 1.000 | \boldsymbol{y} \in \mathbb{R}^{n+1} | | |
| lyche-numerical-linear-algebra_FO0883 | 68 | 1.000 | \boldsymbol{x}=\left[x_{1}, \ldots, x_{n}\right] | | |
| lyche-numerical-linear-algebra_FO0884 | 69 | 1.000 | \boldsymbol{C} \in \mathbb{R}^{n \times 4} | | |
| lyche-numerical-linear-algebra_FO0885 | 69 | 1.000 | g\left(r_{j}\right) | | |
| lyche-numerical-linear-algebra_FO0886 | 69 | 1.000 | \boldsymbol{r}=\left[r_{1}, \ldots, r_{m}\right] \in | | |
| lyche-numerical-linear-algebra_FO0887 | 69 | 1.000 | i_{j} | | |
| lyche-numerical-linear-algebra_FO0888 | 69 | 1.000 | g\left(r_{j}\right)=p_{i_{j}}\left(r_{j}\right) | | |
| lyche-numerical-linear-algebra_FO0889 | 69 | 1.000 | k \in \mathbb{N}, \boldsymbol{t}=\left[t_{1}, \ldots, t_{k}\right] | | |
| lyche-numerical-linear-algebra_FO0890 | 69 | 1.000 | i=1 | | |
| lyche-numerical-linear-algebra_FO0891 | 69 | 1.000 | x<t_{2}, i=k | | |
| lyche-numerical-linear-algebra_FO0892 | 69 | 1.000 | x \geq t_{k} | | |
| lyche-numerical-linear-algebra_FO0893 | 69 | 1.000 | t_{i} \leq | | |
| lyche-numerical-linear-algebra_FO0894 | 69 | 1.000 | x<t_{i+1} | | |
| lyche-numerical-linear-algebra_FO0895 | 69 | 1.000 | \boldsymbol{x} \in \mathbb{R}^{m} | | |
| lyche-numerical-linear-algebra_FO0896 | 69 | 1.000 | \boldsymbol{i} | | |
| lyche-numerical-linear-algebra_FO0897 | 69 | 1.000 | \boldsymbol{i}=\left[i_{1}, \ldots, i_{m}\right] | | |
| lyche-numerical-linear-algebra_FO0898 | 70 | 1.000 | \boldsymbol{x}, \boldsymbol{C} | | |
| lyche-numerical-linear-algebra_FO0899 | 70 | 1.000 | \boldsymbol{G}=g(\boldsymbol{X}) | | |
| lyche-numerical-linear-algebra_FO0900 | 71 | 1.000 | f \in C^{2}[x-h, x+h] | | |
| lyche-numerical-linear-algebra_FO0901 | 71 | 1.000 | \eta_{2} | | |
| lyche-numerical-linear-algebra_FO0902 | 71 | 1.000 | f \in C^{4}[x-h, x+h] | | |
| lyche-numerical-linear-algebra_FO0903 | 71 | 1.000 | \eta_{4} | | |
| lyche-numerical-linear-algebra_FO0904 | 71 | 1.000 | \delta^{2} f(x) | | |
| lyche-numerical-linear-algebra_FO0905 | 71 | 1.000 | f, q | | |
| lyche-numerical-linear-algebra_FO0906 | 71 | 1.000 | q(x) \geq 0 | | |
| lyche-numerical-linear-algebra_FO0907 | 71 | 1.000 | x \in[a, b] | | |
| lyche-numerical-linear-algebra_FO0908 | 71 | 1.000 | m \in \mathbb{N}, h=(b-a) /(m+1), x_{j}=a+j h | | |
| lyche-numerical-linear-algebra_FO0909 | 71 | 0.993 | v_{0}=g_{0} | | |
| lyche-numerical-linear-algebra_FO0910 | 71 | 0.993 | v_{m+1}=g_{1} | | |
| lyche-numerical-linear-algebra_FO0911 | 71 | 1.000 | \boldsymbol{A} \boldsymbol{v}=\boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO0912 | 71 | 0.523 | \operatorname{tridiag}\left(a_{j}, d_{j} . c_{j}\right) \in \mathbb{R}^{m \times m} | | |
| lyche-numerical-linear-algebra_FO0913 | 72 | 1.000 | \frac{h}{2}|r(x)|<1 | | |
| lyche-numerical-linear-algebra_FO0914 | 72 | 1.000 | v_{1}, \ldots, v_{m} | | |
| lyche-numerical-linear-algebra_FO0915 | 72 | 1.000 | r=0, f=q=1 | | |
| lyche-numerical-linear-algebra_FO0916 | 72 | 1.000 | u(0)=1, u(1)=0 | | |
| lyche-numerical-linear-algebra_FO0917 | 72 | 1.000 | u(x)=1-\sinh x / \sinh 1 | | |
| lyche-numerical-linear-algebra_FO0918 | 72 | 1.000 | h=0.1,0.05,0.025,0.0125 | | |
| lyche-numerical-linear-algebra_FO0919 | 72 | 1.000 | \max _{1 \leq j \leq m}\left|u\left(x_{j}\right)-v_{j}\right| | | |
| lyche-numerical-linear-algebra_FO0920 | 72 | 1.000 | v_{j}, j= | | |
| lyche-numerical-linear-algebra_FO0921 | 72 | 1.000 | 0, \ldots, m+1 | | |
| lyche-numerical-linear-algebra_FO0922 | 72 | 1.000 | h=0.1 | | |
| lyche-numerical-linear-algebra_FO0923 | 72 | 1.000 | h^{p} | | |
| lyche-numerical-linear-algebra_FO0924 | 72 | 1.000 | a_{i, i+1} a_{i+1, i}>0 | | |
| lyche-numerical-linear-algebra_FO0925 | 72 | 1.000 | \boldsymbol{D}=\operatorname{diag}\left(d_{1}, \ldots, d_{n}\right) | | |
| lyche-numerical-linear-algebra_FO0926 | 72 | 1.000 | d_{i}>0 | | |
| lyche-numerical-linear-algebra_FO0927 | 72 | 1.000 | \boldsymbol{B}:=\boldsymbol{D} \boldsymbol{A} \boldsymbol{D}^{-1} | | |
| lyche-numerical-linear-algebra_FO0928 | 73 | 1.000 | \boldsymbol{T}=\boldsymbol{L} \boldsymbol{U} | | |
| lyche-numerical-linear-algebra_FO0929 | 73 | 0.998 | \boldsymbol{S} \in \mathbb{R}^{m \times m} | | |
| lyche-numerical-linear-algebra_FO0930 | 73 | 1.000 | s_{i j} | | |
| lyche-numerical-linear-algebra_FO0931 | 73 | 1.000 | \boldsymbol{S} \boldsymbol{T}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO0932 | 73 | 1.000 | \boldsymbol{T}^{-1}=\boldsymbol{S} | | |
| lyche-numerical-linear-algebra_FO0933 | 73 | 1.000 | a_{i j}=\boldsymbol{e}_{i}^{T} \boldsymbol{A} \boldsymbol{e}_{j} | | |
| lyche-numerical-linear-algebra_FO0934 | 73 | 1.000 | \boldsymbol{A}=\sum_{i=1}^{m} \sum_{j=1}^{n} a_{i j} \boldsymbol{e}_{i} \boldsymbol{e}_{j}^{T} | | |
| lyche-numerical-linear-algebra_FO0935 | 73 | 0.942 | \boldsymbol{A}^{T} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO0936 | 73 | 0.942 | \boldsymbol{B}=\boldsymbol{A}^{T} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO0937 | 73 | 1.000 | b_{i j}=\boldsymbol{a}_{: i}^{T} \boldsymbol{a}_{: j} | | |
| lyche-numerical-linear-algebra_FO0938 | 73 | 1.000 | \boldsymbol{B} \in \mathbb{R}^{p \times n} | | |
| lyche-numerical-linear-algebra_FO0939 | 73 | 1.000 | \boldsymbol{A} \in \mathbb{R}^{m \times n}, \boldsymbol{B} \in \mathbb{R}^{m \times p} | | |
| lyche-numerical-linear-algebra_FO0940 | 73 | 1.000 | \boldsymbol{X} \in \mathbb{R}^{n \times p} | | |
| lyche-numerical-linear-algebra_FO0941 | 74 | 0.989 | \boldsymbol{B}= | | |
| lyche-numerical-linear-algebra_FO0942 | 74 | 1.000 | \left[\begin{array}{c}\boldsymbol{B}_{1} \\ \mathbf{0}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO0943 | 74 | 1.000 | \boldsymbol{A} \boldsymbol{B}=\boldsymbol{A}_{1} \boldsymbol{B}_{1} | | |
| lyche-numerical-linear-algebra_FO0944 | 74 | 0.999 | \boldsymbol{A}, \boldsymbol{B}, \boldsymbol{C} \in | | |
| lyche-numerical-linear-algebra_FO0945 | 74 | 1.000 | \boldsymbol{A}_{1}, \boldsymbol{B}_{1}, \boldsymbol{C}_{1} \in \mathbb{R}^{(n-1) \times(n-1)} | | |
| lyche-numerical-linear-algebra_FO0946 | 74 | 0.921 | \boldsymbol{T} \boldsymbol{x}=\boldsymbol{b} ? | | |
| lyche-numerical-linear-algebra_FO0947 | 74 | 1.000 | \left[\begin{array}{cc}1 & 1+i \\ 1+i & 2\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO0948 | 75 | 1.000 | \boldsymbol{P} | | |
| lyche-numerical-linear-algebra_FO0949 | 75 | 0.951 | 3 \times 3 | | |
| lyche-numerical-linear-algebra_FO0950 | 76 | 1.000 | a_{11}^{(1)} \neq 0 | | |
| lyche-numerical-linear-algebra_FO0951 | 76 | 1.000 | l_{21}^{(1)}:= | | |
| lyche-numerical-linear-algebra_FO0952 | 76 | 0.997 | a_{21}^{(1)} / a_{11}^{(1)} | | |
| lyche-numerical-linear-algebra_FO0953 | 76 | 0.997 | l_{31}^{(1)}:=a_{31}^{(1)} / a_{11}^{(1)} | | |
| lyche-numerical-linear-algebra_FO0954 | 76 | 1.000 | b_{i}^{(2)}=b_{i}^{(1)}-l_{i 1}^{(1)} b_{i}^{(1)} | | |
| lyche-numerical-linear-algebra_FO0955 | 76 | 1.000 | i=2,3 | | |
| lyche-numerical-linear-algebra_FO0956 | 76 | 1.000 | a_{i j}^{(2)}=a_{i j}^{(1)}-l_{i, 1}^{(1)} a_{1 j}^{(1)} | | |
| lyche-numerical-linear-algebra_FO0957 | 76 | 1.000 | i, j=2,3 | | |
| lyche-numerical-linear-algebra_FO0958 | 76 | 1.000 | a_{11}^{(1)}=0 | | |
| lyche-numerical-linear-algebra_FO0959 | 76 | 1.000 | a_{21}^{(1)} \neq 0 | | |
| lyche-numerical-linear-algebra_FO0960 | 76 | 1.000 | a_{11}^{(1)}= | | |
| lyche-numerical-linear-algebra_FO0961 | 76 | 0.997 | a_{21}^{(1)}=0 | | |
| lyche-numerical-linear-algebra_FO0962 | 76 | 0.705 | a_{22}^{(2)} \neq 0 | | |
| lyche-numerical-linear-algebra_FO0963 | 76 | 0.705 | l_{32}^{(2)}:=a_{32}^{(2)} / a_{22}^{(2)} | | |
| lyche-numerical-linear-algebra_FO0964 | 76 | 0.705 | { }^{\prime} | | |
| lyche-numerical-linear-algebra_FO0965 | 76 | 1.000 | a_{33}^{(3)}=a_{33}^{(2)}-l_{32}^{(2)} a_{23}^{(2)} | | |
| lyche-numerical-linear-algebra_FO0966 | 76 | 1.000 | b_{3}^{(3)}=b_{3}^{(2)}-l_{32}^{(2)} b_{2}^{(2)} | | |
| lyche-numerical-linear-algebra_FO0967 | 76 | 1.000 | a_{22}^{(2)}=0 | | |
| lyche-numerical-linear-algebra_FO0968 | 76 | 1.000 | a_{32}^{(2)} \neq 0 | | |
| lyche-numerical-linear-algebra_FO0969 | 76 | 1.000 | a_{k k}^{(k)} \neq 0, k=1,2 | | |
| lyche-numerical-linear-algebra_FO0970 | 77 | 1.000 | \boldsymbol{A}^{(1)} | | |
| lyche-numerical-linear-algebra_FO0971 | 77 | 1.000 | \boldsymbol{A}^{(k)} \boldsymbol{x}=\boldsymbol{b}^{(k)} | | |
| lyche-numerical-linear-algebra_FO0972 | 77 | 1.000 | k= | | |
| lyche-numerical-linear-algebra_FO0973 | 77 | 1.000 | 1, \ldots, n | | |
| lyche-numerical-linear-algebra_FO0974 | 77 | 1.000 | \boldsymbol{A}^{(1)}=\boldsymbol{A}, \boldsymbol{b}^{(1)}=\boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO0975 | 77 | 1.000 | \boldsymbol{A}^{(k)} | | |
| lyche-numerical-linear-algebra_FO0976 | 77 | 1.000 | k-1 | | |
| lyche-numerical-linear-algebra_FO0977 | 77 | 1.000 | \boldsymbol{A}^{(n)} | | |
| lyche-numerical-linear-algebra_FO0978 | 77 | 1.000 | \boldsymbol{A}^{(n)} \boldsymbol{x}=\boldsymbol{b}^{(n)} | | |
| lyche-numerical-linear-algebra_FO0979 | 77 | 1.000 | \boldsymbol{A}^{(k+1)} | | |
| lyche-numerical-linear-algebra_FO0980 | 78 | 1.000 | j=k | | |
| lyche-numerical-linear-algebra_FO0981 | 78 | 1.000 | a_{i k}^{(k+1)}=a_{i k}^{(k)}-\frac{a_{i k}^{(k)}}{a_{k k}^{(k)}} a_{k k}^{(k)}=0 | | |
| lyche-numerical-linear-algebra_FO0982 | 78 | 1.000 | k+1, \ldots, n | | |
| lyche-numerical-linear-algebra_FO0983 | 78 | 1.000 | l_{i k}^{(k)} | | |
| lyche-numerical-linear-algebra_FO0984 | 78 | 1.000 | a_{k k}^{(k)} | | |
| lyche-numerical-linear-algebra_FO0985 | 78 | 1.000 | \boldsymbol{A}_{[k]} \in \mathbb{C}^{k \times k} | | |
| lyche-numerical-linear-algebra_FO0986 | 78 | 0.997 | \boldsymbol{B} \in \mathbb{C}^{k \times k} | | |
| lyche-numerical-linear-algebra_FO0987 | 78 | 0.997 | \boldsymbol{B}=\boldsymbol{A}(\boldsymbol{r}, \boldsymbol{r}) | | |
| lyche-numerical-linear-algebra_FO0988 | 78 | 1.000 | \boldsymbol{r}=\left[r_{1}, \ldots, r_{k}\right] | | |
| lyche-numerical-linear-algebra_FO0989 | 78 | 1.000 | 1 \leq r_{1}<\cdots<r_{k} \leq n | | |
| lyche-numerical-linear-algebra_FO0990 | 78 | 1.000 | r_{j}=j | | |
| lyche-numerical-linear-algebra_FO0991 | 78 | 1.000 | j=1, \ldots, k | | |
| lyche-numerical-linear-algebra_FO0992 | 78 | 1.000 | \boldsymbol{A}=\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO0993 | 78 | 1.000 | a_{k, k}^{(k)} \neq 0 | | |
| lyche-numerical-linear-algebra_FO0994 | 78 | 1.000 | \boldsymbol{A}_{[k]} | | |
| lyche-numerical-linear-algebra_FO0995 | 79 | 1.000 | \boldsymbol{B}_{k}=\boldsymbol{A}_{k-1}^{(k)} | | |
| lyche-numerical-linear-algebra_FO0996 | 79 | 1.000 | \boldsymbol{B}_{k} | | |
| lyche-numerical-linear-algebra_FO0997 | 79 | 1.000 | \boldsymbol{A}_{[k-1]} | | |
| lyche-numerical-linear-algebra_FO0998 | 79 | 1.000 | \boldsymbol{B}_{k+1} | | |
| lyche-numerical-linear-algebra_FO0999 | 79 | 1.000 | a_{11}^{(1)} \cdots a_{k k}^{(k)} \neq 0 | | |
| lyche-numerical-linear-algebra_FO1000 | 79 | 1.000 | 1, \ldots, n-1 | | |
| lyche-numerical-linear-algebra_FO1001 | 79 | 1.000 | \operatorname{det}\left(\boldsymbol{A}_{[k]}\right) \neq 0 | | |
| lyche-numerical-linear-algebra_FO1002 | 79 | 1.000 | l_{i j}^{(j)} | | |
| lyche-numerical-linear-algebra_FO1003 | 79 | 1.000 | a_{i j}^{(i)} | | |
| lyche-numerical-linear-algebra_FO1004 | 79 | 1.000 | i \leq j | | |
| lyche-numerical-linear-algebra_FO1005 | 79 | 1.000 | a_{n n}^{(n)}=0 | | |
| lyche-numerical-linear-algebra_FO1006 | 80 | 1.000 | \boldsymbol{L} \boldsymbol{U} \boldsymbol{x}=\boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO1007 | 80 | 1.000 | \boldsymbol{L} \boldsymbol{y}=\boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO1008 | 80 | 1.000 | \boldsymbol{y}:=\boldsymbol{U} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO1009 | 80 | 1.000 | \boldsymbol{U} \boldsymbol{x}=\boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO1010 | 80 | 0.999 | 2.5 \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO1011 | 80 | 1.000 | x_{1}=b_{1} / a_{11} | | |
| lyche-numerical-linear-algebra_FO1012 | 80 | 1.000 | x_{2} | | |
| lyche-numerical-linear-algebra_FO1013 | 80 | 0.999 | x_{2}=\left(b_{2}-a_{21} x_{1}\right) / a_{22} | | |
| lyche-numerical-linear-algebra_FO1014 | 80 | 0.999 | x_{3}=\left(b_{3}-a_{31} x_{1}-\right. | | |
| lyche-numerical-linear-algebra_FO1015 | 80 | 1.000 | \left.a_{32} x_{2}\right) / a_{33} | | |
| lyche-numerical-linear-algebra_FO1016 | 80 | 1.000 | a_{k, j}=0 | | |
| lyche-numerical-linear-algebra_FO1017 | 80 | 1.000 | j \notin\left\{l_{k}, l_{k}+1, \ldots, k\right. | | |
| lyche-numerical-linear-algebra_FO1018 | 80 | 1.000 | k=1,2, \ldots, n | | |
| lyche-numerical-linear-algebra_FO1019 | 80 | 1.000 | l_{k}:=\max (1, k-d) | | |
| lyche-numerical-linear-algebra_FO1020 | 80 | 1.000 | d=n | | |
| lyche-numerical-linear-algebra_FO1021 | 80 | 1.000 | A\left(k, l_{k}:(k-1)\right) * x\left(l_{k}:(k-1)\right) | | |
| lyche-numerical-linear-algebra_FO1022 | 80 | 0.995 | \sum_{j=l_{k}}^{k-1} a_{k j} x_{j} | | |
| lyche-numerical-linear-algebra_FO1023 | 81 | 1.000 | x_{n} | | |
| lyche-numerical-linear-algebra_FO1024 | 81 | 1.000 | \boldsymbol{A} \in \mathbb{C}^{n \times n}, \boldsymbol{b} \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO1025 | 81 | 1.000 | n-k+1 | | |
| lyche-numerical-linear-algebra_FO1026 | 81 | 1.000 | x_{k}, \ldots, x_{n} | | |
| lyche-numerical-linear-algebra_FO1027 | 82 | 1.000 | x_{k}=b_{k} / a_{k, k} | | |
| lyche-numerical-linear-algebra_FO1028 | 82 | 1.000 | x_{k} | | |
| lyche-numerical-linear-algebra_FO1029 | 82 | 1.000 | n-k | | |
| lyche-numerical-linear-algebra_FO1030 | 82 | 1.000 | x_{k+1}, \ldots, x_{n} | | |
| lyche-numerical-linear-algebra_FO1031 | 82 | 1.000 | k=1,2, \ldots n | | |
| lyche-numerical-linear-algebra_FO1032 | 82 | 1.000 | x_{k}=b_{k} / A(k, k) | | |
| lyche-numerical-linear-algebra_FO1033 | 82 | 1.000 | b | | |
| lyche-numerical-linear-algebra_FO1034 | 82 | 1.000 | M, D, A, S | | |
| lyche-numerical-linear-algebra_FO1035 | 82 | 1.000 | a_{i j}^{k+1}=a_{i j}^{(k)}-l_{i k}^{(k)} a_{k j}^{(k)} | | |
| lyche-numerical-linear-algebra_FO1036 | 82 | 1.000 | (n-k)^{2} | | |
| lyche-numerical-linear-algebra_FO1037 | 83 | 1.000 | M+ | | |
| lyche-numerical-linear-algebra_FO1038 | 83 | 1.000 | S=2 \sum_{k=1}^{n-1}(n-k)^{2}=2 \sum_{m=1}^{n-1} m^{2}=\frac{2}{3} n(n-1)\left(n-\frac{1}{2}\right) | | |
| lyche-numerical-linear-algebra_FO1039 | 83 | 1.000 | \sum_{k=1}^{n-1}(n-k)=\frac{1}{2} n(n-1) | | |
| lyche-numerical-linear-algebra_FO1040 | 83 | 1.000 | N_{L U} | | |
| lyche-numerical-linear-algebra_FO1041 | 83 | 1.000 | \frac{2}{3} n^{3} | | |
| lyche-numerical-linear-algebra_FO1042 | 83 | 1.000 | G_{n} | | |
| lyche-numerical-linear-algebra_FO1043 | 83 | 0.997 | 3.2\left(G_{n}:=\frac{2}{3} n^{3}\right) | | |
| lyche-numerical-linear-algebra_FO1044 | 83 | 0.997 | G_{n}:=\frac{2}{3} n^{3} | | |
| lyche-numerical-linear-algebra_FO1045 | 83 | 1.000 | 2 n^{3} / 3 | | |
| lyche-numerical-linear-algebra_FO1046 | 83 | 1.000 | M+S | | |
| lyche-numerical-linear-algebra_FO1047 | 83 | 1.000 | N_{S} | | |
| lyche-numerical-linear-algebra_FO1048 | 83 | 1.000 | n=10^{6} | | |
| lyche-numerical-linear-algebra_FO1049 | 83 | 1.000 | c=10^{-14} | | |
| lyche-numerical-linear-algebra_FO1050 | 83 | 1.000 | c n^{3}= | | |
| lyche-numerical-linear-algebra_FO1051 | 83 | 1.000 | 10^{4} | | |
| lyche-numerical-linear-algebra_FO1052 | 83 | 1.000 | \approx 3 | | |
| lyche-numerical-linear-algebra_FO1053 | 83 | 1.000 | c n^{2}=0.01 | | |
| lyche-numerical-linear-algebra_FO1054 | 84 | 1.000 | \left[\begin{array}{ll}0 & 1 \\ 1 & 1\end{array}\right]\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]=\left[\begin{array}{l}1 \\ 1\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1055 | 84 | 0.980 | (k, k) | | |
| lyche-numerical-linear-algebra_FO1056 | 84 | 1.000 | r_{k} \geq k | | |
| lyche-numerical-linear-algebra_FO1057 | 84 | 1.000 | a_{r_{k}, k}^{(k)} \neq 0 | | |
| lyche-numerical-linear-algebra_FO1058 | 84 | 1.000 | r_{k} | | |
| lyche-numerical-linear-algebra_FO1059 | 84 | 1.000 | a_{i j}^{(k+1)} | | |
| lyche-numerical-linear-algebra_FO1060 | 84 | 1.000 | \boldsymbol{A}^{(1)}=\boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO1061 | 84 | 1.000 | \boldsymbol{e}_{i_{1}}, \ldots, \boldsymbol{e}_{i_{n}} | | |
| lyche-numerical-linear-algebra_FO1062 | 84 | 1.000 | \boldsymbol{e}_{1}, \ldots, \boldsymbol{e}_{n} \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO1063 | 84 | 1.000 | \boldsymbol{p}=\left[i_{1}, \ldots, i_{n}\right]^{T} | | |
| lyche-numerical-linear-algebra_FO1064 | 85 | 0.997 | \boldsymbol{A} \boldsymbol{P}=\left(\boldsymbol{A} \boldsymbol{e}_{i_{1}}, \ldots, \boldsymbol{A} \boldsymbol{e}_{i_{n}}\right)=\boldsymbol{A}(:, \boldsymbol{p}) | | |
| lyche-numerical-linear-algebra_FO1065 | 85 | 0.997 | \boldsymbol{P}^{T} \boldsymbol{A}=\left(\boldsymbol{A}^{T} \boldsymbol{P}\right)^{T}=\left(\boldsymbol{A}^{T}(\right. | | |
| lyche-numerical-linear-algebra_FO1066 | 85 | 0.773 | , \boldsymbol{p}))^{T}=\boldsymbol{A}(\boldsymbol{p},:) | | |
| lyche-numerical-linear-algebra_FO1067 | 85 | 1.000 | \boldsymbol{P}^{T} \boldsymbol{P}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO1068 | 85 | 1.000 | \boldsymbol{P}^{-1}=\boldsymbol{P}^{T} | | |
| lyche-numerical-linear-algebra_FO1069 | 85 | 1.000 | \boldsymbol{P} \boldsymbol{P}^{T}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO1070 | 85 | 0.997 | (j, k) | | |
| lyche-numerical-linear-algebra_FO1071 | 85 | 0.997 | \boldsymbol{I}_{j k} | | |
| lyche-numerical-linear-algebra_FO1072 | 85 | 0.999 | \boldsymbol{I}_{j k}=\boldsymbol{I}_{k j} | | |
| lyche-numerical-linear-algebra_FO1073 | 85 | 1.000 | \boldsymbol{I}_{j k}^{2}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO1074 | 85 | 1.000 | \boldsymbol{p}_{k} | | |
| lyche-numerical-linear-algebra_FO1075 | 85 | 1.000 | \boldsymbol{p}_{k+1} | | |
| lyche-numerical-linear-algebra_FO1076 | 85 | 0.686 | \boldsymbol{P}_{k}:=\boldsymbol{I}_{r_{k}, k} | | |
| lyche-numerical-linear-algebra_FO1077 | 85 | 0.686 | \boldsymbol{P}_{k} \boldsymbol{I}\left(\boldsymbol{p}_{k},:\right)=\boldsymbol{I}\left(\boldsymbol{P}_{k} \boldsymbol{p}_{k},:\right)=\boldsymbol{I}\left(\boldsymbol{p}_{k+1},:\right) | | |
| lyche-numerical-linear-algebra_FO1078 | 85 | 1.000 | a_{i j}^{(1)}=a_{i j} | | |
| lyche-numerical-linear-algebra_FO1079 | 86 | 0.994 | \boldsymbol{A}=\boldsymbol{P} \boldsymbol{L} \boldsymbol{U} | | |
| lyche-numerical-linear-algebra_FO1080 | 86 | 0.976 | \boldsymbol{P}=\boldsymbol{I}(:, \boldsymbol{p}) | | |
| lyche-numerical-linear-algebra_FO1081 | 86 | 0.976 | \boldsymbol{p}=\boldsymbol{I}_{r_{n-1}, n-1} \cdots \boldsymbol{I}_{r_{1}, 1}[1, \ldots, n]^{T} | | |
| lyche-numerical-linear-algebra_FO1082 | 86 | 1.000 | \boldsymbol{P}^{T} \boldsymbol{A}=\boldsymbol{L} \boldsymbol{U} | | |
| lyche-numerical-linear-algebra_FO1083 | 87 | 0.998 | \mathrm{fl}\left(x_{1}\right) | | |
| lyche-numerical-linear-algebra_FO1084 | 87 | 0.998 | \mathrm{fl}\left(x_{2}\right) | | |
| lyche-numerical-linear-algebra_FO1085 | 87 | 1.000 | x_{1} | | |
| lyche-numerical-linear-algebra_FO1086 | 88 | 1.000 | \mathrm{fl}\left(x_{1}\right)=1 | | |
| lyche-numerical-linear-algebra_FO1087 | 88 | 1.000 | \mathrm{fl}\left(x_{2}\right)=2 | | |
| lyche-numerical-linear-algebra_FO1088 | 88 | 1.000 | s_{k} | | |
| lyche-numerical-linear-algebra_FO1089 | 88 | 1.000 | r_{k}, s_{k} | | |
| lyche-numerical-linear-algebra_FO1090 | 88 | 0.967 | \mathbf{L U} | | |
| lyche-numerical-linear-algebra_FO1091 | 88 | 0.967 | \boldsymbol{L} \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO1092 | 88 | 1.000 | \boldsymbol{U} \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO1093 | 88 | 0.963 | n^{2} | | |
| lyche-numerical-linear-algebra_FO1094 | 88 | 1.000 | n^{2}+n | | |
| lyche-numerical-linear-algebra_FO1095 | 89 | 0.701 | \quad l_{i i}=1 | | |
| lyche-numerical-linear-algebra_FO1096 | 89 | 0.992 | \quad u_{i i}=1 | | |
| lyche-numerical-linear-algebra_FO1097 | 89 | 0.998 | \quad \boldsymbol{A}=\boldsymbol{L} \boldsymbol{D} \boldsymbol{U}, l_{i i}=u_{i i}=1 | | |
| lyche-numerical-linear-algebra_FO1098 | 89 | 0.998 | i, \boldsymbol{D}=\operatorname{diag}\left(d_{11}, \ldots, d_{n n}\right) | | |
| lyche-numerical-linear-algebra_FO1099 | 89 | 0.889 | a, b, c, d \in \mathbb{C} | | |
| lyche-numerical-linear-algebra_FO1100 | 89 | 0.855 | \boldsymbol{A}=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1101 | 89 | 0.984 | l_{1} | | |
| lyche-numerical-linear-algebra_FO1102 | 89 | 0.984 | u_{1}, u_{2}, u_{3} | | |
| lyche-numerical-linear-algebra_FO1103 | 89 | 0.999 | a l_{1}=c | | |
| lyche-numerical-linear-algebra_FO1104 | 89 | 1.000 | a \neq 0 | | |
| lyche-numerical-linear-algebra_FO1105 | 89 | 1.000 | a=c=0 | | |
| lyche-numerical-linear-algebra_FO1106 | 89 | 1.000 | a=0, c \neq 0 | | |
| lyche-numerical-linear-algebra_FO1107 | 89 | 1.000 | \boldsymbol{A}_{4} | | |
| lyche-numerical-linear-algebra_FO1108 | 90 | 1.000 | \boldsymbol{A}_{[k]}, \boldsymbol{L}_{[k]}, \boldsymbol{U}_{[k]} | | |
| lyche-numerical-linear-algebra_FO1109 | 90 | 0.986 | \boldsymbol{A}, \boldsymbol{L}, \boldsymbol{U} | | |
| lyche-numerical-linear-algebra_FO1110 | 90 | 0.986 | \boldsymbol{A}_{[k]}=\boldsymbol{L}_{[k]} \boldsymbol{U}_{[k]} | | |
| lyche-numerical-linear-algebra_FO1111 | 90 | 1.000 | \boldsymbol{F}_{k}, \boldsymbol{N}_{k}, \boldsymbol{T}_{k} \in \mathbb{C}^{n-k, n-k} | | |
| lyche-numerical-linear-algebra_FO1112 | 90 | 1.000 | \boldsymbol{L}_{[k]} | | |
| lyche-numerical-linear-algebra_FO1113 | 90 | 1.000 | \boldsymbol{U}_{[k]} | | |
| lyche-numerical-linear-algebra_FO1114 | 90 | 0.597 | 1 \times 1 | | |
| lyche-numerical-linear-algebra_FO1115 | 90 | 1.000 | \left[a_{11}\right]=[1]\left[a_{11}\right] | | |
| lyche-numerical-linear-algebra_FO1116 | 90 | 1.000 | \boldsymbol{A}_{[n-1]} | | |
| lyche-numerical-linear-algebra_FO1117 | 90 | 1.000 | \boldsymbol{A}_{[n-1]}= | | |
| lyche-numerical-linear-algebra_FO1118 | 90 | 0.992 | \boldsymbol{L}_{n-1} \boldsymbol{U}_{n-1} | | |
| lyche-numerical-linear-algebra_FO1119 | 90 | 0.992 | \boldsymbol{A}_{[1]}, \ldots, \boldsymbol{A}_{[n-1]} | | |
| lyche-numerical-linear-algebra_FO1120 | 90 | 1.000 | \boldsymbol{A}_{[n-1]}=\boldsymbol{L}_{n-1} \boldsymbol{U}_{n-1} | | |
| lyche-numerical-linear-algebra_FO1121 | 90 | 1.000 | \boldsymbol{l}_{n}, \boldsymbol{u}_{n} \in \mathbb{C}^{n-1} | | |
| lyche-numerical-linear-algebra_FO1122 | 90 | 1.000 | u_{n n} \in \mathbb{C} | | |
| lyche-numerical-linear-algebra_FO1123 | 90 | 1.000 | \boldsymbol{L}_{n-1} | | |
| lyche-numerical-linear-algebra_FO1124 | 90 | 1.000 | \boldsymbol{U}_{n-1} | | |
| lyche-numerical-linear-algebra_FO1125 | 90 | 1.000 | \boldsymbol{l}_{n}, \boldsymbol{u}_{n} | | |
| lyche-numerical-linear-algebra_FO1126 | 90 | 1.000 | u_{n n} | | |
| lyche-numerical-linear-algebra_FO1127 | 91 | 1.000 | k \leq n-1 | | |
| lyche-numerical-linear-algebra_FO1128 | 91 | 1.000 | \boldsymbol{A}_{[j]} | | |
| lyche-numerical-linear-algebra_FO1129 | 91 | 0.916 | \boldsymbol{j} \leq k-1 | | |
| lyche-numerical-linear-algebra_FO1130 | 91 | 1.000 | \boldsymbol{U}_{[k]}^{T} \boldsymbol{M}_{k}^{T}=\boldsymbol{C}_{k}^{T} | | |
| lyche-numerical-linear-algebra_FO1131 | 91 | 1.000 | \boldsymbol{M}_{k}^{T} | | |
| lyche-numerical-linear-algebra_FO1132 | 91 | 1.000 | \boldsymbol{U}_{[k]}^{T} | | |
| lyche-numerical-linear-algebra_FO1133 | 91 | 1.000 | \boldsymbol{M}_{k} | | |
| lyche-numerical-linear-algebra_FO1134 | 91 | 1.000 | k=n | | |
| lyche-numerical-linear-algebra_FO1135 | 91 | 1.000 | k-d | | |
| lyche-numerical-linear-algebra_FO1136 | 91 | 1.000 | \boldsymbol{r}_{k} | | |
| lyche-numerical-linear-algebra_FO1137 | 91 | 1.000 | \boldsymbol{c}_{k} | | |
| lyche-numerical-linear-algebra_FO1138 | 91 | 1.000 | \boldsymbol{U}_{k-1}^{T} \boldsymbol{l}_{k}=\boldsymbol{r}_{k} | | |
| lyche-numerical-linear-algebra_FO1139 | 91 | 1.000 | \boldsymbol{l}_{k}^{T} | | |
| lyche-numerical-linear-algebra_FO1140 | 91 | 0.699 | \left[\begin{array}{c}\boldsymbol{u}_{k} \\ u_{k}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1141 | 91 | 0.699 | k=2, \ldots, n | | |
| lyche-numerical-linear-algebra_FO1142 | 91 | 1.000 | d \geq 1 | | |
| lyche-numerical-linear-algebra_FO1143 | 91 | 1.000 | O\left(d^{2} n\right) | | |
| lyche-numerical-linear-algebra_FO1144 | 91 | 0.996 | k<n | | |
| lyche-numerical-linear-algebra_FO1145 | 91 | 1.000 | \boldsymbol{A}=\boldsymbol{I} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO1146 | 91 | 1.000 | a_{k k} | | |
| lyche-numerical-linear-algebra_FO1147 | 92 | 1.000 | \boldsymbol{A}_{i i} | | |
| lyche-numerical-linear-algebra_FO1148 | 92 | 0.434 | \mathbf{L} | | |
| lyche-numerical-linear-algebra_FO1149 | 92 | 0.434 | \boldsymbol{U}_{i i} | | |
| lyche-numerical-linear-algebra_FO1150 | 92 | 0.999 | k=1, \ldots, m-1 | | |
| lyche-numerical-linear-algebra_FO1151 | 92 | 1.000 | \boldsymbol{A}_{\{k\}} | | |
| lyche-numerical-linear-algebra_FO1152 | 92 | 0.993 | \boldsymbol{A}_{\{m-1\}} | | |
| lyche-numerical-linear-algebra_FO1153 | 92 | 0.993 | \boldsymbol{A}_{\{m-1\}}= | | |
| lyche-numerical-linear-algebra_FO1154 | 92 | 0.999 | \boldsymbol{L}_{\{m-1\}} \boldsymbol{U}_{\{m-1\}} | | |
| lyche-numerical-linear-algebra_FO1155 | 92 | 0.999 | \boldsymbol{A}_{\{1\}}, \ldots, \boldsymbol{A}_{\{m-1\}} | | |
| lyche-numerical-linear-algebra_FO1156 | 92 | 0.999 | \boldsymbol{L}_{\{m-1\}} | | |
| lyche-numerical-linear-algebra_FO1157 | 92 | 0.523 | \boldsymbol{U}_{\{m-1\}} | | |
| lyche-numerical-linear-algebra_FO1158 | 93 | 0.934 | \boldsymbol{A}_{\{k\}}=\boldsymbol{L}_{\{k\}} \boldsymbol{U}_{\{k\}} | | |
| lyche-numerical-linear-algebra_FO1159 | 93 | 0.934 | k=1, \ldots, m | | |
| lyche-numerical-linear-algebra_FO1160 | 93 | 1.000 | \boldsymbol{U}_{i i}=\tilde{\boldsymbol{L}}_{i i} \tilde{\boldsymbol{U}}_{i i} | | |
| lyche-numerical-linear-algebra_FO1161 | 93 | 1.000 | \boldsymbol{A}=\hat{\boldsymbol{L}} \hat{\boldsymbol{U}} | | |
| lyche-numerical-linear-algebra_FO1162 | 93 | 1.000 | \hat{\boldsymbol{L}}:=\boldsymbol{L} \operatorname{diag}\left(\tilde{\boldsymbol{L}}_{i i}\right) | | |
| lyche-numerical-linear-algebra_FO1163 | 93 | 0.987 | \hat{\boldsymbol{U}}:=\operatorname{diag}\left(\tilde{\boldsymbol{L}}_{i i}^{-1}\right) \boldsymbol{U} | | |
| lyche-numerical-linear-algebra_FO1164 | 93 | 0.997 | \left[\boldsymbol{b}_{1}, \ldots, \boldsymbol{b}_{n}\right] | | |
| lyche-numerical-linear-algebra_FO1165 | 93 | 0.997 | \boldsymbol{b}_{k} | | |
| lyche-numerical-linear-algebra_FO1166 | 93 | 1.000 | \boldsymbol{A} \boldsymbol{b}_{k}=\boldsymbol{e}_{k} | | |
| lyche-numerical-linear-algebra_FO1167 | 93 | 1.000 | b_{k}(k)=1 / a(k, k) | | |
| lyche-numerical-linear-algebra_FO1168 | 94 | 1.000 | k=n, n-1, \ldots, 1 | | |
| lyche-numerical-linear-algebra_FO1169 | 94 | 1.000 | \frac{1}{3} n\left(2 n^{2}+1\right) \approx G_{n} | | |
| lyche-numerical-linear-algebra_FO1170 | 94 | 1.000 | \boldsymbol{B} \in \mathbb{R}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO1171 | 94 | 1.000 | \boldsymbol{B} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO1172 | 94 | 1.000 | \boldsymbol{P}, \boldsymbol{L}, \boldsymbol{U} | | |
| lyche-numerical-linear-algebra_FO1173 | 94 | 1.000 | \boldsymbol{A}^{*} \boldsymbol{x}=\boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO1174 | 94 | 1.000 | 2 G_{n} | | |
| lyche-numerical-linear-algebra_FO1175 | 94 | 1.000 | \boldsymbol{A}^{-1}=\boldsymbol{U}^{-1} \boldsymbol{L}^{-1} \boldsymbol{P}^{T} | | |
| lyche-numerical-linear-algebra_FO1176 | 94 | 1.000 | \boldsymbol{P}^{T} | | |
| lyche-numerical-linear-algebra_FO1177 | 95 | 0.993 | k=1,2, \ldots, n-1 | | |
| lyche-numerical-linear-algebra_FO1178 | 95 | 1.000 | a_{k, j} | | |
| lyche-numerical-linear-algebra_FO1179 | 95 | 1.000 | a_{k+1, j} | | |
| lyche-numerical-linear-algebra_FO1180 | 95 | 1.000 | j=k, k+1, \ldots, n | | |
| lyche-numerical-linear-algebra_FO1181 | 95 | 1.000 | i=k+1, k+2, \ldots, n | | |
| lyche-numerical-linear-algebra_FO1182 | 95 | 1.000 | a_{i, k}=m_{i, k}=a_{i, k} / a_{k, k} | | |
| lyche-numerical-linear-algebra_FO1183 | 95 | 1.000 | a_{i, j}=a_{i, j}-m_{i, k} a_{k, j} | | |
| lyche-numerical-linear-algebra_FO1184 | 95 | 1.000 | j=k+1, k+2, \ldots, n | | |
| lyche-numerical-linear-algebra_FO1185 | 95 | 1.000 | b_{k} | | |
| lyche-numerical-linear-algebra_FO1186 | 95 | 1.000 | b_{k+1} | | |
| lyche-numerical-linear-algebra_FO1187 | 95 | 1.000 | b_{i}=b_{i}-a_{i, k} b_{k} | | |
| lyche-numerical-linear-algebra_FO1188 | 95 | 1.000 | x_{n}=b_{n} / a_{n, n} | | |
| lyche-numerical-linear-algebra_FO1189 | 95 | 0.981 | =0 | | |
| lyche-numerical-linear-algebra_FO1190 | 95 | 0.958 | +a_{k, j} x_{j} | | |
| lyche-numerical-linear-algebra_FO1191 | 95 | 1.000 | x_{k}=\left(b_{k}-\right. | | |
| lyche-numerical-linear-algebra_FO1192 | 95 | 1.000 | ) / a_{k, k} | | |
| lyche-numerical-linear-algebra_FO1193 | 95 | 1.000 | \boldsymbol{H} \in \mathbb{R}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO1194 | 95 | 1.000 | h_{i, i-1} \neq 0 | | |
| lyche-numerical-linear-algebra_FO1195 | 95 | 0.983 | \boldsymbol{H} \boldsymbol{x}=\boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO1196 | 95 | 1.000 | \boldsymbol{U} \in \mathbb{R}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO1197 | 95 | 1.000 | \boldsymbol{v} \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO1198 | 95 | 1.000 | \boldsymbol{e}_{1} | | |
| lyche-numerical-linear-algebra_FO1199 | 95 | 0.999 | \boldsymbol{I}_{i, j} | | |
| lyche-numerical-linear-algebra_FO1200 | 95 | 0.822 | \boldsymbol{E}:=\boldsymbol{C} \boldsymbol{P} | | |
| lyche-numerical-linear-algebra_FO1201 | 96 | 1.000 | \boldsymbol{W} \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO1202 | 96 | 1.000 | \boldsymbol{B}=\boldsymbol{L} \boldsymbol{H} \boldsymbol{P}^{T} | | |
| lyche-numerical-linear-algebra_FO1203 | 96 | 1.000 | \boldsymbol{H} | | |
| lyche-numerical-linear-algebra_FO1204 | 96 | 1.000 | \boldsymbol{B} \boldsymbol{x}=\boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO1205 | 96 | 0.999 | 1 \leq | | |
| lyche-numerical-linear-algebra_FO1206 | 96 | 1.000 | d \leq n | | |
| lyche-numerical-linear-algebra_FO1207 | 96 | 1.000 | A(k, k)=1 | | |
| lyche-numerical-linear-algebra_FO1208 | 96 | 1.000 | N_{L U}(n, d):=\left(2 d^{2}+d\right) n-\left(d^{2}+d\right)(8 d+1) / 6=O\left(d^{2} n\right) | | |
| lyche-numerical-linear-algebra_FO1209 | 96 | 1.000 | d=n-1 | | |
| lyche-numerical-linear-algebra_FO1210 | 96 | 1.000 | N_{L U}(n, n)=\frac{2}{3} n^{3}-\frac{1}{2} n^{2}-\frac{1}{6} n \approx | | |
| lyche-numerical-linear-algebra_FO1211 | 96 | 1.000 | N_{L U}(n, 1)=3 n-3=O(n) | | |
| lyche-numerical-linear-algebra_FO1212 | 96 | 0.985 | \boldsymbol{A}=\left[\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]\right. | | |
| lyche-numerical-linear-algebra_FO1213 | 96 | 0.999 | 2 \leq k \leq d | | |
| lyche-numerical-linear-algebra_FO1214 | 96 | 0.999 | d+1 \leq k \leq n | | |
| lyche-numerical-linear-algebra_FO1215 | 97 | 1.000 | u_{k k} | | |
| lyche-numerical-linear-algebra_FO1216 | 97 | 1.000 | \boldsymbol{L}_{2,2}, \boldsymbol{U}_{2,2} \in \mathbb{R}^{(n-1) \times(n-1)} | | |
| lyche-numerical-linear-algebra_FO1217 | 97 | 1.000 | \boldsymbol{A}_{2,2}:=\boldsymbol{L}_{2,2} \boldsymbol{U}_{2,2} | | |
| lyche-numerical-linear-algebra_FO1218 | 97 | 1.000 | \boldsymbol{B}_{2,2}:=\boldsymbol{A}_{2,2}^{-1} | | |
| lyche-numerical-linear-algebra_FO1219 | 97 | 1.000 | l_{i, j}, i>j | | |
| lyche-numerical-linear-algebra_FO1220 | 97 | 1.000 | u_{i, j}, j \geq i | | |
| lyche-numerical-linear-algebra_FO1221 | 97 | 1.000 | a_{i, j} | | |
| lyche-numerical-linear-algebra_FO1222 | 97 | 1.000 | \boldsymbol{s} \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO1223 | 97 | 1.000 | \boldsymbol{T} \boldsymbol{H} \boldsymbol{x}=\boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO1224 | 97 | 1.000 | \boldsymbol{T}, \boldsymbol{H}, \boldsymbol{S} \in \mathbb{R}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO1225 | 97 | 1.000 | \boldsymbol{T}=\left(t_{i j}\right) | | |
| lyche-numerical-linear-algebra_FO1226 | 97 | 1.000 | \boldsymbol{H}=\left(h_{i j}\right) | | |
| lyche-numerical-linear-algebra_FO1227 | 97 | 1.000 | \boldsymbol{S}:=\boldsymbol{T} \boldsymbol{H} | | |
| lyche-numerical-linear-algebra_FO1228 | 97 | 1.000 | \boldsymbol{H}=\boldsymbol{L} \boldsymbol{U} | | |
| lyche-numerical-linear-algebra_FO1229 | 97 | 1.000 | \|\boldsymbol{L}\|_{\infty}\|\boldsymbol{U}\|_{\infty} \leq K\|\boldsymbol{H}\|_{\infty} | | |
| lyche-numerical-linear-algebra_FO1230 | 97 | 1.000 | \boldsymbol{S} | | |
| lyche-numerical-linear-algebra_FO1231 | 97 | 0.999 | \left(t_{i j}, i>j\right) | | |
| lyche-numerical-linear-algebra_FO1232 | 97 | 0.999 | \left(h_{i j}, i>j+1\right) | | |
| lyche-numerical-linear-algebra_FO1233 | 98 | 0.987 | \boldsymbol{T}, \boldsymbol{H} | | |
| lyche-numerical-linear-algebra_FO1234 | 98 | 1.000 | \boldsymbol{S} \boldsymbol{x}=\boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO1235 | 98 | 0.779 | \boldsymbol{S}=\boldsymbol{T} \boldsymbol{H} | | |
| lyche-numerical-linear-algebra_FO1236 | 98 | 0.991 | \boldsymbol{T} \boldsymbol{z}=\boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO1237 | 98 | 0.894 | \boldsymbol{H} \boldsymbol{x}=\boldsymbol{z} | | |
| lyche-numerical-linear-algebra_FO1238 | 98 | 1.000 | \boldsymbol{a}_{1}, \boldsymbol{a}_{2}, \ldots, \boldsymbol{a}_{n} | | |
| lyche-numerical-linear-algebra_FO1239 | 98 | 1.000 | p \leq n | | |
| lyche-numerical-linear-algebra_FO1240 | 98 | 1.000 | \boldsymbol{H}:=\boldsymbol{L}^{-1} \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO1241 | 98 | 1.000 | \boldsymbol{H}=\boldsymbol{L}_{H} \boldsymbol{U}_{H} | | |
| lyche-numerical-linear-algebra_FO1242 | 98 | 1.000 | \boldsymbol{H}:=\boldsymbol{L}_{H} \boldsymbol{U}_{H} | | |
| lyche-numerical-linear-algebra_FO1243 | 98 | 1.000 | \boldsymbol{L}_{H} | | |
| lyche-numerical-linear-algebra_FO1244 | 98 | 1.000 | \boldsymbol{U}_{H} | | |
| lyche-numerical-linear-algebra_FO1245 | 98 | 0.999 | \boldsymbol{A}=\boldsymbol{U} \boldsymbol{L} | | |
| lyche-numerical-linear-algebra_FO1246 | 98 | 1.000 | \boldsymbol{P} \in \mathbb{R}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO1247 | 99 | 1.000 | \boldsymbol{P}^{T}=\boldsymbol{P} | | |
| lyche-numerical-linear-algebra_FO1248 | 99 | 1.000 | \boldsymbol{P}^{2}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO1249 | 99 | 1.000 | \boldsymbol{P} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO1250 | 99 | 1.000 | \boldsymbol{B}:=\boldsymbol{P} \boldsymbol{A} \boldsymbol{P} | | |
| lyche-numerical-linear-algebra_FO1251 | 99 | 1.000 | r, s | | |
| lyche-numerical-linear-algebra_FO1252 | 99 | 1.000 | i, j, n | | |
| lyche-numerical-linear-algebra_FO1253 | 99 | 1.000 | b_{i, j}=a_{r, s} | | |
| lyche-numerical-linear-algebra_FO1254 | 99 | 1.000 | b_{i, j} | | |
| lyche-numerical-linear-algebra_FO1255 | 99 | 1.000 | w \in \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO1256 | 99 | 1.000 | \boldsymbol{P} \boldsymbol{A} \boldsymbol{P}=\boldsymbol{M} \boldsymbol{R} | | |
| lyche-numerical-linear-algebra_FO1257 | 99 | 1.000 | \boldsymbol{P} \boldsymbol{A} \boldsymbol{P} | | |
| lyche-numerical-linear-algebra_FO1258 | 99 | 1.000 | \boldsymbol{M} | | |
| lyche-numerical-linear-algebra_FO1259 | 99 | 1.000 | \boldsymbol{M}, \boldsymbol{R} | | |
| lyche-numerical-linear-algebra_FO1260 | 99 | 0.504 | \hat{\boldsymbol{L}} | | |
| lyche-numerical-linear-algebra_FO1261 | 99 | 1.000 | \hat{\boldsymbol{U}} | | |
| lyche-numerical-linear-algebra_FO1262 | 100 | 0.997 | a_{j i}=\bar{a}_{i j} | | |
| lyche-numerical-linear-algebra_FO1263 | 100 | 1.000 | a_{i i}=\bar{a}_{i i} | | |
| lyche-numerical-linear-algebra_FO1264 | 100 | 1.000 | \boldsymbol{U}=\boldsymbol{L}^{*} | | |
| lyche-numerical-linear-algebra_FO1265 | 100 | 1.000 | l_{i i}>0 | | |
| lyche-numerical-linear-algebra_FO1266 | 100 | 0.999 | \boldsymbol{A}^{*}=\left(\boldsymbol{L} \boldsymbol{D} \boldsymbol{L}^{*}\right)^{*}=\boldsymbol{L} \boldsymbol{D}^{*} \boldsymbol{L}^{*}=\boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO1267 | 100 | 0.966 | b \in \mathbb{C} | | |
| lyche-numerical-linear-algebra_FO1268 | 101 | 1.000 | d_{1}, d_{2} | | |
| lyche-numerical-linear-algebra_FO1269 | 101 | 1.000 | d_{1} | | |
| lyche-numerical-linear-algebra_FO1270 | 101 | 1.000 | d_{2} | | |
| lyche-numerical-linear-algebra_FO1271 | 101 | 1.000 | a=b=0 | | |
| lyche-numerical-linear-algebra_FO1272 | 101 | 1.000 | a=0, b \neq 0 | | |
| lyche-numerical-linear-algebra_FO1273 | 101 | 0.830 | \boldsymbol{A}=\boldsymbol{L} \boldsymbol{D L}^{*} | | |
| lyche-numerical-linear-algebra_FO1274 | 101 | 0.954 | L D L^{*} | | |
| lyche-numerical-linear-algebra_FO1275 | 101 | 0.954 | \boldsymbol{A}_{[k]}, \boldsymbol{L}_{[k]} | | |
| lyche-numerical-linear-algebra_FO1276 | 101 | 0.954 | \boldsymbol{D}_{[k]} | | |
| lyche-numerical-linear-algebra_FO1277 | 101 | 1.000 | \boldsymbol{A}, \boldsymbol{L} | | |
| lyche-numerical-linear-algebra_FO1278 | 101 | 1.000 | \boldsymbol{A}_{[k]}= | | |
| lyche-numerical-linear-algebra_FO1279 | 101 | 1.000 | \boldsymbol{L}_{[k]} \boldsymbol{D}_{[k]} \boldsymbol{L}_{[k]}^{*} | | |
| lyche-numerical-linear-algebra_FO1280 | 101 | 1.000 | \boldsymbol{A}=\boldsymbol{L} \boldsymbol{D} \boldsymbol{L}^{*} | | |
| lyche-numerical-linear-algebra_FO1281 | 101 | 1.000 | \boldsymbol{F}_{k}, \boldsymbol{N}_{k}, \boldsymbol{E}_{k} \in \mathbb{C}^{n-k, n-k} | | |
| lyche-numerical-linear-algebra_FO1282 | 101 | 1.000 | \boldsymbol{A}_{[k]}=\boldsymbol{L}_{[k]} \boldsymbol{D}_{[k]} \boldsymbol{L}_{[k]}^{*} | | |
| lyche-numerical-linear-algebra_FO1283 | 101 | 1.000 | \boldsymbol{A}=\boldsymbol{A}^{*} | | |
| lyche-numerical-linear-algebra_FO1284 | 101 | 1.000 | \boldsymbol{A}_{[k]}^{*}=\boldsymbol{A}_{[k]} | | |
| lyche-numerical-linear-algebra_FO1285 | 101 | 0.983 | \left[a_{11}\right]=[1]\left[a_{11}\right][1] | | |
| lyche-numerical-linear-algebra_FO1286 | 101 | 0.997 | \boldsymbol{L}_{n-1} \boldsymbol{D}_{n-1} \boldsymbol{L}_{n-1}^{*} | | |
| lyche-numerical-linear-algebra_FO1287 | 101 | 0.997 | \boldsymbol{D}_{n-1} | | |
| lyche-numerical-linear-algebra_FO1288 | 102 | 1.000 | \boldsymbol{A}_{[n-1]}=\boldsymbol{L}_{n-1} \boldsymbol{D}_{n-1} \boldsymbol{L}_{n-1}^{*} | | |
| lyche-numerical-linear-algebra_FO1289 | 102 | 1.000 | d_{n n} | | |
| lyche-numerical-linear-algebra_FO1290 | 102 | 1.000 | a_{n n} | | |
| lyche-numerical-linear-algebra_FO1291 | 102 | 1.000 | \frac{1}{2} G_{n} | | |
| lyche-numerical-linear-algebra_FO1292 | 102 | 1.000 | \boldsymbol{m} | | |
| lyche-numerical-linear-algebra_FO1293 | 102 | 1.000 | f: \mathbb{C}^{n} \rightarrow \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO1294 | 103 | 0.998 | \overline{f(\boldsymbol{x})}=\overline{\boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{x}}=\left(\boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{x}\right)^{*}=\boldsymbol{x}^{*} \boldsymbol{A}^{*} \boldsymbol{x}=f(\boldsymbol{x}) | | |
| lyche-numerical-linear-algebra_FO1295 | 103 | 1.000 | \boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{x}>0 | | |
| lyche-numerical-linear-algebra_FO1296 | 103 | 1.000 | \boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{x} \geq 0 | | |
| lyche-numerical-linear-algebra_FO1297 | 103 | 1.000 | -\boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO1298 | 103 | 1.000 | \boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{x}=0 \Longrightarrow \boldsymbol{x}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO1299 | 103 | 0.999 | \boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{x}=0 | | |
| lyche-numerical-linear-algebra_FO1300 | 103 | 1.000 | \boldsymbol{A} \in \mathbb{R}^{n \times n}, \boldsymbol{A}^{T}=\boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO1301 | 103 | 1.000 | \boldsymbol{x}^{T} \boldsymbol{A} \boldsymbol{x}>0 | | |
| lyche-numerical-linear-algebra_FO1302 | 103 | 1.000 | \boldsymbol{z}^{*} \boldsymbol{A} \boldsymbol{z}>0 | | |
| lyche-numerical-linear-algebra_FO1303 | 103 | 0.999 | \boldsymbol{z} \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO1304 | 103 | 0.999 | \boldsymbol{z}=\boldsymbol{x}+i \boldsymbol{y} \neq \mathbf{0} | | |
| lyche-numerical-linear-algebra_FO1305 | 103 | 0.991 | \boldsymbol{x}, \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO1306 | 103 | 1.000 | \nabla f | | |
| lyche-numerical-linear-algebra_FO1307 | 103 | 0.658 | H f | | |
| lyche-numerical-linear-algebra_FO1308 | 103 | 0.658 | f: \Omega \subset \mathbb{R}^{n} \rightarrow \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO1309 | 103 | 1.000 | \Omega | | |
| lyche-numerical-linear-algebra_FO1310 | 103 | 1.000 | \nabla f(\boldsymbol{x})=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO1311 | 103 | 1.000 | H f(\boldsymbol{x}) | | |
| lyche-numerical-linear-algebra_FO1312 | 103 | 0.933 | \boldsymbol{A}^{*} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO1313 | 103 | 1.000 | m, n \in \mathbb{N} | | |
| lyche-numerical-linear-algebra_FO1314 | 104 | 0.986 | \boldsymbol{z}:=\boldsymbol{A} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO1315 | 104 | 1.000 | \boldsymbol{x}^{*} \boldsymbol{A}^{*} \boldsymbol{A} \boldsymbol{x}=\boldsymbol{z}^{*} \boldsymbol{z}=\|\boldsymbol{z}\|_{2}^{2}=\|\boldsymbol{A} \boldsymbol{x}\|_{2}^{2} \geq 0 | | |
| lyche-numerical-linear-algebra_FO1316 | 104 | 0.515 | \boldsymbol{T}= | | |
| lyche-numerical-linear-algebra_FO1317 | 104 | 0.983 | \operatorname{tridiag}(-1,2,-1) \in \mathbb{R}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO1318 | 104 | 0.994 | \boldsymbol{x}^{T} \boldsymbol{T} \boldsymbol{x} \geq 0 | | |
| lyche-numerical-linear-algebra_FO1319 | 104 | 0.994 | \boldsymbol{x}^{T} \boldsymbol{T} \boldsymbol{x}=0 | | |
| lyche-numerical-linear-algebra_FO1320 | 104 | 0.994 | x_{1}=x_{n}=0 | | |
| lyche-numerical-linear-algebra_FO1321 | 104 | 0.994 | x_{i}=x_{i+1} | | |
| lyche-numerical-linear-algebra_FO1322 | 104 | 0.999 | \boldsymbol{x}=0 | | |
| lyche-numerical-linear-algebra_FO1323 | 104 | 1.000 | \boldsymbol{B}=\boldsymbol{A}(\boldsymbol{r}, \boldsymbol{r}) \in \mathbb{C}^{k \times k} | | |
| lyche-numerical-linear-algebra_FO1324 | 104 | 1.000 | b_{i, j}=a_{r_{i}, r_{j}} | | |
| lyche-numerical-linear-algebra_FO1325 | 104 | 1.000 | i, j=1, \ldots, k | | |
| lyche-numerical-linear-algebra_FO1326 | 104 | 1.000 | \boldsymbol{r}=[1,2, \ldots, k]^{T} | | |
| lyche-numerical-linear-algebra_FO1327 | 104 | 1.000 | \boldsymbol{B}:=\boldsymbol{A}(\boldsymbol{r}, \boldsymbol{r}) | | |
| lyche-numerical-linear-algebra_FO1328 | 104 | 1.000 | \boldsymbol{y}^{*} \boldsymbol{B} \boldsymbol{y}=0 | | |
| lyche-numerical-linear-algebra_FO1329 | 105 | 1.000 | \boldsymbol{A}^{-*}:=\left(\boldsymbol{A}^{-1}\right)^{*}=\left(\boldsymbol{A}^{*}\right)^{-1} | | |
| lyche-numerical-linear-algebra_FO1330 | 105 | 1.000 | 1 \Longrightarrow 2 \Longrightarrow 3 \Longrightarrow 1 | | |
| lyche-numerical-linear-algebra_FO1331 | 105 | 0.999 | 1 \Longrightarrow | | |
| lyche-numerical-linear-algebra_FO1332 | 105 | 1.000 | \boldsymbol{x}_{i}:=\boldsymbol{L}^{-*} \boldsymbol{e}_{i} | | |
| lyche-numerical-linear-algebra_FO1333 | 105 | 1.000 | \boldsymbol{L}^{-*} | | |
| lyche-numerical-linear-algebra_FO1334 | 105 | 1.000 | d_{i i}=\boldsymbol{e}_{i}^{*} \boldsymbol{D} \boldsymbol{e}_{i}=\boldsymbol{e}_{i}^{*} \boldsymbol{L}^{-1} \boldsymbol{A} \boldsymbol{L}^{-*} \boldsymbol{e}_{i}=\boldsymbol{x}_{i}^{*} \boldsymbol{A} \boldsymbol{x}_{i}>0 | | |
| lyche-numerical-linear-algebra_FO1335 | 105 | 0.913 | 2 \Longrightarrow | | |
| lyche-numerical-linear-algebra_FO1336 | 105 | 0.913 | \mathrm{LDL}^{*} | | |
| lyche-numerical-linear-algebra_FO1337 | 105 | 1.000 | d_{i i} | | |
| lyche-numerical-linear-algebra_FO1338 | 105 | 1.000 | \boldsymbol{A}=\boldsymbol{S} \boldsymbol{S}^{*} | | |
| lyche-numerical-linear-algebra_FO1339 | 105 | 1.000 | \boldsymbol{S}:=\boldsymbol{L} \boldsymbol{D}^{1 / 2} | | |
| lyche-numerical-linear-algebra_FO1340 | 105 | 1.000 | \boldsymbol{D}^{1 / 2}:= | | |
| lyche-numerical-linear-algebra_FO1341 | 105 | 1.000 | \operatorname{diag}\left(\sqrt{d_{11}}, \ldots, \sqrt{d_{n n}}\right) | | |
| lyche-numerical-linear-algebra_FO1342 | 105 | 1.000 | 3 \Longrightarrow | | |
| lyche-numerical-linear-algebra_FO1343 | 105 | 1.000 | \boldsymbol{A}=\boldsymbol{L} \boldsymbol{L}^{*} | | |
| lyche-numerical-linear-algebra_FO1344 | 105 | 1.000 | \boldsymbol{L} \boldsymbol{L}^{*}=\boldsymbol{S} \boldsymbol{S}^{*} | | |
| lyche-numerical-linear-algebra_FO1345 | 105 | 1.000 | \boldsymbol{S}^{-1} \boldsymbol{L}=\boldsymbol{S}^{*} \boldsymbol{L}^{-*} | | |
| lyche-numerical-linear-algebra_FO1346 | 105 | 1.000 | 2.5 \boldsymbol{S}^{-1} \boldsymbol{L} | | |
| lyche-numerical-linear-algebra_FO1347 | 105 | 1.000 | \boldsymbol{S}^{*} \boldsymbol{L}^{-*} | | |
| lyche-numerical-linear-algebra_FO1348 | 105 | 1.000 | \ell_{i i} / s_{i i} | | |
| lyche-numerical-linear-algebra_FO1349 | 105 | 1.000 | s_{i i} / \ell_{i i} | | |
| lyche-numerical-linear-algebra_FO1350 | 105 | 1.000 | \ell_{i i}^{2}=s_{i i}^{2} | | |
| lyche-numerical-linear-algebra_FO1351 | 105 | 1.000 | \ell_{i i}=s_{i i} | | |
| lyche-numerical-linear-algebra_FO1352 | 105 | 1.000 | \boldsymbol{S}^{-1} \boldsymbol{L}=\boldsymbol{I}=\boldsymbol{S}^{*} \boldsymbol{L}^{-*} | | |
| lyche-numerical-linear-algebra_FO1353 | 105 | 1.000 | \boldsymbol{L}=\boldsymbol{S} | | |
| lyche-numerical-linear-algebra_FO1354 | 105 | 0.993 | \boldsymbol{A}=\boldsymbol{R}^{*} \boldsymbol{R} | | |
| lyche-numerical-linear-algebra_FO1355 | 105 | 1.000 | \boldsymbol{R}=\boldsymbol{L}^{*} | | |
| lyche-numerical-linear-algebra_FO1356 | 105 | 0.993 | 4.4(2 \times 2) | | |
| lyche-numerical-linear-algebra_FO1357 | 105 | 0.993 | \boldsymbol{A}=\left[\begin{array}{cc}2 & -1 \\ -1 & 2\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1358 | 106 | 1.000 | \frac{1}{2} G_{n}=n^{3} / 3 | | |
| lyche-numerical-linear-algebra_FO1359 | 106 | 1.000 | a_{i i}>0,\left(a_{i i} \geq 0\right) | | |
| lyche-numerical-linear-algebra_FO1360 | 106 | 1.000 | \left|\operatorname{Re}\left(a_{i j}\right)\right|<\left(a_{i i}+a_{j j}\right) / 2,\left(\left|\operatorname{Re}\left(a_{i j}\right)\right| \leq\left(a_{i i}+a_{j j}\right) / 2\right) | | |
| lyche-numerical-linear-algebra_FO1361 | 106 | 1.000 | \left|a_{i j}\right|<\sqrt{a_{i i} a_{j j}},\left(\left|a_{i j}\right| \leq \sqrt{a_{i i} a_{j j}}\right) | | |
| lyche-numerical-linear-algebra_FO1362 | 106 | 1.000 | a_{i i}=0 | | |
| lyche-numerical-linear-algebra_FO1363 | 106 | 1.000 | a_{i j}=a_{j i}=0 | | |
| lyche-numerical-linear-algebra_FO1364 | 106 | 1.000 | a_{i i}=\boldsymbol{e}_{i}^{T} \boldsymbol{A} \boldsymbol{e}_{i}>(\geq) 0 | | |
| lyche-numerical-linear-algebra_FO1365 | 106 | 1.000 | \alpha, \beta \in \mathbb{C} | | |
| lyche-numerical-linear-algebra_FO1366 | 106 | 1.000 | \alpha \boldsymbol{e}_{i}+\beta \boldsymbol{e}_{j} \neq | | |
| lyche-numerical-linear-algebra_FO1367 | 106 | 0.999 | \alpha=1, \beta= \pm 1 | | |
| lyche-numerical-linear-algebra_FO1368 | 106 | 0.999 | a_{i i}+a_{j j} \pm 2 \operatorname{Re} a_{i j}>0 | | |
| lyche-numerical-linear-algebra_FO1369 | 106 | 0.834 | \alpha=-a_{i j}, \beta=a_{i i} | | |
| lyche-numerical-linear-algebra_FO1370 | 106 | 1.000 | a_{i i}>0 | | |
| lyche-numerical-linear-algebra_FO1371 | 106 | 1.000 | \varepsilon>0 | | |
| lyche-numerical-linear-algebra_FO1372 | 106 | 1.000 | \boldsymbol{B}:=\boldsymbol{A}+\varepsilon \boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO1373 | 106 | 1.000 | \boldsymbol{x}^{*} \boldsymbol{B} \boldsymbol{x} \geq \varepsilon\|\boldsymbol{x}\|_{2}^{2}>0 | | |
| lyche-numerical-linear-algebra_FO1374 | 107 | 1.000 | \left[\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1375 | 107 | 1.000 | \boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{x}=\lambda \boldsymbol{x}^{*} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO1376 | 107 | 1.000 | \lambda>0(\geq 0) | | |
| lyche-numerical-linear-algebra_FO1377 | 107 | 1.000 | \boldsymbol{x}^{*} \boldsymbol{x}=\|\boldsymbol{x}\|_{2}^{2}>0 | | |
| lyche-numerical-linear-algebra_FO1378 | 107 | 1.000 | \boldsymbol{U}^{*} \boldsymbol{U}= | | |
| lyche-numerical-linear-algebra_FO1379 | 107 | 1.000 | \boldsymbol{U} \boldsymbol{U}^{*}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO1380 | 107 | 1.000 | \boldsymbol{U}^{*} \boldsymbol{A} \boldsymbol{U}=\operatorname{diag}\left(\lambda_{1}, \ldots, \lambda_{n}\right) | | |
| lyche-numerical-linear-algebra_FO1381 | 107 | 1.000 | \boldsymbol{z}:= | | |
| lyche-numerical-linear-algebra_FO1382 | 107 | 1.000 | \boldsymbol{U}^{*} \boldsymbol{x}=\left[z_{1}, \ldots, z_{n}\right]^{T} \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO1383 | 107 | 1.000 | \boldsymbol{x}=\boldsymbol{U} \boldsymbol{U}^{*} \boldsymbol{x}=\boldsymbol{U} \boldsymbol{z} | | |
| lyche-numerical-linear-algebra_FO1384 | 107 | 0.982 | \boldsymbol{U}^{*} | | |
| lyche-numerical-linear-algebra_FO1385 | 107 | 0.982 | \boldsymbol{z}= | | |
| lyche-numerical-linear-algebra_FO1386 | 107 | 1.000 | \boldsymbol{U}^{*} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO1387 | 108 | 1.000 | \boldsymbol{A}=\boldsymbol{B} \boldsymbol{B}^{*} | | |
| lyche-numerical-linear-algebra_FO1388 | 108 | 0.793 | \boldsymbol{B}=\boldsymbol{L} | | |
| lyche-numerical-linear-algebra_FO1389 | 108 | 0.973 | 4 \Longrightarrow | | |
| lyche-numerical-linear-algebra_FO1390 | 108 | 1.000 | \boldsymbol{A}:=\left[\begin{array}{ll}3 & 1 \\ 1 & 3\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1391 | 108 | 0.991 | \boldsymbol{x}^{T} \boldsymbol{A} \boldsymbol{x}=2 x_{1}^{2}+2 x_{2}^{2}+\left(x_{1}+x_{2}\right)^{2}>0 | | |
| lyche-numerical-linear-algebra_FO1392 | 108 | 1.000 | \lambda_{1}=2 | | |
| lyche-numerical-linear-algebra_FO1393 | 108 | 1.000 | \lambda_{2}=4 | | |
| lyche-numerical-linear-algebra_FO1394 | 108 | 0.998 | \operatorname{det}\left(\boldsymbol{A}_{[1]}\right)=3 | | |
| lyche-numerical-linear-algebra_FO1395 | 108 | 0.998 | \operatorname{det}\left(\boldsymbol{A}_{[2]}\right)=8 | | |
| lyche-numerical-linear-algebra_FO1396 | 109 | 1.000 | \boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{x}=\left\|\boldsymbol{L}^{*} \boldsymbol{x}\right\|_{2}^{2} \geq 0 | | |
| lyche-numerical-linear-algebra_FO1397 | 109 | 1.000 | \alpha=\boldsymbol{e}_{1}^{*} \boldsymbol{A} \boldsymbol{e}_{1}>0 | | |
| lyche-numerical-linear-algebra_FO1398 | 109 | 1.000 | \boldsymbol{C}:=\boldsymbol{B}- | | |
| lyche-numerical-linear-algebra_FO1399 | 109 | 1.000 | \boldsymbol{v} \boldsymbol{v}^{*} / \alpha | | |
| lyche-numerical-linear-algebra_FO1400 | 109 | 1.000 | \boldsymbol{y} \in \mathbb{C}^{n-1} | | |
| lyche-numerical-linear-algebra_FO1401 | 109 | 1.000 | \boldsymbol{x}^{*}:=\left[-\boldsymbol{y}^{*} \boldsymbol{v} / \alpha, \boldsymbol{y}^{*}\right] \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO1402 | 109 | 1.000 | \boldsymbol{C} \in \mathbb{C}^{(n-1) \times(n-1)} | | |
| lyche-numerical-linear-algebra_FO1403 | 109 | 1.000 | \boldsymbol{C}=\boldsymbol{L}_{1} \boldsymbol{L}_{1}^{*} | | |
| lyche-numerical-linear-algebra_FO1404 | 109 | 1.000 | \alpha=0 | | |
| lyche-numerical-linear-algebra_FO1405 | 109 | 1.000 | \boldsymbol{v}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO1406 | 109 | 1.000 | \boldsymbol{B} \in | | |
| lyche-numerical-linear-algebra_FO1407 | 109 | 1.000 | \mathbb{C}^{(n-1) \times(n-1)} | | |
| lyche-numerical-linear-algebra_FO1408 | 110 | 0.554 | \boldsymbol{B}=\boldsymbol{L}_{1} \boldsymbol{L}_{1}^{*} | | |
| lyche-numerical-linear-algebra_FO1409 | 110 | 0.554 | \boldsymbol{L}^{*} | | |
| lyche-numerical-linear-algebra_FO1410 | 110 | 0.554 | \boldsymbol{L}=\left[\begin{array}{cc}0 & \mathbf{0}^{*} \\ \mathbf{0} & \boldsymbol{L}_{1}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1411 | 110 | 1.000 | \boldsymbol{v}^{*}=\left[\boldsymbol{u}^{*}, \mathbf{0}^{*}\right] | | |
| lyche-numerical-linear-algebra_FO1412 | 110 | 1.000 | \boldsymbol{u} \in \mathbb{C}^{d} | | |
| lyche-numerical-linear-algebra_FO1413 | 110 | 1.000 | \boldsymbol{C}:=\boldsymbol{B}-\boldsymbol{v} \boldsymbol{v}^{*} / \alpha | | |
| lyche-numerical-linear-algebra_FO1414 | 110 | 1.000 | d \times d | | |
| lyche-numerical-linear-algebra_FO1415 | 110 | 1.000 | n, \boldsymbol{C}=\boldsymbol{L}_{1} \boldsymbol{L}_{1}^{*} | | |
| lyche-numerical-linear-algebra_FO1416 | 110 | 0.999 | \boldsymbol{L}_{1}^{*} | | |
| lyche-numerical-linear-algebra_FO1417 | 110 | 1.000 | \left[\beta, \boldsymbol{v}^{*} / \beta\right]^{*} | | |
| lyche-numerical-linear-algebra_FO1418 | 110 | 1.000 | \alpha>0 | | |
| lyche-numerical-linear-algebra_FO1419 | 110 | 0.947 | \boldsymbol{C}=\boldsymbol{B}-\boldsymbol{v} \boldsymbol{v}^{*} / \alpha | | |
| lyche-numerical-linear-algebra_FO1420 | 110 | 0.947 | A(2: n, 2: n) | | |
| lyche-numerical-linear-algebra_FO1421 | 110 | 1.000 | \min (i+d, n) | | |
| lyche-numerical-linear-algebra_FO1422 | 111 | 1.000 | \ell_{k k}=0 | | |
| lyche-numerical-linear-algebra_FO1423 | 111 | 0.910 | 1 \Longleftrightarrow 2 | | |
| lyche-numerical-linear-algebra_FO1424 | 111 | 0.858 | 1 \Longleftrightarrow | | |
| lyche-numerical-linear-algebra_FO1425 | 111 | 0.933 | 1 \Longleftrightarrow 3 | | |
| lyche-numerical-linear-algebra_FO1426 | 111 | 0.933 | 4 \Longrightarrow 3 | | |
| lyche-numerical-linear-algebra_FO1427 | 111 | 0.835 | 1 \Longrightarrow 3 | | |
| lyche-numerical-linear-algebra_FO1428 | 111 | 0.835 | 1 \Longleftrightarrow 4 | | |
| lyche-numerical-linear-algebra_FO1429 | 111 | 0.835 | 3 \Longrightarrow 1 | | |
| lyche-numerical-linear-algebra_FO1430 | 112 | 0.997 | \boldsymbol{A}:=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1431 | 112 | 0.996 | \boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{x}=x_{1}^{2}+x_{2}^{2}+x_{1} x_{2}+x_{2} x_{1}=\left(x_{1}+x_{2}\right)^{2} \geq 0 | | |
| lyche-numerical-linear-algebra_FO1432 | 112 | 0.996 | \boldsymbol{x} \in \mathbb{R}^{2} | | |
| lyche-numerical-linear-algebra_FO1433 | 112 | 1.000 | \lambda_{2}=0 | | |
| lyche-numerical-linear-algebra_FO1434 | 112 | 0.998 | \operatorname{det}\left(\left[a_{11}\right]\right)= | | |
| lyche-numerical-linear-algebra_FO1435 | 112 | 0.971 | \operatorname{det}\left(\left[a_{22}\right]\right)=1 | | |
| lyche-numerical-linear-algebra_FO1436 | 112 | 0.575 | \boldsymbol{A}=\boldsymbol{B B}^{*} | | |
| lyche-numerical-linear-algebra_FO1437 | 112 | 0.575 | \boldsymbol{B}=\left[\begin{array}{ll}1 & 0 \\ 1 & 0\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1438 | 112 | 0.546 | \boldsymbol{A}:=\left[\begin{array}{rr}0 & 0 \\ 0 & -1\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1439 | 112 | 1.000 | \operatorname{det}(\boldsymbol{A})>0 | | |
| lyche-numerical-linear-algebra_FO1440 | 112 | 1.000 | a_{i i} a_{j j}>a_{i j} a_{j i} | | |
| lyche-numerical-linear-algebra_FO1441 | 113 | 0.752 | \lambda=\frac{\boldsymbol{x}^{T} \boldsymbol{A} \boldsymbol{x}}{\boldsymbol{x}^{T} \boldsymbol{x}}>0 | | |
| lyche-numerical-linear-algebra_FO1442 | 113 | 0.981 | \left[\begin{array}{cc}a_{i i} & a_{i j} \\ a_{j i} & a_{j j}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1443 | 113 | 1.000 | a \in \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO1444 | 113 | 1.000 | \boldsymbol{A}[a] | | |
| lyche-numerical-linear-algebra_FO1445 | 113 | 1.000 | a \in\left[1-\frac{\sqrt{5}}{2}, 1+\frac{\sqrt{5}}{2}\right] | | |
| lyche-numerical-linear-algebra_FO1446 | 113 | 1.000 | a | | |
| lyche-numerical-linear-algebra_FO1447 | 114 | 1.000 | \boldsymbol{A}:=\left[\begin{array}{cc}1 & 0 \\ -2 & 1\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1448 | 114 | 1.000 | \boldsymbol{E} \in \mathbb{R}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO1449 | 114 | 1.000 | \boldsymbol{E}=\boldsymbol{I}+\boldsymbol{u} \boldsymbol{u}^{T} | | |
| lyche-numerical-linear-algebra_FO1450 | 114 | 1.000 | \boldsymbol{u} \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO1451 | 114 | 1.000 | \boldsymbol{E} | | |
| lyche-numerical-linear-algebra_FO1452 | 114 | 0.834 | \boldsymbol{E}^{-1} .1 | | |
| lyche-numerical-linear-algebra_FO1453 | 114 | 1.000 | \boldsymbol{A}=\boldsymbol{B}+\boldsymbol{u} \boldsymbol{u}^{T} | | |
| lyche-numerical-linear-algebra_FO1454 | 114 | 1.000 | \boldsymbol{A}=\boldsymbol{L} \boldsymbol{L}^{T} | | |
| lyche-numerical-linear-algebra_FO1455 | 114 | 0.998 | \boldsymbol{A}_{+} | | |
| lyche-numerical-linear-algebra_FO1456 | 114 | 0.998 | (n+1) \times(n+1) | | |
| lyche-numerical-linear-algebra_FO1457 | 114 | 1.000 | \alpha | | |
| lyche-numerical-linear-algebra_FO1458 | 114 | 1.000 | \boldsymbol{A}_{+}=\boldsymbol{L}_{+} \boldsymbol{L}_{+}^{T} | | |
| lyche-numerical-linear-algebra_FO1459 | 114 | 1.000 | \boldsymbol{L}_{+} | | |
| lyche-numerical-linear-algebra_FO1460 | 114 | 1.000 | \alpha>\left\|\boldsymbol{L}^{-1} \boldsymbol{a}\right\|_{2}^{2} | | |
| lyche-numerical-linear-algebra_FO1461 | 114 | 1.000 | \boldsymbol{E}^{-1} | | |
| lyche-numerical-linear-algebra_FO1462 | 114 | 1.000 | \boldsymbol{E}^{-1}=\boldsymbol{I}+a \boldsymbol{u} \boldsymbol{u}^{T} | | |
| lyche-numerical-linear-algebra_FO1463 | 115 | 0.493 | \left[\begin{array}{lll}10 & 4 & 3 \\ 4 & 0 & 2 \\ 3 & 2 & 5\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1464 | 117 | 1.000 | \mathcal{V} \times \mathcal{V} \rightarrow \mathbb{C} | | |
| lyche-numerical-linear-algebra_FO1465 | 117 | 1.000 | \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z} \in \mathcal{V} | | |
| lyche-numerical-linear-algebra_FO1466 | 117 | 1.000 | a, b \in \mathbb{C} | | |
| lyche-numerical-linear-algebra_FO1467 | 117 | 1.000 | \langle\boldsymbol{x}, \boldsymbol{x}\rangle \geq 0 | | |
| lyche-numerical-linear-algebra_FO1468 | 117 | 1.000 | \langle\boldsymbol{x}, \boldsymbol{y}\rangle=\overline{\langle\boldsymbol{y}, \boldsymbol{x}\rangle} | | |
| lyche-numerical-linear-algebra_FO1469 | 117 | 1.000 | \langle a \boldsymbol{x}+b \boldsymbol{y}, \boldsymbol{z}\rangle=a\langle\boldsymbol{x}, \boldsymbol{z}\rangle+b\langle\boldsymbol{y}, \boldsymbol{z}\rangle | | |
| lyche-numerical-linear-algebra_FO1470 | 117 | 0.996 | (\mathcal{V},\langle\cdot, \cdot\rangle) | | |
| lyche-numerical-linear-algebra_FO1471 | 117 | 0.975 | \|\boldsymbol{x}\|=\|\boldsymbol{x}\|_{2}=\sqrt{\boldsymbol{x}^{*} \boldsymbol{x}} | | |
| lyche-numerical-linear-algebra_FO1472 | 118 | 1.000 | 0 \boldsymbol{x}+\boldsymbol{y}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO1473 | 118 | 1.000 | \langle\boldsymbol{x}, \boldsymbol{y}\rangle=\langle\boldsymbol{x}, 0 \boldsymbol{y}\rangle=0\langle\boldsymbol{x}, \boldsymbol{y}\rangle=0 | | |
| lyche-numerical-linear-algebra_FO1474 | 118 | 1.000 | \|\boldsymbol{y}\|=0 | | |
| lyche-numerical-linear-algebra_FO1475 | 118 | 1.000 | \boldsymbol{y} \neq \mathbf{0} | | |
| lyche-numerical-linear-algebra_FO1476 | 118 | 0.862 | \langle\boldsymbol{z}, \boldsymbol{y}\rangle=\langle\boldsymbol{x}, \boldsymbol{y}\rangle-a\langle\boldsymbol{y}, \boldsymbol{y}\rangle=0 | | |
| lyche-numerical-linear-algebra_FO1477 | 118 | 1.000 | \|\boldsymbol{y}\|^{2} | | |
| lyche-numerical-linear-algebra_FO1478 | 118 | 1.000 | \boldsymbol{z}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO1479 | 118 | 1.000 | \|\boldsymbol{x}\| \geq 0 | | |
| lyche-numerical-linear-algebra_FO1480 | 118 | 1.000 | \|a \boldsymbol{x}\|=|a|\|\boldsymbol{x}\| | | |
| lyche-numerical-linear-algebra_FO1481 | 118 | 1.000 | \|\boldsymbol{x}+\boldsymbol{y}\| \leq\|\boldsymbol{x}\|+\|\boldsymbol{y}\| | | |
| lyche-numerical-linear-algebra_FO1482 | 118 | 1.000 | \|\boldsymbol{x}\|:=\sqrt{\langle\boldsymbol{x}, \boldsymbol{x}\rangle} | | |
| lyche-numerical-linear-algebra_FO1483 | 118 | 1.000 | \left\|\|: \mathbb{C}^{n} \rightarrow \mathbb{R}\right. | | |
| lyche-numerical-linear-algebra_FO1484 | 119 | 1.000 | \|\boldsymbol{x}+a \boldsymbol{y}\|^{2}=\langle\boldsymbol{x}+a \boldsymbol{y}, \boldsymbol{x}+a \boldsymbol{y}\rangle | | |
| lyche-numerical-linear-algebra_FO1485 | 119 | 1.000 | a=1 | | |
| lyche-numerical-linear-algebra_FO1486 | 119 | 0.999 | -1 \leq \frac{\langle\boldsymbol{x}, \boldsymbol{y}\rangle}{\|\boldsymbol{x}\|\|\boldsymbol{y}\|} \leq 1 | | |
| lyche-numerical-linear-algebra_FO1487 | 119 | 1.000 | \theta | | |
| lyche-numerical-linear-algebra_FO1488 | 119 | 1.000 | [0, \pi] | | |
| lyche-numerical-linear-algebra_FO1489 | 119 | 1.000 | \boldsymbol{x} \perp \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO1490 | 119 | 1.000 | \langle\boldsymbol{x}, \boldsymbol{y}\rangle=0 | | |
| lyche-numerical-linear-algebra_FO1491 | 119 | 1.000 | \|\boldsymbol{x}\|=\|\boldsymbol{y}\|=1 | | |
| lyche-numerical-linear-algebra_FO1492 | 119 | 1.000 | \theta=\pi / 2 | | |
| lyche-numerical-linear-algebra_FO1493 | 119 | 0.997 | \left\langle\boldsymbol{v}_{i}, \boldsymbol{v}_{j}\right\rangle=0 | | |
| lyche-numerical-linear-algebra_FO1494 | 119 | 0.734 | \left\langle\boldsymbol{v}_{i}, \boldsymbol{v}_{j}\right\rangle=\delta_{i j} | | |
| lyche-numerical-linear-algebra_FO1495 | 120 | 0.999 | \left\{\boldsymbol{s}_{1}, \ldots, \boldsymbol{s}_{k}\right\} | | |
| lyche-numerical-linear-algebra_FO1496 | 120 | 0.833 | (\mathcal{S},\langle\cdot, \cdot\rangle) | | |
| lyche-numerical-linear-algebra_FO1497 | 120 | 0.999 | S_{j}:=\operatorname{span}\left\{\boldsymbol{s}_{1}, \ldots, \boldsymbol{s}_{j}\right\} | | |
| lyche-numerical-linear-algebra_FO1498 | 120 | 0.999 | \boldsymbol{v}_{1}=\boldsymbol{s}_{1} | | |
| lyche-numerical-linear-algebra_FO1499 | 120 | 1.000 | S_{1} | | |
| lyche-numerical-linear-algebra_FO1500 | 120 | 1.000 | j \geq 2 | | |
| lyche-numerical-linear-algebra_FO1501 | 120 | 1.000 | \boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{j-1} | | |
| lyche-numerical-linear-algebra_FO1502 | 120 | 1.000 | S_{j-1} | | |
| lyche-numerical-linear-algebra_FO1503 | 120 | 1.000 | \boldsymbol{v}_{j} | | |
| lyche-numerical-linear-algebra_FO1504 | 120 | 1.000 | \boldsymbol{v}_{i} | | |
| lyche-numerical-linear-algebra_FO1505 | 120 | 1.000 | \boldsymbol{s}_{1}, \ldots, \boldsymbol{s}_{i} | | |
| lyche-numerical-linear-algebra_FO1506 | 120 | 1.000 | \boldsymbol{v}_{j}=\sum_{i=1}^{j} a_{i} \boldsymbol{s}_{i} | | |
| lyche-numerical-linear-algebra_FO1507 | 120 | 1.000 | a_{0}, \ldots, a_{j} | | |
| lyche-numerical-linear-algebra_FO1508 | 120 | 1.000 | a_{j}=1 | | |
| lyche-numerical-linear-algebra_FO1509 | 120 | 1.000 | \boldsymbol{s}_{1}, \ldots, \boldsymbol{s}_{j} | | |
| lyche-numerical-linear-algebra_FO1510 | 120 | 1.000 | a_{j} \neq 0 | | |
| lyche-numerical-linear-algebra_FO1511 | 120 | 1.000 | \boldsymbol{v}_{j} \neq 0 | | |
| lyche-numerical-linear-algebra_FO1512 | 120 | 1.000 | l=1, \ldots, j-1 | | |
| lyche-numerical-linear-algebra_FO1513 | 120 | 1.000 | \boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{j} | | |
| lyche-numerical-linear-algebra_FO1514 | 120 | 1.000 | S_{j} | | |
| lyche-numerical-linear-algebra_FO1515 | 120 | 1.000 | \left\{\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{k}\right\} | | |
| lyche-numerical-linear-algebra_FO1516 | 121 | 1.000 | \mathcal{S} \subset \mathcal{T} | | |
| lyche-numerical-linear-algebra_FO1517 | 121 | 1.000 | \operatorname{dim} \mathcal{S}:=k<\operatorname{dim} \mathcal{T}=n | | |
| lyche-numerical-linear-algebra_FO1518 | 121 | 1.000 | \boldsymbol{s}_{1}, \ldots, \boldsymbol{s}_{k} | | |
| lyche-numerical-linear-algebra_FO1519 | 121 | 1.000 | \boldsymbol{s}_{1}, \ldots, \boldsymbol{s}_{k}, \boldsymbol{s}_{k+1}, \ldots, \boldsymbol{s}_{n} | | |
| lyche-numerical-linear-algebra_FO1520 | 121 | 0.955 | \boldsymbol{v}_{i}=\boldsymbol{s}_{i} | | |
| lyche-numerical-linear-algebra_FO1521 | 121 | 1.000 | 1, \ldots, k | | |
| lyche-numerical-linear-algebra_FO1522 | 121 | 1.000 | 2 \leq r<k | | |
| lyche-numerical-linear-algebra_FO1523 | 121 | 1.000 | \boldsymbol{v}_{j}=\boldsymbol{s}_{j} | | |
| lyche-numerical-linear-algebra_FO1524 | 121 | 1.000 | j=1, \ldots, r-1 | | |
| lyche-numerical-linear-algebra_FO1525 | 121 | 1.000 | j=r | | |
| lyche-numerical-linear-algebra_FO1526 | 121 | 1.000 | \left\langle\boldsymbol{s}_{r}, \boldsymbol{v}_{i}\right\rangle= | | |
| lyche-numerical-linear-algebra_FO1527 | 121 | 0.999 | \left\langle\boldsymbol{s}_{r}, \boldsymbol{s}_{i}\right\rangle=0 | | |
| lyche-numerical-linear-algebra_FO1528 | 121 | 0.999 | i<r | | |
| lyche-numerical-linear-algebra_FO1529 | 121 | 0.999 | \boldsymbol{v}_{r}=\boldsymbol{s}_{r} | | |
| lyche-numerical-linear-algebra_FO1530 | 121 | 0.999 | \mathcal{S}=\operatorname{span}\left(\boldsymbol{s}_{1}, \ldots, \boldsymbol{s}_{k}\right) | | |
| lyche-numerical-linear-algebra_FO1531 | 121 | 1.000 | 1 \leq k<n | | |
| lyche-numerical-linear-algebra_FO1532 | 121 | 1.000 | \langle\boldsymbol{x}, \boldsymbol{y}\rangle | | |
| lyche-numerical-linear-algebra_FO1533 | 121 | 1.000 | \mathcal{S}+\mathcal{T}:=\{\boldsymbol{s}+\boldsymbol{t}: \boldsymbol{s} \in \mathcal{S} | | |
| lyche-numerical-linear-algebra_FO1534 | 121 | 1.000 | \boldsymbol{t} \in \mathcal{T}\} | | |
| lyche-numerical-linear-algebra_FO1535 | 121 | 0.967 | \boldsymbol{\operatorname { s u m }} \mathcal{S} \oplus \mathcal{T} | | |
| lyche-numerical-linear-algebra_FO1536 | 121 | 0.998 | \mathcal{S} \stackrel{\perp}{\oplus} \mathcal{T} | | |
| lyche-numerical-linear-algebra_FO1537 | 121 | 0.998 | \langle\boldsymbol{s}, \boldsymbol{t}\rangle=0 | | |
| lyche-numerical-linear-algebra_FO1538 | 121 | 0.998 | \boldsymbol{s} \in \mathcal{S} | | |
| lyche-numerical-linear-algebra_FO1539 | 121 | 0.998 | \boldsymbol{t} \in \mathcal{T} | | |
| lyche-numerical-linear-algebra_FO1540 | 121 | 1.000 | \boldsymbol{v} \in \mathcal{S} \oplus \mathcal{T} | | |
| lyche-numerical-linear-algebra_FO1541 | 121 | 1.000 | \boldsymbol{v}=\boldsymbol{s}+\boldsymbol{t} | | |
| lyche-numerical-linear-algebra_FO1542 | 121 | 1.000 | \boldsymbol{v}=\boldsymbol{s}_{1}+\boldsymbol{t}_{1}=\boldsymbol{s}_{2}+\boldsymbol{t}_{2} | | |
| lyche-numerical-linear-algebra_FO1543 | 121 | 1.000 | \boldsymbol{s}_{1}, \boldsymbol{s}_{2} \in \mathcal{S} | | |
| lyche-numerical-linear-algebra_FO1544 | 121 | 1.000 | \boldsymbol{t}_{1}, \boldsymbol{t}_{2} \in \mathcal{T} | | |
| lyche-numerical-linear-algebra_FO1545 | 121 | 0.969 | \mathbf{0}=\boldsymbol{s}_{1}-\boldsymbol{s}_{2}+\boldsymbol{t}_{1}-\boldsymbol{t}_{2} | | |
| lyche-numerical-linear-algebra_FO1546 | 121 | 0.969 | \boldsymbol{s}_{1}-\boldsymbol{s}_{2}=\boldsymbol{t}_{2}-\boldsymbol{t}_{1} | | |
| lyche-numerical-linear-algebra_FO1547 | 121 | 0.969 | \boldsymbol{s}_{1}-\boldsymbol{s}_{2} | | |
| lyche-numerical-linear-algebra_FO1548 | 121 | 0.969 | \boldsymbol{t}_{2}-\boldsymbol{t}_{1} | | |
| lyche-numerical-linear-algebra_FO1549 | 121 | 1.000 | \boldsymbol{s}_{1}-\boldsymbol{s}_{2}=\boldsymbol{t}_{2}-\boldsymbol{t}_{1}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO1550 | 121 | 1.000 | \boldsymbol{s}_{1}=\boldsymbol{s}_{2} | | |
| lyche-numerical-linear-algebra_FO1551 | 121 | 1.000 | \boldsymbol{t}_{2}=\boldsymbol{t}_{1} | | |
| lyche-numerical-linear-algebra_FO1552 | 121 | 1.000 | \boldsymbol{v} \in \mathcal{S} \cap \mathcal{T} | | |
| lyche-numerical-linear-algebra_FO1553 | 121 | 1.000 | \langle\boldsymbol{v}, \boldsymbol{v}\rangle=0 | | |
| lyche-numerical-linear-algebra_FO1554 | 121 | 1.000 | \boldsymbol{v}=0 | | |
| lyche-numerical-linear-algebra_FO1555 | 121 | 1.000 | \mathcal{T}:=\mathcal{S}^{\perp} | | |
| lyche-numerical-linear-algebra_FO1556 | 121 | 1.000 | \boldsymbol{v}=\boldsymbol{s}_{0}+\boldsymbol{t}_{0} \in \mathcal{S} \oplus \mathcal{T} | | |
| lyche-numerical-linear-algebra_FO1557 | 121 | 1.000 | \boldsymbol{s}_{0} \in \mathcal{S} | | |
| lyche-numerical-linear-algebra_FO1558 | 121 | 1.000 | \boldsymbol{t}_{0} \in \mathcal{T} | | |
| lyche-numerical-linear-algebra_FO1559 | 121 | 1.000 | \boldsymbol{s}_{0} | | |
| lyche-numerical-linear-algebra_FO1560 | 121 | 1.000 | \boldsymbol{t}_{0} | | |
| lyche-numerical-linear-algebra_FO1561 | 122 | 1.000 | \mathcal{T}=\mathcal{S}^{\perp} | | |
| lyche-numerical-linear-algebra_FO1562 | 122 | 1.000 | \langle\cdot, \cdot\rangle | | |
| lyche-numerical-linear-algebra_FO1563 | 122 | 0.984 | \boldsymbol{v} \in \mathcal{S} \stackrel{\perp}{\oplus} \mathcal{T} | | |
| lyche-numerical-linear-algebra_FO1564 | 122 | 1.000 | \boldsymbol{v}=\boldsymbol{s}_{0}+\boldsymbol{t}_{0} | | |
| lyche-numerical-linear-algebra_FO1565 | 122 | 1.000 | \left\langle\boldsymbol{s}_{0}, \boldsymbol{s}\right\rangle=\left\langle\boldsymbol{v}-\boldsymbol{t}_{0}, \boldsymbol{s}\right\rangle=\langle\boldsymbol{v}, \boldsymbol{s}\rangle | | |
| lyche-numerical-linear-algebra_FO1566 | 122 | 1.000 | \left\langle\boldsymbol{t}_{0}, \boldsymbol{s}\right\rangle=0 | | |
| lyche-numerical-linear-algebra_FO1567 | 123 | 1.000 | \|\boldsymbol{v}\|:=\sqrt{\langle\boldsymbol{v}, \boldsymbol{v}\rangle} | | |
| lyche-numerical-linear-algebra_FO1568 | 123 | 1.000 | \boldsymbol{v} \in \mathcal{V} | | |
| lyche-numerical-linear-algebra_FO1569 | 123 | 0.975 | s_{0} \neq s \in \mathcal{S} | | |
| lyche-numerical-linear-algebra_FO1570 | 123 | 0.975 | 0 \neq \boldsymbol{u}:=s_{0}-s \in \mathcal{S} | | |
| lyche-numerical-linear-algebra_FO1571 | 123 | 1.000 | \left\langle\boldsymbol{v}-\boldsymbol{s}_{0}, \boldsymbol{u}\right\rangle=0 | | |
| lyche-numerical-linear-algebra_FO1572 | 123 | 1.000 | \langle\boldsymbol{x}, \boldsymbol{y}\rangle:=\boldsymbol{y}^{*} \boldsymbol{x}=\sum_{j=1}^{n} x_{j} \overline{y_{j}} | | |
| lyche-numerical-linear-algebra_FO1573 | 123 | 1.000 | \boldsymbol{A}^{T}=\boldsymbol{A} \Longleftrightarrow\langle\boldsymbol{A} \boldsymbol{x}, \boldsymbol{y}\rangle=\langle\boldsymbol{x}, \overline{\boldsymbol{A}} \boldsymbol{y}\rangle | | |
| lyche-numerical-linear-algebra_FO1574 | 123 | 1.000 | \boldsymbol{A}^{*}=\boldsymbol{A} \Longleftrightarrow\langle\boldsymbol{A} \boldsymbol{x}, \boldsymbol{y}\rangle=\langle\boldsymbol{x}, \boldsymbol{A} \boldsymbol{y}\rangle | | |
| lyche-numerical-linear-algebra_FO1575 | 123 | 1.000 | \boldsymbol{x}=\boldsymbol{e}_{j} | | |
| lyche-numerical-linear-algebra_FO1576 | 123 | 1.000 | \boldsymbol{y}=\boldsymbol{e}_{i} | | |
| lyche-numerical-linear-algebra_FO1577 | 123 | 0.815 | \boldsymbol{A}=\boldsymbol{A}^{T} | | |
| lyche-numerical-linear-algebra_FO1578 | 123 | 1.000 | \boldsymbol{U}^{*} \boldsymbol{U}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO1579 | 123 | 1.000 | \boldsymbol{U}^{T} \boldsymbol{U}= | | |
| lyche-numerical-linear-algebra_FO1580 | 123 | 1.000 | \boldsymbol{U}^{-1}=\boldsymbol{U}^{*} | | |
| lyche-numerical-linear-algebra_FO1581 | 123 | 1.000 | \boldsymbol{U} \boldsymbol{U}^{*}= | | |
| lyche-numerical-linear-algebra_FO1582 | 123 | 1.000 | \boldsymbol{U} \boldsymbol{U}^{-1}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO1583 | 123 | 1.000 | \boldsymbol{U}_{1}^{*} \boldsymbol{U}_{1}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO1584 | 123 | 1.000 | \boldsymbol{U}_{2}^{*} \boldsymbol{U}_{2}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO1585 | 123 | 1.000 | \left(\boldsymbol{U}_{1} \boldsymbol{U}_{2}\right)^{*}\left(\boldsymbol{U}_{1} \boldsymbol{U}_{2}\right)= | | |
| lyche-numerical-linear-algebra_FO1586 | 123 | 0.984 | \boldsymbol{U}_{2}^{*} \boldsymbol{U}_{1}^{*} \boldsymbol{U}_{1} \boldsymbol{U}_{2}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO1587 | 124 | 0.997 | \langle\boldsymbol{U} \boldsymbol{x}, \boldsymbol{U} \boldsymbol{y}\rangle=\langle\boldsymbol{x}, \boldsymbol{y}\rangle | | |
| lyche-numerical-linear-algebra_FO1588 | 124 | 0.997 | \|\boldsymbol{U} \boldsymbol{x}\|_{2}= | | |
| lyche-numerical-linear-algebra_FO1589 | 124 | 1.000 | \|\boldsymbol{x}\|_{2} | | |
| lyche-numerical-linear-algebra_FO1590 | 124 | 1.000 | i, j=1, \ldots, n | | |
| lyche-numerical-linear-algebra_FO1591 | 124 | 1.000 | \left(\boldsymbol{U}^{*} \boldsymbol{U}\right)_{i, j}=\delta_{i, j} | | |
| lyche-numerical-linear-algebra_FO1592 | 124 | 1.000 | \boldsymbol{y}=\boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO1593 | 124 | 1.000 | \boldsymbol{H} \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO1594 | 124 | 1.000 | \boldsymbol{H}^{*}=(\boldsymbol{I}- | | |
| lyche-numerical-linear-algebra_FO1595 | 124 | 1.000 | \left.\boldsymbol{u} \boldsymbol{u}^{*}\right)^{*}=\boldsymbol{H} | | |
| lyche-numerical-linear-algebra_FO1596 | 124 | 1.000 | \boldsymbol{I}-2 \boldsymbol{u} \boldsymbol{u}^{*} | | |
| lyche-numerical-linear-algebra_FO1597 | 124 | 1.000 | \boldsymbol{u}^{*} \boldsymbol{u}=1 | | |
| lyche-numerical-linear-algebra_FO1598 | 124 | 1.000 | \boldsymbol{v} \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO1599 | 125 | 1.000 | \boldsymbol{H}=\boldsymbol{I}-\boldsymbol{u} \boldsymbol{u}^{*} | | |
| lyche-numerical-linear-algebra_FO1600 | 125 | 1.000 | \boldsymbol{u}:=\sqrt{2} \frac{\boldsymbol{v}}{\|\boldsymbol{v}\|_{2}} | | |
| lyche-numerical-linear-algebra_FO1601 | 125 | 1.000 | \sqrt{2} | | |
| lyche-numerical-linear-algebra_FO1602 | 125 | 1.000 | \|\boldsymbol{x}\|_{2}=\|\boldsymbol{y}\|_{2}, \boldsymbol{y}^{*} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO1603 | 125 | 0.993 | \boldsymbol{v}:=\boldsymbol{x}-\boldsymbol{y} \neq \mathbf{0} | | |
| lyche-numerical-linear-algebra_FO1604 | 125 | 0.993 | \left(\boldsymbol{I}-2 \frac{\boldsymbol{v} \boldsymbol{v}^{*}}{\boldsymbol{v}^{*} \boldsymbol{v}}\right) \boldsymbol{x}=\boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO1605 | 125 | 1.000 | \boldsymbol{x}^{*} \boldsymbol{x}=\boldsymbol{y}^{*} \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO1606 | 125 | 1.000 | \operatorname{Re}\left(\boldsymbol{y}^{*} \boldsymbol{x}\right)=\boldsymbol{y}^{*} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO1607 | 125 | 0.998 | \left(\boldsymbol{I}-2 \frac{\boldsymbol{v} \boldsymbol{v}^{*}}{\boldsymbol{v}^{*} \boldsymbol{v}}\right) \boldsymbol{x}=\boldsymbol{x}-\frac{2 \boldsymbol{v}^{*} \boldsymbol{x}}{\boldsymbol{v}^{*} \boldsymbol{v}} \boldsymbol{v}=\boldsymbol{x}-\boldsymbol{v}=\boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO1608 | 125 | 1.000 | \boldsymbol{H} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO1609 | 125 | 1.000 | \mathcal{M}:= | | |
| lyche-numerical-linear-algebra_FO1610 | 125 | 1.000 | \left\{\boldsymbol{w} \in \mathbb{R}^{n}: \boldsymbol{w}^{*} \boldsymbol{v}=0\right\} | | |
| lyche-numerical-linear-algebra_FO1611 | 125 | 1.000 | (\boldsymbol{x}+\boldsymbol{y}) / 2 | | |
| lyche-numerical-linear-algebra_FO1612 | 125 | 1.000 | \boldsymbol{x}-\boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO1613 | 126 | 1.000 | \boldsymbol{x}:=[1,0,1]^{T} | | |
| lyche-numerical-linear-algebra_FO1614 | 126 | 1.000 | \boldsymbol{y}:=[-1,0,1]^{T} | | |
| lyche-numerical-linear-algebra_FO1615 | 126 | 1.000 | \boldsymbol{v}= | | |
| lyche-numerical-linear-algebra_FO1616 | 126 | 1.000 | \boldsymbol{x}-\boldsymbol{y}=[2,0,0]^{T} | | |
| lyche-numerical-linear-algebra_FO1617 | 126 | 0.999 | y z | | |
| lyche-numerical-linear-algebra_FO1618 | 126 | 0.999 | \boldsymbol{H} \boldsymbol{x}=[-1,0,1]^{T}=\boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO1619 | 126 | 0.999 | \boldsymbol{P} \boldsymbol{x}=[0,0,1]^{T}= | | |
| lyche-numerical-linear-algebra_FO1620 | 126 | 1.000 | (\boldsymbol{x}+\boldsymbol{y}) / 2 \in \mathcal{M} | | |
| lyche-numerical-linear-algebra_FO1621 | 126 | 1.000 | \boldsymbol{H} \boldsymbol{x}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO1622 | 126 | 1.000 | a=0 | | |
| lyche-numerical-linear-algebra_FO1623 | 126 | 1.000 | \|\boldsymbol{u}\|_{2}=\sqrt{2} | | |
| lyche-numerical-linear-algebra_FO1624 | 126 | 1.000 | \boldsymbol{u}:=\sqrt{2} \boldsymbol{e}_{1} | | |
| lyche-numerical-linear-algebra_FO1625 | 126 | 1.000 | \boldsymbol{x} \neq \mathbf{0} | | |
| lyche-numerical-linear-algebra_FO1626 | 126 | 0.971 | |\rho|=1 | | |
| lyche-numerical-linear-algebra_FO1627 | 126 | 0.971 | \rho\|\boldsymbol{x}\|_{2} \boldsymbol{z}=|\rho|^{2} \boldsymbol{x}=\boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO1628 | 126 | 0.971 | \|\boldsymbol{z}\|_{2}=1 | | |
| lyche-numerical-linear-algebra_FO1629 | 126 | 0.971 | z_{1}= | | |
| lyche-numerical-linear-algebra_FO1630 | 126 | 0.969 | \left|x_{1}\right| /\|\boldsymbol{x}\|_{2} | | |
| lyche-numerical-linear-algebra_FO1631 | 126 | 0.969 | \boldsymbol{u}^{*} \boldsymbol{u}=\frac{\left(z+\boldsymbol{e}_{1}\right)^{*}\left(z+\boldsymbol{e}_{1}\right)}{1+z_{1}}=\frac{2+2 z_{1}}{1+z_{1}}=2 | | |
| lyche-numerical-linear-algebra_FO1632 | 127 | 1.000 | \boldsymbol{u}^{*} \boldsymbol{u}=2 | | |
| lyche-numerical-linear-algebra_FO1633 | 127 | 0.986 | \left(\boldsymbol{I}-\boldsymbol{u u}^{*}\right) \boldsymbol{x}=a \boldsymbol{e}_{1} | | |
| lyche-numerical-linear-algebra_FO1634 | 127 | 0.999 | z_{1}=\left|x_{1}\right| /\|\boldsymbol{x}\|_{2} \geq 0 | | |
| lyche-numerical-linear-algebra_FO1635 | 127 | 0.999 | \|\boldsymbol{z}\|_{2}= | | |
| lyche-numerical-linear-algebra_FO1636 | 127 | 1.000 | 1 \leq 1+z_{1} \leq 2 | | |
| lyche-numerical-linear-algebra_FO1637 | 127 | 1.000 | 1+z_{1} | | |
| lyche-numerical-linear-algebra_FO1638 | 127 | 0.887 | \boldsymbol{H} \boldsymbol{x}=\left(\boldsymbol{I}-\boldsymbol{u} \boldsymbol{u}^{*}\right) \boldsymbol{x}=\boldsymbol{x}-\left(\boldsymbol{u}^{*} \boldsymbol{x}\right) \boldsymbol{u} | | |
| lyche-numerical-linear-algebra_FO1639 | 127 | 0.887 | \boldsymbol{u}, \boldsymbol{x} \in \mathbb{R}^{m} | | |
| lyche-numerical-linear-algebra_FO1640 | 127 | 0.887 | 2 m | | |
| lyche-numerical-linear-algebra_FO1641 | 127 | 0.994 | \boldsymbol{u}^{T} \boldsymbol{x}, m | | |
| lyche-numerical-linear-algebra_FO1642 | 127 | 0.994 | \left(\boldsymbol{u}^{T} \boldsymbol{x}\right) \boldsymbol{u} | | |
| lyche-numerical-linear-algebra_FO1643 | 127 | 1.000 | 4 m | | |
| lyche-numerical-linear-algebra_FO1644 | 127 | 1.000 | 4 m n | | |
| lyche-numerical-linear-algebra_FO1645 | 127 | 1.000 | \boldsymbol{H} \boldsymbol{A}=\boldsymbol{A}-\left(\boldsymbol{u}^{T} \boldsymbol{A}\right) \boldsymbol{u} | | |
| lyche-numerical-linear-algebra_FO1646 | 127 | 1.000 | \boldsymbol{x}^{T}:=[\boldsymbol{y}, \boldsymbol{z}]^{T} | | |
| lyche-numerical-linear-algebra_FO1647 | 127 | 1.000 | \boldsymbol{y} \in \mathbb{C}^{k}, \boldsymbol{z} \in \mathbb{C}^{n-k} | | |
| lyche-numerical-linear-algebra_FO1648 | 127 | 1.000 | 1 \leq k< | | |
| lyche-numerical-linear-algebra_FO1649 | 127 | 0.915 | [\hat{\boldsymbol{u}}, a]:=\operatorname{housegen}(\boldsymbol{z}) | | |
| lyche-numerical-linear-algebra_FO1650 | 127 | 0.384 | \hat{\boldsymbol{H}}=\boldsymbol{I}-\hat{\boldsymbol{u}} \hat{\boldsymbol{u}}^{*} | | |
| lyche-numerical-linear-algebra_FO1651 | 127 | 0.384 | \hat{\boldsymbol{H}} z=a \boldsymbol{e}_{1} \in \mathbb{C}^{n-k} | | |
| lyche-numerical-linear-algebra_FO1652 | 127 | 0.384 | \boldsymbol{u}:=\left[\begin{array}{c}\mathbf{0} \\ \hat{\boldsymbol{u}}\end{array}\right] \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO1653 | 127 | 1.000 | \boldsymbol{u}^{*} \boldsymbol{u}=\hat{\boldsymbol{u}}^{*} \hat{\boldsymbol{u}}=2 | | |
| lyche-numerical-linear-algebra_FO1654 | 128 | 1.000 | \boldsymbol{R} \in \mathbb{C}^{m \times n} | | |
| lyche-numerical-linear-algebra_FO1655 | 128 | 1.000 | r_{i, j}=0 | | |
| lyche-numerical-linear-algebra_FO1656 | 128 | 1.000 | j<i | | |
| lyche-numerical-linear-algebra_FO1657 | 128 | 1.000 | i=2,3 \ldots, m | | |
| lyche-numerical-linear-algebra_FO1658 | 128 | 1.000 | m \leq n | | |
| lyche-numerical-linear-algebra_FO1659 | 128 | 0.996 | \boldsymbol{H}_{1}, \ldots, \boldsymbol{H}_{n} | | |
| lyche-numerical-linear-algebra_FO1660 | 128 | 1.000 | \boldsymbol{R}_{1} | | |
| lyche-numerical-linear-algebra_FO1661 | 129 | 1.000 | \hat{\boldsymbol{H}}_{k}:=\boldsymbol{I}-\hat{\boldsymbol{u}}_{k} \hat{\boldsymbol{u}}_{k}^{*} | | |
| lyche-numerical-linear-algebra_FO1662 | 129 | 1.000 | \left[a_{k, k}^{(k)}, \ldots, a_{m, k}^{(k)}\right]^{T} | | |
| lyche-numerical-linear-algebra_FO1663 | 129 | 1.000 | \boldsymbol{D}_{k} | | |
| lyche-numerical-linear-algebra_FO1664 | 129 | 1.000 | \boldsymbol{e}_{1}, \hat{\boldsymbol{H}}_{k}\left(\boldsymbol{D}_{k} \boldsymbol{e}_{1}\right)=a_{k} \boldsymbol{e}_{1} | | |
| lyche-numerical-linear-algebra_FO1665 | 129 | 0.865 | \left[\hat{\boldsymbol{u}}_{k}, a_{k}\right]=\operatorname{housegen}\left(\boldsymbol{D}_{k} \boldsymbol{e}_{1}\right) | | |
| lyche-numerical-linear-algebra_FO1666 | 129 | 0.865 | \boldsymbol{H}_{k}:=\left[\begin{array}{cc}\boldsymbol{I}_{k-1} & \mathbf{0} \\ \mathbf{0} & \hat{\boldsymbol{H}}_{k}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1667 | 129 | 1.000 | \boldsymbol{B}_{k+1} \in \mathbb{C}^{k \times k} | | |
| lyche-numerical-linear-algebra_FO1668 | 129 | 1.000 | \boldsymbol{D}_{k+1} \in \mathbb{C}^{(m-k) \times(n-k)} | | |
| lyche-numerical-linear-algebra_FO1669 | 129 | 1.000 | \boldsymbol{A}_{k+1} | | |
| lyche-numerical-linear-algebra_FO1670 | 129 | 1.000 | \boldsymbol{R}:=\boldsymbol{A}_{n+1}=\left[\begin{array}{c}\boldsymbol{R}_{1} \\ \mathbf{0}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1671 | 129 | 1.000 | m>1 | | |
| lyche-numerical-linear-algebra_FO1672 | 129 | 1.000 | m-1 | | |
| lyche-numerical-linear-algebra_FO1673 | 129 | 0.999 | \boldsymbol{H}_{m-1} \cdots \boldsymbol{H}_{1} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO1674 | 129 | 1.000 | \hat{\boldsymbol{u}}_{k}=\left[u_{k k}, \ldots, u_{m k}\right]^{T} | | |
| lyche-numerical-linear-algebra_FO1675 | 129 | 1.000 | u_{k, k} | | |
| lyche-numerical-linear-algebra_FO1676 | 129 | 1.000 | a_{k}=r_{k, k} | | |
| lyche-numerical-linear-algebra_FO1677 | 129 | 1.000 | m=4 | | |
| lyche-numerical-linear-algebra_FO1678 | 129 | 1.000 | \boldsymbol{B} \in \mathbb{C}^{m \times r} | | |
| lyche-numerical-linear-algebra_FO1679 | 129 | 1.000 | \boldsymbol{H}_{1}, \ldots, \boldsymbol{H}_{s} | | |
| lyche-numerical-linear-algebra_FO1680 | 129 | 1.000 | \boldsymbol{R}=\boldsymbol{H}_{s} \cdots \boldsymbol{H}_{1} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO1681 | 129 | 1.000 | \boldsymbol{C}=\boldsymbol{H}_{s} \cdots \boldsymbol{H}_{1} \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO1682 | 129 | 1.000 | r_{k, k} | | |
| lyche-numerical-linear-algebra_FO1683 | 129 | 0.961 | \operatorname{side}(\mathrm{s}) \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO1684 | 129 | 0.999 | \boldsymbol{B}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO1685 | 130 | 0.998 | v=\hat{\boldsymbol{u}}_{k} | | |
| lyche-numerical-linear-algebra_FO1686 | 130 | 0.998 | \hat{\boldsymbol{H}}_{k} \boldsymbol{C}=\left(\boldsymbol{I}-\boldsymbol{v} \boldsymbol{v}^{*}\right) \boldsymbol{C}=\boldsymbol{C}-\boldsymbol{v}\left(\boldsymbol{v}^{*} \boldsymbol{C}\right) | | |
| lyche-numerical-linear-algebra_FO1687 | 130 | 1.000 | n+1, \ldots, m | | |
| lyche-numerical-linear-algebra_FO1688 | 130 | 1.000 | \boldsymbol{C}-\boldsymbol{v} *\left(\boldsymbol{v}^{*} * \boldsymbol{C}\right) | | |
| lyche-numerical-linear-algebra_FO1689 | 130 | 1.000 | \boldsymbol{C} \in \mathbb{C}^{(m-k+1) \times(n+r-k)} | | |
| lyche-numerical-linear-algebra_FO1690 | 130 | 1.000 | m \geq n | | |
| lyche-numerical-linear-algebra_FO1691 | 130 | 1.000 | \boldsymbol{C}-\boldsymbol{v} *\left(\boldsymbol{v}^{T} * \boldsymbol{C}\right) | | |
| lyche-numerical-linear-algebra_FO1692 | 130 | 1.000 | 4(m-k+1)(n+r-k) | | |
| lyche-numerical-linear-algebra_FO1693 | 130 | 1.000 | r=0 | | |
| lyche-numerical-linear-algebra_FO1694 | 130 | 1.000 | 4 n^{3} / 3=2 G_{n} | | |
| lyche-numerical-linear-algebra_FO1695 | 130 | 1.000 | \boldsymbol{R} \boldsymbol{x}=\boldsymbol{c} | | |
| lyche-numerical-linear-algebra_FO1696 | 131 | 1.000 | 4 n^{3} / 3= | | |
| lyche-numerical-linear-algebra_FO1697 | 131 | 0.998 | \boldsymbol{R}=\boldsymbol{H}_{n-1} \cdots \boldsymbol{H}_{1} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO1698 | 131 | 1.000 | \boldsymbol{Q}=\boldsymbol{H}_{1} \cdots \boldsymbol{H}_{n-1} | | |
| lyche-numerical-linear-algebra_FO1699 | 131 | 0.982 | \mathbf{Q R} | | |
| lyche-numerical-linear-algebra_FO1700 | 131 | 0.982 | \boldsymbol{Q} \in \mathbb{C}^{m, m} | | |
| lyche-numerical-linear-algebra_FO1701 | 131 | 1.000 | \boldsymbol{R}_{1} \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO1702 | 131 | 1.000 | \mathbf{0}_{m-n, n} | | |
| lyche-numerical-linear-algebra_FO1703 | 131 | 1.000 | m-n | | |
| lyche-numerical-linear-algebra_FO1704 | 131 | 0.961 | \boldsymbol{A}=\boldsymbol{Q}_{1} \boldsymbol{R}_{1} | | |
| lyche-numerical-linear-algebra_FO1705 | 131 | 1.000 | \boldsymbol{Q}_{1} \in \mathbb{C}^{m \times n} | | |
| lyche-numerical-linear-algebra_FO1706 | 131 | 1.000 | \boldsymbol{Q} \boldsymbol{R} | | |
| lyche-numerical-linear-algebra_FO1707 | 131 | 1.000 | \left[\boldsymbol{Q}_{1}, \boldsymbol{Q}_{2}\right] | | |
| lyche-numerical-linear-algebra_FO1708 | 131 | 1.000 | \boldsymbol{R}=\left[\begin{array}{c}\boldsymbol{R}_{1} \\ \mathbf{0}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1709 | 131 | 1.000 | \boldsymbol{Q}_{1} \in \mathbb{R}^{m \times n} | | |
| lyche-numerical-linear-algebra_FO1710 | 131 | 1.000 | \boldsymbol{R}_{1} \in \mathbb{R}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO1711 | 131 | 1.000 | \left\{\boldsymbol{q}_{1}, \ldots, \boldsymbol{q}_{n}\right\} | | |
| lyche-numerical-linear-algebra_FO1712 | 131 | 1.000 | \boldsymbol{Q}_{1} | | |
| lyche-numerical-linear-algebra_FO1713 | 131 | 1.000 | \left\{\boldsymbol{q}_{1}, \ldots, \boldsymbol{q}_{n}, \boldsymbol{q}_{n+1}, \ldots, \boldsymbol{q}_{m}\right\} | | |
| lyche-numerical-linear-algebra_FO1714 | 131 | 1.000 | \boldsymbol{Q}=\left[\boldsymbol{q}_{1}, \ldots, \boldsymbol{q}_{m}\right] | | |
| lyche-numerical-linear-algebra_FO1715 | 132 | 0.743 | \boldsymbol{Q}^{T} \boldsymbol{Q}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO1716 | 132 | 1.000 | Q R | | |
| lyche-numerical-linear-algebra_FO1717 | 132 | 1.000 | \boldsymbol{A}=[1] \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO1718 | 132 | 1.000 | s:=\min (m-1, n) | | |
| lyche-numerical-linear-algebra_FO1719 | 132 | 0.999 | \boldsymbol{R}=\boldsymbol{C} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO1720 | 132 | 0.999 | \boldsymbol{C}= | | |
| lyche-numerical-linear-algebra_FO1721 | 132 | 0.615 | \boldsymbol{H}_{s} \cdots \boldsymbol{H}_{2} \boldsymbol{H}_{1} | | |
| lyche-numerical-linear-algebra_FO1722 | 132 | 0.615 | \boldsymbol{Q}:=\boldsymbol{C}^{*}= | | |
| lyche-numerical-linear-algebra_FO1723 | 132 | 1.000 | \boldsymbol{H}_{1} \cdots \boldsymbol{H}_{s} | | |
| lyche-numerical-linear-algebra_FO1724 | 132 | 1.000 | \boldsymbol{A}^{*} \boldsymbol{A}= | | |
| lyche-numerical-linear-algebra_FO1725 | 132 | 1.000 | \boldsymbol{R}_{1}^{*} \boldsymbol{Q}_{1}^{*} \boldsymbol{Q}_{1} \boldsymbol{R}_{1}=\boldsymbol{R}_{1}^{*} \boldsymbol{R}_{1} | | |
| lyche-numerical-linear-algebra_FO1726 | 132 | 1.000 | \boldsymbol{R}_{1}^{*} \boldsymbol{R}_{1} | | |
| lyche-numerical-linear-algebra_FO1727 | 132 | 1.000 | \boldsymbol{Q}_{1}=\boldsymbol{A} \boldsymbol{R}_{1}^{-1} | | |
| lyche-numerical-linear-algebra_FO1728 | 132 | 1.000 | \boldsymbol{B}:=\boldsymbol{A}^{T} \boldsymbol{A}=\left[\begin{array}{cc}5 & -4 \\ -4 & 5\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1729 | 132 | 1.000 | \boldsymbol{B}=\boldsymbol{R}^{T} \boldsymbol{R} | | |
| lyche-numerical-linear-algebra_FO1730 | 132 | 1.000 | \boldsymbol{R}=\frac{1}{\sqrt{5}}\left[\begin{array}{cc}5 & -4 \\ 0 & 3\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1731 | 132 | 1.000 | \boldsymbol{R}^{-1}=\frac{1}{3 \sqrt{5}}\left[\begin{array}{ll}3 & 4 \\ 0 & 5\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1732 | 132 | 0.889 | \boldsymbol{Q}=\boldsymbol{A} \boldsymbol{R}^{-1}=\frac{1}{\sqrt{5}}\left[\begin{array}{cc}2 & 1 \\ -1 & 2\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1733 | 133 | 0.992 | \boldsymbol{a}_{1}, \ldots, \boldsymbol{a}_{n} | | |
| lyche-numerical-linear-algebra_FO1734 | 133 | 1.000 | \mathcal{R}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO1735 | 133 | 1.000 | \boldsymbol{A}=\left[\begin{array}{cc}2 & -1 \\ -1 & 2\end{array}\right]=\left[\boldsymbol{a}_{1}, \boldsymbol{a}_{2}\right] | | |
| lyche-numerical-linear-algebra_FO1736 | 133 | 0.999 | \boldsymbol{v}_{1}=\boldsymbol{a}_{1} | | |
| lyche-numerical-linear-algebra_FO1737 | 133 | 0.999 | \boldsymbol{v}_{2}=\boldsymbol{a}_{2}-\frac{\boldsymbol{a}_{2}^{T} \boldsymbol{v}_{1}}{\boldsymbol{v}_{1}^{T} \boldsymbol{v}_{1}} \boldsymbol{v}_{1}=\frac{3}{5}\left[\begin{array}{l}1 \\ 2\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1738 | 133 | 0.999 | \boldsymbol{Q}=\left[\boldsymbol{q}_{1}, \boldsymbol{q}_{2}\right] | | |
| lyche-numerical-linear-algebra_FO1739 | 133 | 1.000 | \boldsymbol{q}_{1}=\frac{1}{\sqrt{5}}\left[\begin{array}{c}2 \\ -1\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1740 | 133 | 1.000 | \boldsymbol{q}_{2}=\frac{1}{\sqrt{5}}\left[\begin{array}{l}1 \\ 2\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1741 | 134 | 1.000 | n=4 | | |
| lyche-numerical-linear-algebra_FO1742 | 134 | 1.000 | \boldsymbol{P} \in \mathbb{R}^{2,2} | | |
| lyche-numerical-linear-algebra_FO1743 | 134 | 1.000 | \theta \in[0,2 \pi) | | |
| lyche-numerical-linear-algebra_FO1744 | 134 | 1.000 | c=\cos \theta | | |
| lyche-numerical-linear-algebra_FO1745 | 134 | 1.000 | s=\sin \theta | | |
| lyche-numerical-linear-algebra_FO1746 | 134 | 1.000 | \theta=0 | | |
| lyche-numerical-linear-algebra_FO1747 | 134 | 1.000 | \boldsymbol{P}=\boldsymbol{R} | | |
| lyche-numerical-linear-algebra_FO1748 | 135 | 1.000 | \boldsymbol{P}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO1749 | 135 | 1.000 | 1 \leq i<j \leq n | | |
| lyche-numerical-linear-algebra_FO1750 | 135 | 0.998 | \boldsymbol{P}_{i j}=\left(p_{k l}\right) \in \mathbb{R}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO1751 | 135 | 0.998 | p_{k l}=\delta_{k l} | | |
| lyche-numerical-linear-algebra_FO1752 | 135 | 0.998 | i i, j j, i j, j i | | |
| lyche-numerical-linear-algebra_FO1753 | 135 | 1.000 | \boldsymbol{B}=\boldsymbol{P}_{i j} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO1754 | 135 | 1.000 | \boldsymbol{C}=\boldsymbol{A} \boldsymbol{P}_{i j} | | |
| lyche-numerical-linear-algebra_FO1755 | 135 | 1.000 | \boldsymbol{B}(k,:)=\boldsymbol{A}(k,:) | | |
| lyche-numerical-linear-algebra_FO1756 | 135 | 1.000 | \boldsymbol{C}(:, k)=\boldsymbol{A}(:, k) | | |
| lyche-numerical-linear-algebra_FO1757 | 135 | 1.000 | k \neq i, j | | |
| lyche-numerical-linear-algebra_FO1758 | 135 | 0.999 | 2 n^{3} | | |
| lyche-numerical-linear-algebra_FO1759 | 135 | 1.000 | \boldsymbol{P}_{i, i+1} | | |
| lyche-numerical-linear-algebra_FO1760 | 136 | 1.000 | \langle\boldsymbol{x}, \boldsymbol{y}\rangle:=\boldsymbol{y}^{*} \boldsymbol{A}^{*} \boldsymbol{A} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO1761 | 136 | 0.998 | [0, \pi / 2] | | |
| lyche-numerical-linear-algebra_FO1762 | 136 | 1.000 | \boldsymbol{x}=\boldsymbol{e}_{1} | | |
| lyche-numerical-linear-algebra_FO1763 | 136 | 1.000 | \|\boldsymbol{x}\|_{2}=\|\boldsymbol{y}\|_{2} | | |
| lyche-numerical-linear-algebra_FO1764 | 136 | 0.998 | \left(\boldsymbol{I}-2 \frac{\boldsymbol{v} \boldsymbol{v}^{T}}{\boldsymbol{v}^{T} \boldsymbol{v}}\right) \boldsymbol{x}=\boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO1765 | 136 | 1.000 | \boldsymbol{H} \boldsymbol{x}=\boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO1766 | 136 | 0.995 | \boldsymbol{x}=\left[\begin{array}{l}3 \\ 4\end{array}\right], \quad \boldsymbol{y}=\left[\begin{array}{l}5 \\ 0\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1767 | 136 | 1.000 | \boldsymbol{x}=\left[\begin{array}{l}2 \\ 2 \\ 1\end{array}\right], \quad \boldsymbol{y}=\left[\begin{array}{l}0 \\ 3 \\ 0\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1768 | 137 | 1.000 | \boldsymbol{x}=[\cos \phi, \sin \phi]^{T} | | |
| lyche-numerical-linear-algebra_FO1769 | 137 | 0.997 | \boldsymbol{v}:=\boldsymbol{x}-\boldsymbol{y} \neq 0 | | |
| lyche-numerical-linear-algebra_FO1770 | 137 | 1.000 | \boldsymbol{B} \in \mathbb{R}^{4,4} | | |
| lyche-numerical-linear-algebra_FO1771 | 137 | 1.000 | 0<\epsilon<1 | | |
| lyche-numerical-linear-algebra_FO1772 | 137 | 1.000 | \boldsymbol{B}_{1} | | |
| lyche-numerical-linear-algebra_FO1773 | 137 | 0.857 | \boldsymbol{B}_{1}:=\boldsymbol{H} \boldsymbol{B} \boldsymbol{H} | | |
| lyche-numerical-linear-algebra_FO1774 | 138 | 1.000 | \boldsymbol{H}_{1}, \boldsymbol{H}_{2} \in \mathbb{R}^{3 \times 3} | | |
| lyche-numerical-linear-algebra_FO1775 | 138 | 1.000 | \boldsymbol{H}_{2} \boldsymbol{H}_{1} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO1776 | 138 | 0.999 | \boldsymbol{A}=\left[\boldsymbol{a}_{1}, \ldots, \boldsymbol{a}_{n}\right] \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO1777 | 138 | 1.000 | |\operatorname{det}(\boldsymbol{Q})|=1 | | |
| lyche-numerical-linear-algebra_FO1778 | 138 | 1.000 | \boldsymbol{R}=\left[\boldsymbol{r}_{1}, \ldots, \boldsymbol{r}_{n}\right] | | |
| lyche-numerical-linear-algebra_FO1779 | 138 | 0.986 | \left(\boldsymbol{A}^{*} \boldsymbol{A}\right)_{j j}=\left\|\boldsymbol{a}_{j}\right\|_{2}^{2}=\left(\boldsymbol{R}^{*} \boldsymbol{R}\right)_{j j}=\left\|\boldsymbol{r}_{j}\right\|_{2}^{2} | | |
| lyche-numerical-linear-algebra_FO1780 | 138 | 1.000 | |\operatorname{det}(\boldsymbol{A})|=\prod_{j=1}^{n}\left|r_{j j}\right| \leq \prod_{j=1}^{n}\left\|\boldsymbol{a}_{j}\right\|_{2} | | |
| lyche-numerical-linear-algebra_FO1781 | 138 | 0.949 | \boldsymbol{B}=\boldsymbol{L}^{T} \boldsymbol{L} | | |
| lyche-numerical-linear-algebra_FO1782 | 138 | 0.995 | \boldsymbol{B}=\boldsymbol{L} \boldsymbol{L}^{T} | | |
| lyche-numerical-linear-algebra_FO1783 | 138 | 1.000 | l_{i, j} | | |
| lyche-numerical-linear-algebra_FO1784 | 138 | 1.000 | i=n, n-1, \ldots, 1 | | |
| lyche-numerical-linear-algebra_FO1785 | 138 | 1.000 | j=i, 1,2 \ldots, i-1 | | |
| lyche-numerical-linear-algebra_FO1786 | 138 | 1.000 | \boldsymbol{L}^{T} \boldsymbol{L} | | |
| lyche-numerical-linear-algebra_FO1787 | 138 | 1.000 | \boldsymbol{B} \boldsymbol{x}=\boldsymbol{c} | | |
| lyche-numerical-linear-algebra_FO1788 | 138 | 1.000 | \|\boldsymbol{L}\|_{F} | | |
| lyche-numerical-linear-algebra_FO1789 | 138 | 1.000 | \boldsymbol{Q} \boldsymbol{L} | | |
| lyche-numerical-linear-algebra_FO1790 | 138 | 1.000 | \boldsymbol{Q} \in \mathbb{R}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO1791 | 138 | 1.000 | \boldsymbol{a}:=\left[a_{1}, \ldots, a_{n}\right]^{T} \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO1792 | 138 | 1.000 | \boldsymbol{e}_{n}:=[0, \ldots, 0,1]^{T} | | |
| lyche-numerical-linear-algebra_FO1793 | 138 | 0.724 | \boldsymbol{H}:=\boldsymbol{I}-2 \frac{\boldsymbol{v} \boldsymbol{v}^{*}}{\boldsymbol{v}^{*} \boldsymbol{v}} | | |
| lyche-numerical-linear-algebra_FO1794 | 138 | 0.724 | \boldsymbol{H} \boldsymbol{a}=-s \boldsymbol{e}_{n} | | |
| lyche-numerical-linear-algebra_FO1795 | 138 | 1.000 | |s|=\|\boldsymbol{a}\|_{2} | | |
| lyche-numerical-linear-algebra_FO1796 | 138 | 1.000 | s | | |
| lyche-numerical-linear-algebra_FO1797 | 138 | 0.630 | \Longleftrightarrow \boldsymbol{R} | | |
| lyche-numerical-linear-algebra_FO1798 | 138 | 0.630 | \Longleftrightarrow \boldsymbol{A}^{*} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO1799 | 139 | 1.000 | 1 \leq r \leq n, \boldsymbol{v}_{r} \in \mathbb{R}^{r}, \boldsymbol{v}_{r} \neq \mathbf{0} | | |
| lyche-numerical-linear-algebra_FO1800 | 139 | 1.000 | \boldsymbol{H}:=\left[\begin{array}{cc}\boldsymbol{V}_{r} & \mathbf{0} \\ \mathbf{0} & \boldsymbol{I}_{n-r}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1801 | 139 | 1.000 | a_{i, j}=0 | | |
| lyche-numerical-linear-algebra_FO1802 | 139 | 1.000 | i=1, \ldots, r | | |
| lyche-numerical-linear-algebra_FO1803 | 139 | 1.000 | j=r+1, \ldots, n | | |
| lyche-numerical-linear-algebra_FO1804 | 139 | 1.000 | \boldsymbol{H} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO1805 | 139 | 1.000 | \boldsymbol{H}_{1}, \ldots, \boldsymbol{H}_{n-1} | | |
| lyche-numerical-linear-algebra_FO1806 | 139 | 1.000 | \boldsymbol{H}_{n-1}, \ldots, \boldsymbol{H}_{1} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO1807 | 139 | 1.000 | d \leq n-1 | | |
| lyche-numerical-linear-algebra_FO1808 | 139 | 1.000 | \boldsymbol{B}:=\boldsymbol{A}^{T} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO1809 | 139 | 1.000 | \leq 2 d | | |
| lyche-numerical-linear-algebra_FO1810 | 139 | 0.426 | \mathrm{B}= | | |
| lyche-numerical-linear-algebra_FO1811 | 139 | 0.426 | (\mathrm{A}, \mathrm{d}) | | |
| lyche-numerical-linear-algebra_FO1812 | 139 | 1.000 | \mathcal{O}\left(c n^{2}\right) | | |
| lyche-numerical-linear-algebra_FO1813 | 139 | 1.000 | \boldsymbol{A}^{T} \boldsymbol{A}=\boldsymbol{R}^{T} \boldsymbol{R} | | |
| lyche-numerical-linear-algebra_FO1814 | 139 | 1.000 | 2 d | | |
| lyche-numerical-linear-algebra_FO1815 | 140 | 0.928 | \boldsymbol{x}=\left[\begin{array}{c}r \cos \alpha \\ r \sin \alpha\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1816 | 140 | 0.928 | \boldsymbol{P} \boldsymbol{x}= | | |
| lyche-numerical-linear-algebra_FO1817 | 140 | 0.711 | \left[\begin{array}{l}r \cos (\alpha-\theta) \\ r \sin (\alpha-\theta)\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1818 | 140 | 0.988 | \boldsymbol{H} \in \mathbb{R}^{4,4} | | |
| lyche-numerical-linear-algebra_FO1819 | 140 | 1.000 | \boldsymbol{G}_{1}, \boldsymbol{G}_{2}, \boldsymbol{G}_{3} | | |
| lyche-numerical-linear-algebra_FO1820 | 140 | 1.000 | \boldsymbol{G}_{k}=\boldsymbol{P}_{i, j} | | |
| lyche-numerical-linear-algebra_FO1821 | 140 | 1.000 | i, j, c | | |
| lyche-numerical-linear-algebra_FO1822 | 140 | 0.999 | \boldsymbol{P}_{k, k+1} | | |
| lyche-numerical-linear-algebra_FO1823 | 140 | 0.838 | \boldsymbol{G}:=\left[\begin{array}{rc}c & s \\ -s & c\end{array}\right] \in \mathbb{R}^{2 \times 2} | | |
| lyche-numerical-linear-algebra_FO1824 | 140 | 0.838 | s^{2}+c^{2}=1 | | |
| lyche-numerical-linear-algebra_FO1825 | 141 | 1.000 | \boldsymbol{G} | | |
| lyche-numerical-linear-algebra_FO1826 | 141 | 1.000 | x_{1}, x_{2} \in \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO1827 | 141 | 1.000 | r:=\sqrt{x_{1}^{2}+x_{2}^{2}} | | |
| lyche-numerical-linear-algebra_FO1828 | 141 | 1.000 | y_{1}, y_{2} | | |
| lyche-numerical-linear-algebra_FO1829 | 141 | 1.000 | y_{1}=y_{2} | | |
| lyche-numerical-linear-algebra_FO1830 | 141 | 1.000 | \left[\begin{array}{l}y_{1} \\ y_{2}\end{array}\right]=\boldsymbol{G}\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1831 | 141 | 1.000 | i j | | |
| lyche-numerical-linear-algebra_FO1832 | 141 | 1.000 | \boldsymbol{P}_{i, j} | | |
| lyche-numerical-linear-algebra_FO1833 | 141 | 0.938 | i i, i j, j i, j j | | |
| lyche-numerical-linear-algebra_FO1834 | 141 | 1.000 | \theta \in \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO1835 | 141 | 1.000 | \theta \in(-\pi / 2, \pi / 2] | | |
| lyche-numerical-linear-algebra_FO1836 | 141 | 0.910 | \boldsymbol{P} \boldsymbol{x}= \pm\|\boldsymbol{x}\|_{2} \boldsymbol{e}_{1} | | |
| lyche-numerical-linear-algebra_FO1837 | 141 | 0.910 | \boldsymbol{e}_{1}=(1,0)^{T} | | |
| lyche-numerical-linear-algebra_FO1838 | 141 | 1.000 | \boldsymbol{w} \in \mathbb{R}^{m} | | |
| lyche-numerical-linear-algebra_FO1839 | 141 | 0.481 | (m-1) m | | |
| lyche-numerical-linear-algebra_FO1840 | 141 | 0.989 | \alpha= \pm\|\boldsymbol{w}\|_{2} | | |
| lyche-numerical-linear-algebra_FO1841 | 142 | 1.000 | \boldsymbol{A}_{-} | | |
| lyche-numerical-linear-algebra_FO1842 | 142 | 0.916 | \boldsymbol{A} .{ }^{2} | | |
| lyche-numerical-linear-algebra_FO1843 | 142 | 0.936 | z | | |
| lyche-numerical-linear-algebra_FO1844 | 142 | 0.999 | \boldsymbol{A}+\boldsymbol{z} \boldsymbol{z}^{*} | | |
| lyche-numerical-linear-algebra_FO1845 | 142 | 1.000 | \boldsymbol{R}^{*} \boldsymbol{R} | | |
| lyche-numerical-linear-algebra_FO1846 | 142 | 0.987 | P_{i_{1}, n+1}, P_{i_{2}, n+1}, \ldots, P_{i_{n}, n+1} | | |
| lyche-numerical-linear-algebra_FO1847 | 142 | 1.000 | \boldsymbol{R}^{\prime} | | |
| lyche-numerical-linear-algebra_FO1848 | 142 | 0.996 | \left[\begin{array}{c}\boldsymbol{L}^{*} \\ \boldsymbol{z}^{*}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1849 | 142 | 0.996 | i_{1}, \ldots, i_{n} | | |
| lyche-numerical-linear-algebra_FO1850 | 142 | 1.000 | \boldsymbol{Q}^{T} \boldsymbol{A}_{-} | | |
| lyche-numerical-linear-algebra_FO1851 | 146 | 0.999 | \left(\lambda_{k}, \boldsymbol{x}_{k}\right), k= | | |
| lyche-numerical-linear-algebra_FO1852 | 146 | 0.999 | \left\{\boldsymbol{x}_{1}, \ldots, \boldsymbol{x}_{m}\right\} | | |
| lyche-numerical-linear-algebra_FO1853 | 146 | 0.999 | \sum_{j=1}^{m} c_{j} \boldsymbol{x}_{j}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO1854 | 146 | 1.000 | c_{j} | | |
| lyche-numerical-linear-algebra_FO1855 | 146 | 1.000 | m \geq 2 | | |
| lyche-numerical-linear-algebra_FO1856 | 146 | 1.000 | \sum_{j=1}^{m} c_{j} \lambda_{m} \boldsymbol{x}_{j}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO1857 | 146 | 1.000 | \sum_{j=1}^{m-1} c_{j}\left(\lambda_{j}-\right. | | |
| lyche-numerical-linear-algebra_FO1858 | 146 | 1.000 | \left.\lambda_{m}\right) \boldsymbol{x}_{j}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO1859 | 146 | 1.000 | \lambda_{j}-\lambda_{m} \neq 0 | | |
| lyche-numerical-linear-algebra_FO1860 | 146 | 1.000 | j=1, \ldots, m-1 | | |
| lyche-numerical-linear-algebra_FO1861 | 146 | 1.000 | c_{j} \neq 0 | | |
| lyche-numerical-linear-algebra_FO1862 | 146 | 1.000 | j<m | | |
| lyche-numerical-linear-algebra_FO1863 | 146 | 1.000 | \left\{\boldsymbol{x}_{1}, \ldots, \boldsymbol{x}_{m-1}\right\} | | |
| lyche-numerical-linear-algebra_FO1864 | 146 | 1.000 | \boldsymbol{I} \boldsymbol{x}=\boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO1865 | 146 | 1.000 | \lambda_{1}=\lambda_{2}=1 | | |
| lyche-numerical-linear-algebra_FO1866 | 146 | 1.000 | \boldsymbol{x} \in \mathbb{C}^{2} | | |
| lyche-numerical-linear-algebra_FO1867 | 146 | 1.000 | \boldsymbol{e}_{2} | | |
| lyche-numerical-linear-algebra_FO1868 | 146 | 1.000 | \mathbb{C}^{2} | | |
| lyche-numerical-linear-algebra_FO1869 | 146 | 1.000 | \boldsymbol{J} | | |
| lyche-numerical-linear-algebra_FO1870 | 146 | 1.000 | \boldsymbol{J} \boldsymbol{x}=\boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO1871 | 146 | 1.000 | x_{2}=0 | | |
| lyche-numerical-linear-algebra_FO1872 | 146 | 1.000 | \left[\begin{array}{cc}2 & -1 \\ -1 & 2\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1873 | 146 | 1.000 | \left(1,[1,1]^{T}\right) | | |
| lyche-numerical-linear-algebra_FO1874 | 146 | 1.000 | \left(3,[1,-1]^{T}\right) | | |
| lyche-numerical-linear-algebra_FO1875 | 146 | 1.000 | \boldsymbol{x}=\left[x_{1}, x_{2}\right]^{T} \in \mathbb{C}^{2} | | |
| lyche-numerical-linear-algebra_FO1876 | 147 | 1.000 | \boldsymbol{S} \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO1877 | 147 | 1.000 | \boldsymbol{B}=\boldsymbol{S}^{-1} \boldsymbol{A} \boldsymbol{S} | | |
| lyche-numerical-linear-algebra_FO1878 | 147 | 1.000 | \boldsymbol{A} \rightarrow \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO1879 | 147 | 1.000 | \boldsymbol{s}_{1}, \boldsymbol{s}_{2}, \ldots, \boldsymbol{s}_{n} | | |
| lyche-numerical-linear-algebra_FO1880 | 147 | 1.000 | \operatorname{det}(\boldsymbol{A} \boldsymbol{C})= | | |
| lyche-numerical-linear-algebra_FO1881 | 147 | 1.000 | \operatorname{det}(\boldsymbol{A}) \operatorname{det}(\boldsymbol{C}) | | |
| lyche-numerical-linear-algebra_FO1882 | 147 | 0.866 | \boldsymbol{S}^{-1} \boldsymbol{A} \boldsymbol{S} | | |
| lyche-numerical-linear-algebra_FO1883 | 147 | 0.866 | \lambda, \boldsymbol{S} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO1884 | 147 | 1.000 | \left(\boldsymbol{S}^{-1} \boldsymbol{A} \boldsymbol{S}\right) \boldsymbol{x}=\lambda \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO1885 | 147 | 1.000 | \boldsymbol{A}(\boldsymbol{S} \boldsymbol{x})=\lambda(\boldsymbol{S} \boldsymbol{x}) | | |
| lyche-numerical-linear-algebra_FO1886 | 147 | 1.000 | \boldsymbol{S}^{-1} \boldsymbol{A} \boldsymbol{S}=\boldsymbol{D}=\operatorname{diag}\left(\lambda_{1}, \ldots, \lambda_{n}\right) | | |
| lyche-numerical-linear-algebra_FO1887 | 147 | 1.000 | \boldsymbol{A} \boldsymbol{S}=\boldsymbol{S} \boldsymbol{D} | | |
| lyche-numerical-linear-algebra_FO1888 | 147 | 1.000 | \left[\boldsymbol{A} \boldsymbol{s}_{1}, \ldots, \boldsymbol{A} \boldsymbol{s}_{n}\right]=\left[\lambda_{1} \boldsymbol{s}_{1}, \ldots, \lambda_{n} \boldsymbol{s}_{n}\right] | | |
| lyche-numerical-linear-algebra_FO1889 | 147 | 1.000 | \boldsymbol{A}, \boldsymbol{C} \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO1890 | 147 | 1.000 | \boldsymbol{A} \boldsymbol{C} | | |
| lyche-numerical-linear-algebra_FO1891 | 147 | 1.000 | \boldsymbol{C} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO1892 | 147 | 1.000 | \boldsymbol{C} \in \mathbb{C}^{n \times m} | | |
| lyche-numerical-linear-algebra_FO1893 | 147 | 1.000 | n+m | | |
| lyche-numerical-linear-algebra_FO1894 | 147 | 0.997 | \boldsymbol{S}^{-1}=\left[\begin{array}{cc}\boldsymbol{I} & -\boldsymbol{A} \\ \mathbf{0} & \boldsymbol{I}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1895 | 147 | 0.997 | \boldsymbol{E} \boldsymbol{S}=\boldsymbol{S} \boldsymbol{F} | | |
| lyche-numerical-linear-algebra_FO1896 | 147 | 1.000 | \boldsymbol{F} | | |
| lyche-numerical-linear-algebra_FO1897 | 147 | 1.000 | \pi_{\boldsymbol{E}}(\lambda)=\lambda^{n} \pi_{\boldsymbol{A} \boldsymbol{C}}(\lambda)=\pi_{\boldsymbol{F}}(\lambda)=\lambda^{m} \pi_{\boldsymbol{C} \boldsymbol{A}}(\lambda) | | |
| lyche-numerical-linear-algebra_FO1898 | 148 | 1.000 | \lambda_{1}, \ldots, \lambda_{k} | | |
| lyche-numerical-linear-algebra_FO1899 | 148 | 1.000 | a_{1}, \ldots, a_{k} | | |
| lyche-numerical-linear-algebra_FO1900 | 148 | 0.890 | a_{i}=a\left(\lambda_{i}\right)=a_{\boldsymbol{A}}\left(\lambda_{i}\right) | | |
| lyche-numerical-linear-algebra_FO1901 | 148 | 1.000 | \lambda_{i} | | |
| lyche-numerical-linear-algebra_FO1902 | 148 | 1.000 | a_{i} | | |
| lyche-numerical-linear-algebra_FO1903 | 148 | 1.000 | \lambda \in \sigma(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO1904 | 148 | 1.000 | g=g(\lambda)= | | |
| lyche-numerical-linear-algebra_FO1905 | 148 | 1.000 | g_{\boldsymbol{A}}(\lambda) | | |
| lyche-numerical-linear-algebra_FO1906 | 148 | 1.000 | \mathcal{N}(\boldsymbol{A}-\lambda \boldsymbol{I}) | | |
| lyche-numerical-linear-algebra_FO1907 | 148 | 1.000 | \lambda=1 | | |
| lyche-numerical-linear-algebra_FO1908 | 148 | 1.000 | \pi_{\boldsymbol{I}}(\lambda)=(1-\lambda)^{n} | | |
| lyche-numerical-linear-algebra_FO1909 | 148 | 1.000 | \boldsymbol{I}-\lambda \boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO1910 | 148 | 1.000 | a=g=n | | |
| lyche-numerical-linear-algebra_FO1911 | 148 | 0.985 | \boldsymbol{J}:=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1912 | 148 | 1.000 | a=2 | | |
| lyche-numerical-linear-algebra_FO1913 | 148 | 1.000 | g=1 | | |
| lyche-numerical-linear-algebra_FO1914 | 148 | 0.991 | \operatorname{dim} \mathcal{N}\left(\boldsymbol{S}^{-1} \boldsymbol{A} \boldsymbol{S}-\lambda \boldsymbol{I}\right)=k | | |
| lyche-numerical-linear-algebra_FO1915 | 148 | 0.991 | \operatorname{dim} \mathcal{N}(\boldsymbol{A}-\lambda \boldsymbol{I})=\ell | | |
| lyche-numerical-linear-algebra_FO1916 | 148 | 1.000 | k=\ell | | |
| lyche-numerical-linear-algebra_FO1917 | 148 | 1.000 | \boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{k} | | |
| lyche-numerical-linear-algebra_FO1918 | 148 | 1.000 | \mathcal{N}\left(\boldsymbol{S}^{-1} \boldsymbol{A} \boldsymbol{S}-\lambda \boldsymbol{I}\right) | | |
| lyche-numerical-linear-algebra_FO1919 | 148 | 1.000 | \boldsymbol{S}^{-1} \boldsymbol{A} \boldsymbol{S} \boldsymbol{v}_{i}=\lambda \boldsymbol{v}_{i} | | |
| lyche-numerical-linear-algebra_FO1920 | 148 | 1.000 | \boldsymbol{A} \boldsymbol{S} \boldsymbol{v}_{i}=\lambda \boldsymbol{S} \boldsymbol{v}_{i}, i=1, \ldots, k | | |
| lyche-numerical-linear-algebra_FO1921 | 148 | 1.000 | \left\{\boldsymbol{S} \boldsymbol{v}_{1}, \ldots, \boldsymbol{S} \boldsymbol{v}_{k}\right\} \subset \mathcal{N}(\boldsymbol{A}-\lambda \boldsymbol{I}) | | |
| lyche-numerical-linear-algebra_FO1922 | 148 | 1.000 | k \leq \ell | | |
| lyche-numerical-linear-algebra_FO1923 | 148 | 1.000 | \boldsymbol{w}_{1}, \ldots, \boldsymbol{w}_{\ell} | | |
| lyche-numerical-linear-algebra_FO1924 | 148 | 1.000 | \left\{\boldsymbol{S}^{-1} \boldsymbol{w}_{1}, \ldots, \boldsymbol{S}^{-1} \boldsymbol{w}_{\ell}\right\} \subset \mathcal{N}\left(\boldsymbol{S}^{-1} \boldsymbol{A} \boldsymbol{S}-\lambda \boldsymbol{I}\right) | | |
| lyche-numerical-linear-algebra_FO1925 | 148 | 1.000 | k \geq \ell | | |
| lyche-numerical-linear-algebra_FO1926 | 149 | 1.000 | \boldsymbol{J}_{m}(\lambda) | | |
| lyche-numerical-linear-algebra_FO1927 | 149 | 0.764 | \boldsymbol{J}_{3}(\lambda)=\left[\begin{array}{ccc}\lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1928 | 149 | 1.000 | \boldsymbol{J}_{m}(\lambda) \boldsymbol{v}=\lambda \boldsymbol{v} | | |
| lyche-numerical-linear-algebra_FO1929 | 149 | 1.000 | \boldsymbol{v}=\left[v_{1}, \ldots, v_{m}\right] | | |
| lyche-numerical-linear-algebra_FO1930 | 149 | 1.000 | \lambda v_{i-1}+v_{i}=\lambda v_{i-1} | | |
| lyche-numerical-linear-algebra_FO1931 | 149 | 1.000 | i=2, \ldots, m | | |
| lyche-numerical-linear-algebra_FO1932 | 149 | 1.000 | v_{2}=\cdots=v_{m}=0 | | |
| lyche-numerical-linear-algebra_FO1933 | 149 | 0.997 | a=m | | |
| lyche-numerical-linear-algebra_FO1934 | 149 | 1.000 | g_{1}, \ldots, g_{k} | | |
| lyche-numerical-linear-algebra_FO1935 | 150 | 1.000 | \boldsymbol{U}_{i} | | |
| lyche-numerical-linear-algebra_FO1936 | 150 | 1.000 | g_{i} | | |
| lyche-numerical-linear-algebra_FO1937 | 150 | 1.000 | m_{i, 1}, \ldots, m_{i, g_{i}} | | |
| lyche-numerical-linear-algebra_FO1938 | 150 | 1.000 | m_{i, 1} \geq m_{i, 2} \geq \cdots \geq m_{i, g_{i}} | | |
| lyche-numerical-linear-algebra_FO1939 | 150 | 1.000 | a_{i}=\sum_{j=1}^{g_{i}} m_{i, j} | | |
| lyche-numerical-linear-algebra_FO1940 | 150 | 1.000 | \boldsymbol{A} \boldsymbol{S}=\boldsymbol{S} \boldsymbol{J} | | |
| lyche-numerical-linear-algebra_FO1941 | 150 | 1.000 | \boldsymbol{U}_{1}=\operatorname{diag}\left(\boldsymbol{J}_{3}(2), \boldsymbol{J}_{2}(2), \boldsymbol{J}_{1}(2)\right) | | |
| lyche-numerical-linear-algebra_FO1942 | 150 | 1.000 | \boldsymbol{U}_{2}=\boldsymbol{J}_{2}(3) | | |
| lyche-numerical-linear-algebra_FO1943 | 150 | 1.000 | \lambda=2 | | |
| lyche-numerical-linear-algebra_FO1944 | 150 | 1.000 | \boldsymbol{S}=\left[\boldsymbol{s}_{1}, \ldots, \boldsymbol{s}_{8}\right] | | |
| lyche-numerical-linear-algebra_FO1945 | 151 | 1.000 | \boldsymbol{J}_{2}(3), \boldsymbol{J}_{2}(2), \boldsymbol{J}_{1}(2), \boldsymbol{J}_{3}(2) | | |
| lyche-numerical-linear-algebra_FO1946 | 151 | 1.000 | m_{i, j} \geq 1 | | |
| lyche-numerical-linear-algebra_FO1947 | 151 | 1.000 | g_{i} \leq a_{i} | | |
| lyche-numerical-linear-algebra_FO1948 | 151 | 1.000 | \boldsymbol{e}_{1}, \boldsymbol{e}_{4}, \boldsymbol{e}_{6}, \boldsymbol{e}_{7} | | |
| lyche-numerical-linear-algebra_FO1949 | 151 | 1.000 | k=2 | | |
| lyche-numerical-linear-algebra_FO1950 | 151 | 1.000 | g_{1}+g_{2}= | | |
| lyche-numerical-linear-algebra_FO1951 | 151 | 1.000 | 3+1=4 | | |
| lyche-numerical-linear-algebra_FO1952 | 151 | 1.000 | \sum_{i} a_{i}=n | | |
| lyche-numerical-linear-algebra_FO1953 | 151 | 1.000 | \sum_{i} g_{i}=n | | |
| lyche-numerical-linear-algebra_FO1954 | 151 | 1.000 | a_{i}=g_{i} | | |
| lyche-numerical-linear-algebra_FO1955 | 151 | 1.000 | i=1, \ldots, k | | |
| lyche-numerical-linear-algebra_FO1956 | 151 | 1.000 | \boldsymbol{S}=\boldsymbol{U} | | |
| lyche-numerical-linear-algebra_FO1957 | 151 | 1.000 | \boldsymbol{S}^{-1}=\boldsymbol{U}^{*} | | |
| lyche-numerical-linear-algebra_FO1958 | 151 | 1.000 | \boldsymbol{U}^{*} \boldsymbol{A} \boldsymbol{U} | | |
| lyche-numerical-linear-algebra_FO1959 | 152 | 1.000 | \boldsymbol{R}:=\boldsymbol{U}^{*} \boldsymbol{A} \boldsymbol{U} | | |
| lyche-numerical-linear-algebra_FO1960 | 152 | 1.000 | \boldsymbol{A}=\boldsymbol{U} \boldsymbol{R} \boldsymbol{U}^{*} | | |
| lyche-numerical-linear-algebra_FO1961 | 152 | 1.000 | k \times k | | |
| lyche-numerical-linear-algebra_FO1962 | 152 | 0.993 | n:=k+1 | | |
| lyche-numerical-linear-algebra_FO1963 | 152 | 0.993 | \left(\lambda_{1}, \boldsymbol{v}_{1}\right) | | |
| lyche-numerical-linear-algebra_FO1964 | 152 | 0.993 | \left\|\boldsymbol{v}_{1}\right\|_{2}=1 | | |
| lyche-numerical-linear-algebra_FO1965 | 152 | 1.000 | \left\{\boldsymbol{v}_{1}, \boldsymbol{v}_{2}, \ldots, \boldsymbol{v}_{n}\right\} | | |
| lyche-numerical-linear-algebra_FO1966 | 152 | 1.000 | \boldsymbol{V}:= | | |
| lyche-numerical-linear-algebra_FO1967 | 152 | 1.000 | \left[\boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{n}\right] \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO1968 | 152 | 1.000 | \boldsymbol{W}_{1} \in \mathbb{C}^{(n-1) \times(n-1)} | | |
| lyche-numerical-linear-algebra_FO1969 | 152 | 1.000 | \boldsymbol{W}_{1}^{*} \boldsymbol{M} \boldsymbol{W}_{1} | | |
| lyche-numerical-linear-algebra_FO1970 | 152 | 1.000 | \boldsymbol{W} | | |
| lyche-numerical-linear-algebra_FO1971 | 152 | 0.570 | \boldsymbol{U}^{T} \boldsymbol{A U} | | |
| lyche-numerical-linear-algebra_FO1972 | 152 | 1.000 | \lambda_{1} | | |
| lyche-numerical-linear-algebra_FO1973 | 152 | 1.000 | \boldsymbol{W}^{T} \boldsymbol{W}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO1974 | 152 | 1.000 | \boldsymbol{V} | | |
| lyche-numerical-linear-algebra_FO1975 | 152 | 1.000 | \boldsymbol{V}^{T} \boldsymbol{V}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO1976 | 152 | 1.000 | \boldsymbol{U}=\boldsymbol{V} \boldsymbol{W} | | |
| lyche-numerical-linear-algebra_FO1977 | 152 | 1.000 | \boldsymbol{U}^{T} \boldsymbol{U}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO1978 | 153 | 1.000 | n-1 . \boldsymbol{M} | | |
| lyche-numerical-linear-algebra_FO1979 | 153 | 1.000 | \boldsymbol{T}:=\left[\begin{array}{ccc}2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO1980 | 153 | 0.675 | \left(2, \boldsymbol{x}_{1}\right) | | |
| lyche-numerical-linear-algebra_FO1981 | 153 | 0.675 | \boldsymbol{x}_{1}=[-1,0,1]^{T} | | |
| lyche-numerical-linear-algebra_FO1982 | 153 | 1.000 | \boldsymbol{x}_{1} | | |
| lyche-numerical-linear-algebra_FO1983 | 153 | 1.000 | \left\{\boldsymbol{x}_{1}, \boldsymbol{x}_{2}, \boldsymbol{x}_{3}\right\} | | |
| lyche-numerical-linear-algebra_FO1984 | 153 | 1.000 | \boldsymbol{x}_{2}=[0,1,0]^{T} | | |
| lyche-numerical-linear-algebra_FO1985 | 153 | 1.000 | \boldsymbol{x}_{3}=[1,0,1]^{T} | | |
| lyche-numerical-linear-algebra_FO1986 | 153 | 1.000 | \boldsymbol{A}^{*} \boldsymbol{A}=\boldsymbol{A} \boldsymbol{A}^{*} | | |
| lyche-numerical-linear-algebra_FO1987 | 153 | 1.000 | \boldsymbol{A}^{*}=-\boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO1988 | 153 | 1.000 | \boldsymbol{A}^{*}=\boldsymbol{A}^{-1} | | |
| lyche-numerical-linear-algebra_FO1989 | 153 | 0.999 | \boldsymbol{A}=\operatorname{diag}\left(d_{1}, \ldots, d_{n}\right) | | |
| lyche-numerical-linear-algebra_FO1990 | 154 | 0.989 | \boldsymbol{U}^{*} \boldsymbol{A} \boldsymbol{U}=\boldsymbol{D} | | |
| lyche-numerical-linear-algebra_FO1991 | 154 | 1.000 | \boldsymbol{D}=\operatorname{diag}\left(\lambda_{1}, \ldots, \lambda_{n}\right) | | |
| lyche-numerical-linear-algebra_FO1992 | 154 | 1.000 | \boldsymbol{U}=\left[\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{n}\right] | | |
| lyche-numerical-linear-algebra_FO1993 | 154 | 1.000 | \left(\lambda_{j}, \boldsymbol{u}_{j}\right), j= | | |
| lyche-numerical-linear-algebra_FO1994 | 154 | 0.996 | \boldsymbol{B}=\boldsymbol{U}^{*} \boldsymbol{A} \boldsymbol{U} | | |
| lyche-numerical-linear-algebra_FO1995 | 154 | 0.996 | \boldsymbol{A}=\boldsymbol{U} \boldsymbol{B} \boldsymbol{U}^{*} | | |
| lyche-numerical-linear-algebra_FO1996 | 154 | 1.000 | \boldsymbol{B} \boldsymbol{B}^{*}=\boldsymbol{B}^{*} \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO1997 | 154 | 1.000 | \boldsymbol{B}:=\boldsymbol{U}^{*} \boldsymbol{A} \boldsymbol{U} | | |
| lyche-numerical-linear-algebra_FO1998 | 154 | 0.458 | e_{i i} | | |
| lyche-numerical-linear-algebra_FO1999 | 154 | 0.458 | \boldsymbol{E}:=\boldsymbol{B}^{*} \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO2000 | 154 | 0.458 | f_{i i} | | |
| lyche-numerical-linear-algebra_FO2001 | 154 | 0.458 | \boldsymbol{F}:=\boldsymbol{B B}^{*} | | |
| lyche-numerical-linear-algebra_FO2002 | 154 | 1.000 | \left|b_{11}\right|^{2}=\left|b_{11}\right|^{2}+\left|b_{12}\right|^{2}+\cdots+\left|b_{1 n}\right|^{2} | | |
| lyche-numerical-linear-algebra_FO2003 | 154 | 1.000 | b_{1 k}=0 | | |
| lyche-numerical-linear-algebra_FO2004 | 154 | 1.000 | k=2,3, \ldots, n | | |
| lyche-numerical-linear-algebra_FO2005 | 154 | 1.000 | i-1 | | |
| lyche-numerical-linear-algebra_FO2006 | 154 | 1.000 | b_{j k}=0 | | |
| lyche-numerical-linear-algebra_FO2007 | 154 | 1.000 | j=1, \ldots, i-1 | | |
| lyche-numerical-linear-algebra_FO2008 | 154 | 0.999 | k=j+1, \ldots, n | | |
| lyche-numerical-linear-algebra_FO2009 | 154 | 1.000 | b_{i k}=0, k=i+1, \ldots, n | | |
| lyche-numerical-linear-algebra_FO2010 | 154 | 0.877 | \boldsymbol{U}^{T} \boldsymbol{A} \boldsymbol{U}= | | |
| lyche-numerical-linear-algebra_FO2011 | 154 | 0.987 | \operatorname{diag}(1,3) | | |
| lyche-numerical-linear-algebra_FO2012 | 154 | 0.987 | \boldsymbol{U}=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1 & 1 \\ 1 & -1\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2013 | 155 | 0.835 | R(\boldsymbol{x})=\frac{\boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{x}}{\boldsymbol{x}^{*} \boldsymbol{x}}=\lambda | | |
| lyche-numerical-linear-algebra_FO2014 | 155 | 0.761 | \lambda_{j}, \boldsymbol{u}_{j} | | |
| lyche-numerical-linear-algebra_FO2015 | 155 | 0.761 | j=1,2, \ldots, n | | |
| lyche-numerical-linear-algebra_FO2016 | 155 | 1.000 | \boldsymbol{x}^{*} \boldsymbol{x}=\sum_{i=1}^{n} \sum_{j=1}^{n} \bar{c}_{i} \overline{\boldsymbol{u}}_{i} c_{j} \boldsymbol{u}_{j}= | | |
| lyche-numerical-linear-algebra_FO2017 | 155 | 1.000 | \sum_{j=1}^{n}\left|c_{j}\right|^{2} | | |
| lyche-numerical-linear-algebra_FO2018 | 155 | 1.000 | \boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{x}=\sum_{i=1}^{n} \sum_{j=1}^{n} \bar{c}_{i} \overline{\boldsymbol{u}}_{i} c_{j} \lambda_{j} \boldsymbol{u}_{j}=\sum_{i=1}^{n} \lambda_{i}\left|c_{i}\right|^{2} | | |
| lyche-numerical-linear-algebra_FO2019 | 155 | 1.000 | \sum_{i=1}^{n}\left|c_{i}\right|^{2} / \sum_{j=1}^{n}\left|c_{j}\right|^{2}=1 | | |
| lyche-numerical-linear-algebra_FO2020 | 155 | 0.998 | \lambda=\mu+i \nu, \bar{\lambda}=\mu-i \nu | | |
| lyche-numerical-linear-algebra_FO2021 | 155 | 0.758 | \mu, \nu | | |
| lyche-numerical-linear-algebra_FO2022 | 155 | 0.999 | \lambda=\mu+i \nu | | |
| lyche-numerical-linear-algebra_FO2023 | 155 | 0.999 | \bar{\lambda}=\mu-i \nu | | |
| lyche-numerical-linear-algebra_FO2024 | 156 | 1.000 | \pi_{\boldsymbol{R}}= | | |
| lyche-numerical-linear-algebra_FO2025 | 156 | 0.831 | \boldsymbol{\pi}_{\boldsymbol{D}_{1}} \boldsymbol{\pi}_{\boldsymbol{D}_{2}} \boldsymbol{\pi}_{\boldsymbol{D}_{3}} | | |
| lyche-numerical-linear-algebra_FO2026 | 156 | 1.000 | \boldsymbol{D}_{1} | | |
| lyche-numerical-linear-algebra_FO2027 | 156 | 1.000 | \boldsymbol{D}_{3} | | |
| lyche-numerical-linear-algebra_FO2028 | 156 | 1.000 | \lambda_{1}=2+i, \lambda_{2}=2-i | | |
| lyche-numerical-linear-algebra_FO2029 | 156 | 1.000 | \boldsymbol{D}_{2} | | |
| lyche-numerical-linear-algebra_FO2030 | 156 | 1.000 | \left\{\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{n}\right\} | | |
| lyche-numerical-linear-algebra_FO2031 | 156 | 1.000 | \boldsymbol{A} \boldsymbol{u}_{j}=\lambda_{j} \boldsymbol{u}_{j} | | |
| lyche-numerical-linear-algebra_FO2032 | 156 | 0.995 | \lambda_{1}, \lambda_{2}, \ldots, \lambda_{n} | | |
| lyche-numerical-linear-algebra_FO2033 | 157 | 0.999 | R(\boldsymbol{x})=R_{\boldsymbol{A}}(\boldsymbol{x}):=\frac{\boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{x}}{\boldsymbol{x}^{*} \boldsymbol{x}} | | |
| lyche-numerical-linear-algebra_FO2034 | 157 | 0.924 | \left(\lambda_{j}, \boldsymbol{u}_{j}\right), 1 \leq j \leq n | | |
| lyche-numerical-linear-algebra_FO2035 | 157 | 0.924 | \lambda_{1} \geq \cdots \geq \lambda_{n} | | |
| lyche-numerical-linear-algebra_FO2036 | 157 | 0.924 | 1 \leq k \leq n | | |
| lyche-numerical-linear-algebra_FO2037 | 157 | 1.000 | \mathcal{S}=\tilde{\mathcal{S}}:=\operatorname{span}\left(\boldsymbol{u}_{k}, \ldots, \boldsymbol{u}_{n}\right) | | |
| lyche-numerical-linear-algebra_FO2038 | 157 | 1.000 | \boldsymbol{x}=\boldsymbol{u}_{k} | | |
| lyche-numerical-linear-algebra_FO2039 | 157 | 1.000 | \mathcal{S}^{\prime}:= | | |
| lyche-numerical-linear-algebra_FO2040 | 157 | 1.000 | \operatorname{span}\left(\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{k}\right) | | |
| lyche-numerical-linear-algebra_FO2041 | 157 | 1.000 | \boldsymbol{y} \in \mathcal{S} | | |
| lyche-numerical-linear-algebra_FO2042 | 157 | 1.000 | R(\boldsymbol{y}) \geq \lambda_{k} | | |
| lyche-numerical-linear-algebra_FO2043 | 157 | 1.000 | \mathcal{S}+\mathcal{S}^{\prime}:= | | |
| lyche-numerical-linear-algebra_FO2044 | 157 | 1.000 | \left\{\boldsymbol{s}+\boldsymbol{s}^{\prime}: \boldsymbol{s} \in \mathcal{S}, \boldsymbol{s}^{\prime} \in \mathcal{S}^{\prime}\right\} | | |
| lyche-numerical-linear-algebra_FO2045 | 157 | 0.996 | \mathcal{S} \cap \mathcal{S}^{\prime} | | |
| lyche-numerical-linear-algebra_FO2046 | 157 | 0.996 | \boldsymbol{y} \in \mathcal{S} \cap \mathcal{S}^{\prime}=\sum_{j=1}^{k} c_{j} \boldsymbol{u}_{j} | | |
| lyche-numerical-linear-algebra_FO2047 | 157 | 0.996 | \sum_{j=1}^{k}\left|c_{j}\right|^{2}= | | |
| lyche-numerical-linear-algebra_FO2048 | 157 | 0.997 | c_{j}=0 | | |
| lyche-numerical-linear-algebra_FO2049 | 157 | 0.997 | k+1 \leq j \leq n | | |
| lyche-numerical-linear-algebra_FO2050 | 157 | 0.877 | \boldsymbol{z} \in \mathcal{S}=\tilde{\mathcal{S}} | | |
| lyche-numerical-linear-algebra_FO2051 | 157 | 0.877 | \boldsymbol{z}=\sum_{j=k}^{n} d_{j} \boldsymbol{u}_{j} | | |
| lyche-numerical-linear-algebra_FO2052 | 157 | 0.999 | d_{k}, \ldots, d_{n} | | |
| lyche-numerical-linear-algebra_FO2053 | 157 | 0.999 | \sum_{j=k}^{n}\left|d_{j}\right|^{2}=1 | | |
| lyche-numerical-linear-algebra_FO2054 | 157 | 0.999 | 6.8 R(z)=\sum_{j=k}^{n} \lambda_{j}\left|d_{j}\right|^{2} \leq | | |
| lyche-numerical-linear-algebra_FO2055 | 157 | 0.959 | \lambda_{k} | | |
| lyche-numerical-linear-algebra_FO2056 | 157 | 0.959 | \boldsymbol{z} \in \tilde{\mathcal{S}} | | |
| lyche-numerical-linear-algebra_FO2057 | 157 | 0.959 | \max _{\substack{\boldsymbol{x} \in \tilde{\mathcal{S}} \\ \boldsymbol{x} \neq \mathbf{0}}} R(\boldsymbol{x}) \leq \lambda_{k} | | |
| lyche-numerical-linear-algebra_FO2058 | 157 | 1.000 | \mathcal{S}=\tilde{\mathcal{S}} | | |
| lyche-numerical-linear-algebra_FO2059 | 157 | 1.000 | R\left(\boldsymbol{u}_{k}\right)=\lambda_{k} | | |
| lyche-numerical-linear-algebra_FO2060 | 158 | 1.000 | \mathcal{S}=\tilde{\mathcal{S}}:=\operatorname{span}\left(\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{k}\right) | | |
| lyche-numerical-linear-algebra_FO2061 | 158 | 1.000 | \left(\lambda_{j}, \boldsymbol{u}_{j}\right), 1 \leq j \leq | | |
| lyche-numerical-linear-algebra_FO2062 | 158 | 1.000 | \operatorname{span}\left(\boldsymbol{u}_{k}, \ldots, \boldsymbol{u}_{n}\right) | | |
| lyche-numerical-linear-algebra_FO2063 | 158 | 1.000 | R(\boldsymbol{y}) \leq \lambda_{k} | | |
| lyche-numerical-linear-algebra_FO2064 | 158 | 1.000 | \boldsymbol{y} \in \mathcal{S} \cap \mathcal{S}^{\prime} | | |
| lyche-numerical-linear-algebra_FO2065 | 158 | 1.000 | \boldsymbol{y} \in \tilde{\mathcal{S}} | | |
| lyche-numerical-linear-algebra_FO2066 | 158 | 0.995 | \boldsymbol{A}, \boldsymbol{B} \in | | |
| lyche-numerical-linear-algebra_FO2067 | 158 | 0.998 | \alpha_{1} \geq \alpha_{2} \geq \cdots \geq \alpha_{n} | | |
| lyche-numerical-linear-algebra_FO2068 | 158 | 0.998 | \beta_{1} \geq \beta_{2} \geq \cdots \geq \beta_{n} | | |
| lyche-numerical-linear-algebra_FO2069 | 158 | 1.000 | \varepsilon_{1} \geq \varepsilon_{2} \geq \cdots \geq \varepsilon_{n} | | |
| lyche-numerical-linear-algebra_FO2070 | 158 | 1.000 | \boldsymbol{E}:=\boldsymbol{B}-\boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO2071 | 158 | 0.997 | \left(\alpha_{j}, \boldsymbol{u}_{j}\right), j=1, \ldots, n | | |
| lyche-numerical-linear-algebra_FO2072 | 158 | 1.000 | \mathcal{S}:=\operatorname{span}\left\{\boldsymbol{u}_{k}, \ldots, \boldsymbol{u}_{n}\right\} | | |
| lyche-numerical-linear-algebra_FO2073 | 158 | 1.000 | \boldsymbol{D}:=-\boldsymbol{E} | | |
| lyche-numerical-linear-algebra_FO2074 | 158 | 1.000 | -\varepsilon_{n} | | |
| lyche-numerical-linear-algebra_FO2075 | 158 | 1.000 | \boldsymbol{A}=\boldsymbol{B}+\boldsymbol{D} | | |
| lyche-numerical-linear-algebra_FO2076 | 158 | 1.000 | \alpha_{k} \leq \beta_{k}-\varepsilon_{n} | | |
| lyche-numerical-linear-algebra_FO2077 | 159 | 1.000 | \mu_{1}, \ldots, \mu_{n} | | |
| lyche-numerical-linear-algebra_FO2078 | 159 | 0.655 | \boldsymbol{y} \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO2079 | 159 | 0.655 | \lambda, \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO2080 | 159 | 0.979 | \boldsymbol{y}^{*} \boldsymbol{A}=\lambda \boldsymbol{y}^{*} | | |
| lyche-numerical-linear-algebra_FO2081 | 159 | 0.979 | \boldsymbol{A}^{*} \boldsymbol{y}=\bar{\lambda} \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO2082 | 159 | 0.988 | \boldsymbol{A} \boldsymbol{y}=\lambda \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO2083 | 159 | 0.988 | (\lambda, \boldsymbol{y}) | | |
| lyche-numerical-linear-algebra_FO2084 | 159 | 0.779 | (\lambda, y) | | |
| lyche-numerical-linear-algebra_FO2085 | 159 | 0.999 | \bar{\lambda} | | |
| lyche-numerical-linear-algebra_FO2086 | 159 | 1.000 | \left\{\boldsymbol{y}_{1}, \ldots, \boldsymbol{y}_{n}\right\} | | |
| lyche-numerical-linear-algebra_FO2087 | 159 | 1.000 | \boldsymbol{y}_{i}^{*} \boldsymbol{x}_{j}=\delta_{i, j} | | |
| lyche-numerical-linear-algebra_FO2088 | 160 | 1.000 | \left(\lambda_{1}, \boldsymbol{x}_{1}\right), \ldots,\left(\lambda_{n}, \boldsymbol{x}_{n}\right) | | |
| lyche-numerical-linear-algebra_FO2089 | 160 | 0.998 | \left(\lambda_{1}, \boldsymbol{y}_{1}\right), \ldots,\left(\lambda_{n}, \boldsymbol{y}_{n}\right) | | |
| lyche-numerical-linear-algebra_FO2090 | 160 | 0.998 | \boldsymbol{A} \boldsymbol{X}=\boldsymbol{X} \boldsymbol{D}, \boldsymbol{Y}^{*} \boldsymbol{A}=\boldsymbol{D} \boldsymbol{Y}^{*} | | |
| lyche-numerical-linear-algebra_FO2091 | 160 | 1.000 | \boldsymbol{A} \boldsymbol{X}=\boldsymbol{X} \boldsymbol{D} \Longrightarrow \boldsymbol{A}=\boldsymbol{X} \boldsymbol{D} \boldsymbol{X}^{-1} \Longrightarrow \boldsymbol{X}^{-1} \boldsymbol{A}= | | |
| lyche-numerical-linear-algebra_FO2092 | 160 | 1.000 | \boldsymbol{D} \boldsymbol{X}^{-1} | | |
| lyche-numerical-linear-algebra_FO2093 | 160 | 1.000 | \boldsymbol{Y}^{*}:=\boldsymbol{X}^{-1} | | |
| lyche-numerical-linear-algebra_FO2094 | 160 | 1.000 | \boldsymbol{Y}^{*} \boldsymbol{X}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO2095 | 160 | 1.000 | \boldsymbol{Y} | | |
| lyche-numerical-linear-algebra_FO2096 | 160 | 1.000 | \boldsymbol{A} \boldsymbol{Y}^{-*}=\boldsymbol{Y}^{-*} \boldsymbol{D} | | |
| lyche-numerical-linear-algebra_FO2097 | 160 | 1.000 | \boldsymbol{X}:=\boldsymbol{Y}^{-*} | | |
| lyche-numerical-linear-algebra_FO2098 | 160 | 0.996 | \boldsymbol{v}=\sum_{j=1}^{n} c_{j} \boldsymbol{x}_{j} | | |
| lyche-numerical-linear-algebra_FO2099 | 160 | 0.996 | \boldsymbol{y}_{i}^{*} \boldsymbol{v}=\sum_{j=1}^{n} c_{j} \boldsymbol{y}_{i}^{*} \boldsymbol{x}_{j}=c_{i} \boldsymbol{y}_{i}^{*} \boldsymbol{x}_{i} | | |
| lyche-numerical-linear-algebra_FO2100 | 160 | 0.996 | c_{i}=\boldsymbol{y}_{i}^{*} \boldsymbol{v} / \boldsymbol{y}_{i}^{*} \boldsymbol{x}_{i} | | |
| lyche-numerical-linear-algebra_FO2101 | 160 | 0.995 | \mu, \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO2102 | 160 | 1.000 | \boldsymbol{y}^{*} \boldsymbol{A} \boldsymbol{x}=\lambda \boldsymbol{y}^{*} \boldsymbol{x}=\mu \boldsymbol{y}^{*} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO2103 | 160 | 1.000 | \boldsymbol{y}^{*} \boldsymbol{x} \neq 0 | | |
| lyche-numerical-linear-algebra_FO2104 | 160 | 1.000 | \|\boldsymbol{x}\|_{2}=1 | | |
| lyche-numerical-linear-algebra_FO2105 | 161 | 1.000 | \boldsymbol{V} \boldsymbol{e}_{1}=\boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO2106 | 161 | 1.000 | \boldsymbol{u}:=\boldsymbol{V}^{*} \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO2107 | 161 | 1.000 | (\bar{\lambda}, \boldsymbol{u}) | | |
| lyche-numerical-linear-algebra_FO2108 | 161 | 1.000 | \boldsymbol{V}^{*} \boldsymbol{A}^{*} \boldsymbol{V} | | |
| lyche-numerical-linear-algebra_FO2109 | 161 | 1.000 | \boldsymbol{y}^{*} \boldsymbol{x}=\boldsymbol{u}^{*} \boldsymbol{V}^{*} \boldsymbol{V} \boldsymbol{e}_{1}=\boldsymbol{u}^{*} \boldsymbol{e}_{1} | | |
| lyche-numerical-linear-algebra_FO2110 | 161 | 0.596 | \boldsymbol{u}^{*} \boldsymbol{e}_{1}=0 | | |
| lyche-numerical-linear-algebra_FO2111 | 161 | 0.596 | \boldsymbol{u}=\left[\begin{array}{l}0 \\ \boldsymbol{v}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2112 | 161 | 0.596 | \boldsymbol{v} \in \mathbb{C}^{n-1} | | |
| lyche-numerical-linear-algebra_FO2113 | 161 | 1.000 | \boldsymbol{A}:=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2114 | 161 | 1.000 | \boldsymbol{y}=\boldsymbol{e}_{2} | | |
| lyche-numerical-linear-algebra_FO2115 | 161 | 1.000 | \operatorname{det}\left(\boldsymbol{B}^{T}\right)= | | |
| lyche-numerical-linear-algebra_FO2116 | 161 | 1.000 | \operatorname{det}(\boldsymbol{B}) | | |
| lyche-numerical-linear-algebra_FO2117 | 161 | 1.000 | \operatorname{det}(\overline{\boldsymbol{B}})=\overline{\operatorname{det}(\boldsymbol{B})} | | |
| lyche-numerical-linear-algebra_FO2118 | 161 | 0.991 | \pi_{\boldsymbol{A}^{T}}=\pi_{\boldsymbol{A}} | | |
| lyche-numerical-linear-algebra_FO2119 | 161 | 1.000 | \pi_{A^{*}}(\bar{\lambda})=\overline{\pi_{A}(\lambda)} | | |
| lyche-numerical-linear-algebra_FO2120 | 161 | 1.000 | \left(\lambda^{-1}, \boldsymbol{x}\right) | | |
| lyche-numerical-linear-algebra_FO2121 | 161 | 0.562 | \left(\lambda^{k}, \boldsymbol{x}\right) | | |
| lyche-numerical-linear-algebra_FO2122 | 161 | 0.562 | \boldsymbol{A}^{k} | | |
| lyche-numerical-linear-algebra_FO2123 | 161 | 0.562 | k \in \mathbb{Z} | | |
| lyche-numerical-linear-algebra_FO2124 | 162 | 1.000 | \left(\lambda_{j}, \boldsymbol{x}_{j}\right), j=1, \ldots, n | | |
| lyche-numerical-linear-algebra_FO2125 | 162 | 1.000 | k \in \mathbb{N} | | |
| lyche-numerical-linear-algebra_FO2126 | 162 | 1.000 | \boldsymbol{A}^{2}= | | |
| lyche-numerical-linear-algebra_FO2127 | 162 | 1.000 | \lambda=0 | | |
| lyche-numerical-linear-algebra_FO2128 | 162 | 1.000 | \boldsymbol{A}^{2}=\boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO2129 | 162 | 1.000 | \boldsymbol{A}^{k}=0 | | |
| lyche-numerical-linear-algebra_FO2130 | 162 | 1.000 | \boldsymbol{A}^{*} \boldsymbol{A}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO2131 | 162 | 1.000 | |\lambda|=1 | | |
| lyche-numerical-linear-algebra_FO2132 | 162 | 1.000 | \epsilon_{0}>0 | | |
| lyche-numerical-linear-algebra_FO2133 | 162 | 1.000 | \boldsymbol{A}+\epsilon \boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO2134 | 162 | 0.997 | \epsilon \in \mathbb{C} | | |
| lyche-numerical-linear-algebra_FO2135 | 162 | 0.997 | |\epsilon|<\epsilon_{0} | | |
| lyche-numerical-linear-algebra_FO2136 | 162 | 0.997 | \operatorname{det}(\boldsymbol{A})=\lambda_{1} \lambda_{2} \cdots \lambda_{n} | | |
| lyche-numerical-linear-algebra_FO2137 | 162 | 1.000 | q_{0}, \ldots, q_{n-1} \in \mathbb{C} | | |
| lyche-numerical-linear-algebra_FO2138 | 162 | 1.000 | p(\lambda)=\lambda^{n}+ | | |
| lyche-numerical-linear-algebra_FO2139 | 162 | 1.000 | q_{n-1} \lambda^{n-1}+\cdots+q_{0} | | |
| lyche-numerical-linear-algebra_FO2140 | 162 | 1.000 | (-1)^{n} p | | |
| lyche-numerical-linear-algebra_FO2141 | 162 | 0.998 | p=(-1)^{n} \pi_{\boldsymbol{A}} | | |
| lyche-numerical-linear-algebra_FO2142 | 162 | 1.000 | p=(-1)^{n} \pi_{\boldsymbol{B}} | | |
| lyche-numerical-linear-algebra_FO2143 | 163 | 0.998 | \boldsymbol{A}=\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 2 & 3 \\ 0 & 0 & 2\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2144 | 163 | 1.000 | \boldsymbol{Y} \in \mathbb{R}^{2 \times 2} | | |
| lyche-numerical-linear-algebra_FO2145 | 163 | 1.000 | \boldsymbol{D} \in \mathbb{R}^{2 \times 2} | | |
| lyche-numerical-linear-algebra_FO2146 | 163 | 1.000 | \boldsymbol{A}=\boldsymbol{Y} \boldsymbol{D} \boldsymbol{Y}^{-1} | | |
| lyche-numerical-linear-algebra_FO2147 | 163 | 1.000 | \boldsymbol{A} \in \mathbb{R}^{n, n} | | |
| lyche-numerical-linear-algebra_FO2148 | 163 | 1.000 | a_{i+1, i} a_{i, i+1}>0 | | |
| lyche-numerical-linear-algebra_FO2149 | 163 | 1.000 | \boldsymbol{A}=\left[\begin{array}{rrr}3 & 0 & 1 \\ -4 & 1 & -2 \\ -4 & 0 & -1\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2150 | 163 | 1.000 | \boldsymbol{J}=\left[\begin{array}{lll}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2151 | 163 | 1.000 | \left(\boldsymbol{J}_{m}(\lambda)-\lambda \boldsymbol{I}\right)^{r}=\left[\begin{array}{cc}\mathbf{0} & \boldsymbol{I}_{m-r} \\ \mathbf{0} & \mathbf{0}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2152 | 163 | 0.998 | 1 \leq r \leq m-1 | | |
| lyche-numerical-linear-algebra_FO2153 | 163 | 0.998 | \left(\boldsymbol{J}_{m}(\lambda)-\lambda \boldsymbol{I}\right)^{m}=0 | | |
| lyche-numerical-linear-algebra_FO2154 | 163 | 1.000 | r=0,1,2, \ldots, m=2,3, \ldots | | |
| lyche-numerical-linear-algebra_FO2155 | 163 | 0.998 | \boldsymbol{A}^{r}=\boldsymbol{S} \boldsymbol{J}^{r} \boldsymbol{S}^{-1} | | |
| lyche-numerical-linear-algebra_FO2156 | 163 | 1.000 | \boldsymbol{J}^{r}=\operatorname{diag}\left(\boldsymbol{U}_{1}^{r}, \ldots, \boldsymbol{U}_{k}^{r}\right) | | |
| lyche-numerical-linear-algebra_FO2157 | 163 | 1.000 | \boldsymbol{D}^{-1} \boldsymbol{A} \boldsymbol{D} | | |
| lyche-numerical-linear-algebra_FO2158 | 164 | 0.985 | \boldsymbol{U}_{i}^{r}=\operatorname{diag}\left(\boldsymbol{J}_{m_{i, 1}}\left(\lambda_{i}\right)^{r}, \ldots, \boldsymbol{J}_{m_{i, g_{i}}}\left(\lambda_{i}\right)^{r}\right) | | |
| lyche-numerical-linear-algebra_FO2159 | 164 | 0.968 | \boldsymbol{J}_{m}(\lambda)^{r}=\left(\boldsymbol{E}_{m}+\lambda \boldsymbol{I}_{m}\right)^{r}=\sum_{k=0}^{\min \{r, m-1\}}\binom{r}{k} \lambda^{r-k} \boldsymbol{E}_{m}^{k} | | |
| lyche-numerical-linear-algebra_FO2160 | 164 | 1.000 | \boldsymbol{J}^{100} | | |
| lyche-numerical-linear-algebra_FO2161 | 164 | 1.000 | \boldsymbol{A}^{100} | | |
| lyche-numerical-linear-algebra_FO2162 | 164 | 1.000 | \mu_{\boldsymbol{A}}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO2163 | 164 | 1.000 | \lambda_{i}-\lambda | | |
| lyche-numerical-linear-algebra_FO2164 | 164 | 1.000 | \lambda_{i} \boldsymbol{I}-\boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO2165 | 164 | 1.000 | \pi_{\boldsymbol{A}}(\lambda)=\prod_{i=1}^{k} \prod_{j=1}^{g_{i}}\left(\lambda_{i}-\lambda\right)^{m_{i, j}} | | |
| lyche-numerical-linear-algebra_FO2166 | 164 | 0.629 | \pi_{\boldsymbol{A}}=\mu_{\boldsymbol{A}} \nu_{\boldsymbol{A}} | | |
| lyche-numerical-linear-algebra_FO2167 | 164 | 0.953 | \nu_{\boldsymbol{A}} | | |
| lyche-numerical-linear-algebra_FO2168 | 164 | 0.986 | \mu_{\boldsymbol{A}}(\boldsymbol{A})=\mathbf{0} \Longleftrightarrow \mu_{\boldsymbol{A}}(\boldsymbol{J})=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO2169 | 164 | 1.000 | m_{i} | | |
| lyche-numerical-linear-algebra_FO2170 | 164 | 1.000 | \mu_{\boldsymbol{A}}(\boldsymbol{A})=0 | | |
| lyche-numerical-linear-algebra_FO2171 | 164 | 1.000 | p(\boldsymbol{A})=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO2172 | 164 | 1.000 | \pi_{\boldsymbol{A}}(\boldsymbol{A})=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO2173 | 164 | 1.000 | p(t):=\sum_{j=0}^{r} b_{j} t^{j} | | |
| lyche-numerical-linear-algebra_FO2174 | 164 | 1.000 | b_{j} \in \mathbb{C} | | |
| lyche-numerical-linear-algebra_FO2175 | 164 | 1.000 | p(\boldsymbol{A}) \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO2176 | 164 | 1.000 | \boldsymbol{A}^{0}:=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO2177 | 164 | 1.000 | p(t):=\left(t-\alpha_{1}\right) \cdots\left(t-\alpha_{r}\right) | | |
| lyche-numerical-linear-algebra_FO2178 | 164 | 1.000 | \alpha_{0}, \ldots, \alpha_{r} \in \mathbb{C} | | |
| lyche-numerical-linear-algebra_FO2179 | 164 | 1.000 | p(\boldsymbol{A})=\left(\boldsymbol{A}-\alpha_{1}\right) \cdots\left(\boldsymbol{A}-\alpha_{r}\right) | | |
| lyche-numerical-linear-algebra_FO2180 | 164 | 1.000 | \boldsymbol{U}^{*} \boldsymbol{A} \boldsymbol{U}=\boldsymbol{T} | | |
| lyche-numerical-linear-algebra_FO2181 | 164 | 0.997 | \left[\begin{array}{rr}2 & 1 \\ -1 & 4\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2182 | 164 | 0.997 | \pi(\boldsymbol{A})=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO2183 | 164 | 1.000 | p(\boldsymbol{A})= | | |
| lyche-numerical-linear-algebra_FO2184 | 164 | 0.877 | \boldsymbol{U} p(\boldsymbol{T}) \boldsymbol{U}^{*} | | |
| lyche-numerical-linear-algebra_FO2185 | 164 | 1.000 | n, k \in \mathbb{N} | | |
| lyche-numerical-linear-algebra_FO2186 | 164 | 1.000 | \boldsymbol{C}, \boldsymbol{D} \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO2187 | 164 | 1.000 | c_{i, j}=0 | | |
| lyche-numerical-linear-algebra_FO2188 | 164 | 1.000 | i, j \leq k | | |
| lyche-numerical-linear-algebra_FO2189 | 164 | 1.000 | d_{k+1, k+1}=0 | | |
| lyche-numerical-linear-algebra_FO2190 | 164 | 1.000 | \boldsymbol{E}:=\boldsymbol{C} \boldsymbol{D} | | |
| lyche-numerical-linear-algebra_FO2191 | 164 | 1.000 | e_{i, j}=0 | | |
| lyche-numerical-linear-algebra_FO2192 | 164 | 1.000 | i, j \leq k+1 | | |
| lyche-numerical-linear-algebra_FO2193 | 165 | 0.773 | p:=\pi_{\boldsymbol{A}} | | |
| lyche-numerical-linear-algebra_FO2194 | 165 | 0.773 | p(\boldsymbol{T})=\mathbf{0} .^{3} | | |
| lyche-numerical-linear-algebra_FO2195 | 165 | 1.000 | \boldsymbol{A}=\left[\begin{array}{ll}1 & 2 \\ 3 & 2\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2196 | 165 | 1.000 | \boldsymbol{U}^{T} \boldsymbol{A} \boldsymbol{U}=\left[\begin{array}{cc}-1 & -1 \\ 0 & 4\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2197 | 165 | 1.000 | \boldsymbol{U}=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1 & 1 \\ -1 & 1\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2198 | 165 | 0.998 | \boldsymbol{C}=\boldsymbol{A}+i \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO2199 | 165 | 1.000 | \boldsymbol{A}^{T}=-\boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO2200 | 165 | 1.000 | \boldsymbol{B}^{T}=\boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO2201 | 165 | 1.000 | \left(\lambda_{1}, \boldsymbol{u}_{1}\right), \ldots,\left(\lambda_{n}, \boldsymbol{u}_{n}\right) | | |
| lyche-numerical-linear-algebra_FO2202 | 165 | 0.999 | \boldsymbol{u}_{j}^{*} \boldsymbol{u}_{k}=0 | | |
| lyche-numerical-linear-algebra_FO2203 | 165 | 0.999 | j \neq k | | |
| lyche-numerical-linear-algebra_FO2204 | 165 | 0.999 | \boldsymbol{A} \boldsymbol{x} \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO2205 | 165 | 0.996 | \lambda_{\text {min }} | | |
| lyche-numerical-linear-algebra_FO2206 | 165 | 0.996 | R(\boldsymbol{x}) | | |
| lyche-numerical-linear-algebra_FO2207 | 166 | 0.997 | \boldsymbol{y} \neq 0 | | |
| lyche-numerical-linear-algebra_FO2208 | 166 | 1.000 | t>0 | | |
| lyche-numerical-linear-algebra_FO2209 | 166 | 1.000 | \boldsymbol{x}^{0} \in B_{1} | | |
| lyche-numerical-linear-algebra_FO2210 | 166 | 1.000 | \boldsymbol{A} \boldsymbol{x}^{0}-R\left(\boldsymbol{x}^{0}\right) \boldsymbol{x}^{0} \neq 0 | | |
| lyche-numerical-linear-algebra_FO2211 | 166 | 1.000 | \left(R\left(\boldsymbol{x}^{0}\right), \boldsymbol{x}^{0}\right) | | |
| lyche-numerical-linear-algebra_FO2212 | 166 | 1.000 | \boldsymbol{x}^{1} \in B_{1} | | |
| lyche-numerical-linear-algebra_FO2213 | 166 | 1.000 | R\left(\boldsymbol{x}^{1}\right)< | | |
| lyche-numerical-linear-algebra_FO2214 | 166 | 0.993 | R\left(\boldsymbol{x}^{0}\right) | | |
| lyche-numerical-linear-algebra_FO2215 | 166 | 1.000 | \beta_{i} \geq \alpha_{i} | | |
| lyche-numerical-linear-algebra_FO2216 | 166 | 1.000 | \boldsymbol{A}:=\left[\begin{array}{ll}0 & 0 \\ 0 & 4\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2217 | 166 | 1.000 | \boldsymbol{B}:=\left[\begin{array}{cc}-1 & -1 \\ 1 & 1\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2218 | 166 | 1.000 | \boldsymbol{A}:=\left[\begin{array}{ll}3 & 1 \\ 2 & 2\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2219 | 166 | 1.000 | \boldsymbol{v} \in \mathbb{R}^{2} | | |
| lyche-numerical-linear-algebra_FO2220 | 166 | 1.000 | \boldsymbol{y}, \boldsymbol{x} \in | | |
| lyche-numerical-linear-algebra_FO2221 | 166 | 1.000 | R(\boldsymbol{y}, \boldsymbol{x})=R_{\boldsymbol{A}}(\boldsymbol{y}, \boldsymbol{x}):=\frac{\boldsymbol{y}^{*} \boldsymbol{A} \boldsymbol{x}}{\boldsymbol{y}^{*} \boldsymbol{x}} | | |
| lyche-numerical-linear-algebra_FO2222 | 166 | 1.000 | R(\boldsymbol{y}, \boldsymbol{x})=\lambda | | |
| lyche-numerical-linear-algebra_FO2223 | 166 | 1.000 | \boldsymbol{A}, \boldsymbol{A}^{T} | | |
| lyche-numerical-linear-algebra_FO2224 | 166 | 1.000 | \boldsymbol{A} \boldsymbol{A}^{*} | | |
| lyche-numerical-linear-algebra_FO2225 | 166 | 0.993 | f(t)=R(\boldsymbol{x}-t \boldsymbol{y}) | | |
| lyche-numerical-linear-algebra_FO2226 | 168 | 1.000 | \boldsymbol{U}:=\left[\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{n}\right] \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO2227 | 168 | 1.000 | \boldsymbol{D}:= | | |
| lyche-numerical-linear-algebra_FO2228 | 168 | 1.000 | \operatorname{diag}\left(\lambda_{1}, \ldots, \lambda_{n}\right) | | |
| lyche-numerical-linear-algebra_FO2229 | 168 | 1.000 | \boldsymbol{A}=\boldsymbol{U} \boldsymbol{\Sigma} \boldsymbol{V}^{*} | | |
| lyche-numerical-linear-algebra_FO2230 | 168 | 1.000 | \boldsymbol{\Sigma} | | |
| lyche-numerical-linear-algebra_FO2231 | 168 | 1.000 | \Sigma_{i j}=0 | | |
| lyche-numerical-linear-algebra_FO2232 | 168 | 1.000 | \Sigma_{i i} \geq 0 | | |
| lyche-numerical-linear-algebra_FO2233 | 168 | 1.000 | \sigma_{i}:=\Sigma_{i i} | | |
| lyche-numerical-linear-algebra_FO2234 | 168 | 1.000 | \sigma_{i} \geq \sigma_{i+1} | | |
| lyche-numerical-linear-algebra_FO2235 | 169 | 1.000 | \sigma_{1}=2 | | |
| lyche-numerical-linear-algebra_FO2236 | 169 | 1.000 | \sigma_{2}=1 | | |
| lyche-numerical-linear-algebra_FO2237 | 169 | 1.000 | \mathcal{R}(\boldsymbol{A}), \mathcal{N}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO2238 | 169 | 1.000 | \mathcal{R}\left(\boldsymbol{A}^{*}\right) | | |
| lyche-numerical-linear-algebra_FO2239 | 169 | 1.000 | \mathcal{N}\left(\boldsymbol{A}^{*}\right) | | |
| lyche-numerical-linear-algebra_FO2240 | 169 | 0.978 | \left.\boldsymbol{A}^{*} \boldsymbol{A}, \boldsymbol{A} \boldsymbol{A}^{*}\right) | | |
| lyche-numerical-linear-algebra_FO2241 | 169 | 1.000 | \boldsymbol{A}^{*} \boldsymbol{A} \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO2242 | 169 | 1.000 | \boldsymbol{A} \boldsymbol{A}^{*} \in \mathbb{C}^{m \times m} | | |
| lyche-numerical-linear-algebra_FO2243 | 169 | 0.992 | \left(\lambda_{j}, \boldsymbol{v}_{j}\right) | | |
| lyche-numerical-linear-algebra_FO2244 | 169 | 1.000 | \left\{\boldsymbol{A} \boldsymbol{v}_{1}, \ldots, \boldsymbol{A} \boldsymbol{v}_{r}\right\} | | |
| lyche-numerical-linear-algebra_FO2245 | 169 | 1.000 | \mathcal{R}(\boldsymbol{A}):= | | |
| lyche-numerical-linear-algebra_FO2246 | 169 | 1.000 | \left\{\boldsymbol{A} \boldsymbol{y} \in \mathbb{C}^{m}: \boldsymbol{y} \in \mathbb{C}^{n}\right\} | | |
| lyche-numerical-linear-algebra_FO2247 | 169 | 1.000 | \left\{\boldsymbol{v}_{r+1}, \ldots, \boldsymbol{v}_{n}\right\} | | |
| lyche-numerical-linear-algebra_FO2248 | 169 | 0.899 | \mathcal{N}(\boldsymbol{A}):=\left\{\boldsymbol{y} \in \mathbb{C}^{n}: \boldsymbol{A} \boldsymbol{y}=\mathbf{0}\right\} | | |
| lyche-numerical-linear-algebra_FO2249 | 169 | 0.996 | \left(\lambda_{j}, \boldsymbol{u}_{j}\right) | | |
| lyche-numerical-linear-algebra_FO2250 | 169 | 0.996 | \lambda_{j}>0, j=1, \ldots, r | | |
| lyche-numerical-linear-algebra_FO2251 | 169 | 1.000 | \lambda_{j}=0, j=r+1, \ldots, m | | |
| lyche-numerical-linear-algebra_FO2252 | 169 | 1.000 | \left\{\boldsymbol{A}^{*} \boldsymbol{u}_{1}, \ldots, \boldsymbol{A}^{*} \boldsymbol{u}_{r}\right\} | | |
| lyche-numerical-linear-algebra_FO2253 | 169 | 1.000 | \left\{\boldsymbol{u}_{r+1}, \ldots, \boldsymbol{u}_{m}\right\} | | |
| lyche-numerical-linear-algebra_FO2254 | 169 | 1.000 | \pi_{\boldsymbol{A}^{*} \boldsymbol{A}} | | |
| lyche-numerical-linear-algebra_FO2255 | 169 | 1.000 | \pi_{\boldsymbol{A} \boldsymbol{A}^{*}} | | |
| lyche-numerical-linear-algebra_FO2256 | 170 | 1.000 | \boldsymbol{A}^{*} \boldsymbol{A} \boldsymbol{v}=\lambda \boldsymbol{v} | | |
| lyche-numerical-linear-algebra_FO2257 | 170 | 1.000 | \boldsymbol{v} \neq \mathbf{0} | | |
| lyche-numerical-linear-algebra_FO2258 | 170 | 1.000 | \left(\boldsymbol{A} \boldsymbol{v}_{j}\right)^{*} \boldsymbol{A} \boldsymbol{v}_{k}=\boldsymbol{v}_{j}^{*} \boldsymbol{A}^{*} \boldsymbol{A} \boldsymbol{v}_{k}=\lambda_{k} \boldsymbol{v}_{j}^{*} \boldsymbol{v}_{k} | | |
| lyche-numerical-linear-algebra_FO2259 | 170 | 0.578 | \boldsymbol{A} \boldsymbol{v}_{1}, \ldots, \boldsymbol{A} \boldsymbol{v}_{n} | | |
| lyche-numerical-linear-algebra_FO2260 | 170 | 1.000 | \boldsymbol{A} \boldsymbol{v}_{1}, \ldots, \boldsymbol{A} \boldsymbol{v}_{r} | | |
| lyche-numerical-linear-algebra_FO2261 | 170 | 1.000 | \boldsymbol{A} \boldsymbol{v}_{j}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO2262 | 170 | 1.000 | j=r+ | | |
| lyche-numerical-linear-algebra_FO2263 | 170 | 1.000 | \mathcal{N}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO2264 | 170 | 1.000 | \mathcal{R}(\boldsymbol{A}) \subset \operatorname{span}\left(\boldsymbol{A} \boldsymbol{v}_{1}, \ldots, \boldsymbol{A} \boldsymbol{v}_{r}\right) | | |
| lyche-numerical-linear-algebra_FO2265 | 170 | 1.000 | \mathcal{N}(\boldsymbol{A}) \subset | | |
| lyche-numerical-linear-algebra_FO2266 | 170 | 0.999 | \operatorname{span}\left(\boldsymbol{v}_{r+1}, \ldots, \boldsymbol{v}_{n}\right) | | |
| lyche-numerical-linear-algebra_FO2267 | 170 | 0.999 | \boldsymbol{x} \in \mathcal{R}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO2268 | 170 | 0.999 | \boldsymbol{x}=\boldsymbol{A} \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO2269 | 170 | 1.000 | \boldsymbol{y}=\sum_{j=1}^{n} c_{j} \boldsymbol{v}_{j} | | |
| lyche-numerical-linear-algebra_FO2270 | 170 | 1.000 | \boldsymbol{x}=\boldsymbol{A} \boldsymbol{y}=\sum_{j=1}^{n} c_{j} \boldsymbol{A} \boldsymbol{v}_{j}=\sum_{j=1}^{r} c_{j} \boldsymbol{A} \boldsymbol{v}_{j} \in | | |
| lyche-numerical-linear-algebra_FO2271 | 170 | 1.000 | \operatorname{span}\left(\boldsymbol{A} \boldsymbol{v}_{1}, \ldots, \boldsymbol{A} \boldsymbol{v}_{r}\right) | | |
| lyche-numerical-linear-algebra_FO2272 | 170 | 1.000 | \boldsymbol{y}=\sum_{j=1}^{n} c_{j} \boldsymbol{v}_{j} \in \mathcal{N}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO2273 | 170 | 1.000 | \boldsymbol{A} \boldsymbol{y}=\sum_{j=1}^{r} c_{j} \boldsymbol{A} \boldsymbol{v}_{j}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO2274 | 170 | 1.000 | c_{1}=\cdots=c_{r}=0 | | |
| lyche-numerical-linear-algebra_FO2275 | 170 | 1.000 | \boldsymbol{y}=\sum_{j=r+1}^{n} c_{j} \boldsymbol{v}_{j} \in \operatorname{span}\left(\boldsymbol{v}_{r+1}, \ldots, \boldsymbol{v}_{n}\right) | | |
| lyche-numerical-linear-algebra_FO2276 | 170 | 1.000 | \boldsymbol{A} \boldsymbol{A}^{*}=\boldsymbol{B}^{*} \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO2277 | 170 | 1.000 | \boldsymbol{B}:=\boldsymbol{A}^{*} | | |
| lyche-numerical-linear-algebra_FO2278 | 170 | 1.000 | \boldsymbol{A}=\boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO2279 | 170 | 1.000 | 2 \boldsymbol{A}^{*} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO2280 | 170 | 1.000 | 4 r | | |
| lyche-numerical-linear-algebra_FO2281 | 170 | 1.000 | m, n, r \in \mathbb{N} | | |
| lyche-numerical-linear-algebra_FO2282 | 170 | 1.000 | \lambda_{1} \geq \cdots \geq | | |
| lyche-numerical-linear-algebra_FO2283 | 170 | 1.000 | \lambda_{r}>0=\lambda_{r+1}=\cdots=\lambda_{n} | | |
| lyche-numerical-linear-algebra_FO2284 | 170 | 1.000 | \boldsymbol{V}:=\left[\boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{n}\right] \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO2285 | 170 | 1.000 | \Sigma \in \mathbb{R}^{m \times n} | | |
| lyche-numerical-linear-algebra_FO2286 | 170 | 1.000 | \sigma_{j}:=\sqrt{\lambda_{j}} | | |
| lyche-numerical-linear-algebra_FO2287 | 170 | 1.000 | j= | | |
| lyche-numerical-linear-algebra_FO2288 | 170 | 1.000 | 1, \ldots, \min (m, n) | | |
| lyche-numerical-linear-algebra_FO2289 | 170 | 0.999 | \boldsymbol{U}:=\left[\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{m}\right] \in \mathbb{C}^{m \times m} | | |
| lyche-numerical-linear-algebra_FO2290 | 170 | 0.999 | \boldsymbol{u}_{j}=\sigma_{j}^{-1} \boldsymbol{A} \boldsymbol{v}_{j} | | |
| lyche-numerical-linear-algebra_FO2291 | 170 | 0.999 | j=1, \ldots, r | | |
| lyche-numerical-linear-algebra_FO2292 | 170 | 1.000 | \boldsymbol{u}_{r+1}, \ldots, \boldsymbol{u}_{m} | | |
| lyche-numerical-linear-algebra_FO2293 | 170 | 1.000 | \boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{r} | | |
| lyche-numerical-linear-algebra_FO2294 | 170 | 1.000 | \boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{m} | | |
| lyche-numerical-linear-algebra_FO2295 | 170 | 1.000 | \boldsymbol{U}, \boldsymbol{\Sigma}, \boldsymbol{V} | | |
| lyche-numerical-linear-algebra_FO2296 | 170 | 1.000 | \sigma_{j}=\left\|\boldsymbol{A} \boldsymbol{v}_{j}\right\|_{2}>0, j=1, \ldots, r | | |
| lyche-numerical-linear-algebra_FO2297 | 171 | 1.000 | \boldsymbol{U} \boldsymbol{\Sigma}=\boldsymbol{A} \boldsymbol{V} | | |
| lyche-numerical-linear-algebra_FO2298 | 171 | 1.000 | \boldsymbol{U} \boldsymbol{\Sigma} \boldsymbol{V}^{*}=\boldsymbol{A} \boldsymbol{V} \boldsymbol{V}^{*}=\boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO2299 | 171 | 1.000 | \sigma_{1} \geq \sigma_{2} \geq \cdots \geq \sigma_{r} | | |
| lyche-numerical-linear-algebra_FO2300 | 171 | 1.000 | \boldsymbol{A}=\frac{1}{15}\left[\begin{array}{cc}14 & 2 \\ 4 & 22 \\ 16 & 13\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2301 | 171 | 0.990 | \sigma_{1}=2, \sigma_{2}=1 | | |
| lyche-numerical-linear-algebra_FO2302 | 171 | 0.990 | \boldsymbol{V}=\frac{1}{5}\left[\begin{array}{rr}3 & 4 \\ 4 & -3\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2303 | 171 | 0.990 | \boldsymbol{u}_{1}=\boldsymbol{A} \boldsymbol{v}_{1} / \sigma_{1}=[1,2,2]^{T} / 3 | | |
| lyche-numerical-linear-algebra_FO2304 | 171 | 0.998 | \boldsymbol{u}_{2}=\boldsymbol{A} \boldsymbol{v}_{2} / \sigma_{2}=[2,-2,1]^{T} / 3 | | |
| lyche-numerical-linear-algebra_FO2305 | 171 | 0.998 | \boldsymbol{u}_{3} | | |
| lyche-numerical-linear-algebra_FO2306 | 171 | 1.000 | \boldsymbol{u}_{1} | | |
| lyche-numerical-linear-algebra_FO2307 | 171 | 1.000 | \boldsymbol{u}_{2} . \boldsymbol{u}_{3}=[2,1,-2]^{T} / 3 | | |
| lyche-numerical-linear-algebra_FO2308 | 171 | 1.000 | \Sigma_{1} | | |
| lyche-numerical-linear-algebra_FO2309 | 171 | 1.000 | k, l \geq 0 | | |
| lyche-numerical-linear-algebra_FO2310 | 171 | 1.000 | \mathbf{0}_{k, l}=[] | | |
| lyche-numerical-linear-algebra_FO2311 | 171 | 1.000 | k=0 | | |
| lyche-numerical-linear-algebra_FO2312 | 171 | 1.000 | l=0 | | |
| lyche-numerical-linear-algebra_FO2313 | 172 | 1.000 | l | | |
| lyche-numerical-linear-algebra_FO2314 | 172 | 1.000 | \boldsymbol{A}=\boldsymbol{U}_{1} \boldsymbol{\Sigma}_{1} \boldsymbol{V}_{1}^{*} | | |
| lyche-numerical-linear-algebra_FO2315 | 172 | 1.000 | \boldsymbol{U}_{1} \in \mathbb{C}^{m \times r} | | |
| lyche-numerical-linear-algebra_FO2316 | 172 | 1.000 | \boldsymbol{V}_{1} \in \mathbb{C}^{n \times r} | | |
| lyche-numerical-linear-algebra_FO2317 | 172 | 1.000 | \boldsymbol{\Sigma}_{1} \in \mathbb{R}^{r \times r} | | |
| lyche-numerical-linear-algebra_FO2318 | 172 | 1.000 | \sigma_{1} \geq \cdots \geq \sigma_{r}>0 | | |
| lyche-numerical-linear-algebra_FO2319 | 172 | 1.000 | \boldsymbol{U}_{1}, \boldsymbol{V}_{1} | | |
| lyche-numerical-linear-algebra_FO2320 | 172 | 1.000 | \boldsymbol{U}, \boldsymbol{V} | | |
| lyche-numerical-linear-algebra_FO2321 | 172 | 1.000 | \Sigma_{1} \in \mathbb{R}^{r \times r} | | |
| lyche-numerical-linear-algebra_FO2322 | 172 | 1.000 | \boldsymbol{U}_{1} | | |
| lyche-numerical-linear-algebra_FO2323 | 172 | 1.000 | \boldsymbol{V}_{1} | | |
| lyche-numerical-linear-algebra_FO2324 | 172 | 1.000 | \boldsymbol{U} \in \mathbb{C}^{m \times m} | | |
| lyche-numerical-linear-algebra_FO2325 | 172 | 1.000 | \boldsymbol{V} \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO2326 | 172 | 1.000 | \boldsymbol{A}=\left[\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{r}\right] \operatorname{diag}\left(\sigma_{1}, \ldots, \sigma_{r}\right)\left[\boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{r}\right]^{*} | | |
| lyche-numerical-linear-algebra_FO2327 | 172 | 1.000 | 7.3(r<n<m) | | |
| lyche-numerical-linear-algebra_FO2328 | 173 | 1.000 | \sigma_{1}=2, \sigma_{2}=0 | | |
| lyche-numerical-linear-algebra_FO2329 | 173 | 1.000 | r=1, m=3, n=2 | | |
| lyche-numerical-linear-algebra_FO2330 | 173 | 1.000 | \boldsymbol{u}_{1}=\boldsymbol{A} \boldsymbol{v}_{1} / \sigma_{1}=\boldsymbol{s}_{1} / \sqrt{2} | | |
| lyche-numerical-linear-algebra_FO2331 | 173 | 1.000 | \boldsymbol{s}_{1}=[1,1,0]^{T} | | |
| lyche-numerical-linear-algebra_FO2332 | 173 | 1.000 | \boldsymbol{s}_{1} | | |
| lyche-numerical-linear-algebra_FO2333 | 173 | 1.000 | \left\{\boldsymbol{s}_{1}, \boldsymbol{s}_{2}, \boldsymbol{s}_{3}\right\} | | |
| lyche-numerical-linear-algebra_FO2334 | 173 | 0.979 | \boldsymbol{w}_{1}=\boldsymbol{s}_{1}, \boldsymbol{w}_{2}=\boldsymbol{s}_{2}-\frac{\boldsymbol{s}_{2}^{T} \boldsymbol{w}_{1}}{\boldsymbol{w}_{1}^{T} \boldsymbol{w}_{1}} \boldsymbol{w}_{1}=\left[\begin{array}{c}-1 / 2 \\ 1 / 2 \\ 0\end{array}\right], \boldsymbol{w}_{3}=\boldsymbol{s}_{3}- | | |
| lyche-numerical-linear-algebra_FO2335 | 173 | 0.763 | \frac{\boldsymbol{s}_{3}^{T} \boldsymbol{w}_{1}}{\boldsymbol{w}_{1}^{T} \boldsymbol{w}_{1}} \boldsymbol{w}_{1}-\frac{\boldsymbol{s}_{3}^{T} \boldsymbol{w}_{2}}{\boldsymbol{w}_{2}^{T} \boldsymbol{w}_{2}} \boldsymbol{w}_{2}=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2336 | 173 | 0.763 | \boldsymbol{w}_{i} | | |
| lyche-numerical-linear-algebra_FO2337 | 173 | 0.763 | \boldsymbol{u}_{1}=\boldsymbol{w}_{1} /\left\|\boldsymbol{w}_{1}\right\|_{2}= | | |
| lyche-numerical-linear-algebra_FO2338 | 173 | 0.999 | [1 / \sqrt{2}, 1 / \sqrt{2}, 0]^{T}, \boldsymbol{u}_{2}=\boldsymbol{w}_{2} /\left\|\boldsymbol{w}_{2}\right\|_{2}=[-1 / \sqrt{2}, 1 / \sqrt{2}, 0]^{T} | | |
| lyche-numerical-linear-algebra_FO2339 | 173 | 0.999 | \boldsymbol{u}_{3}=\boldsymbol{s}_{3} /\left\|\boldsymbol{s}_{3}\right\|_{2}= | | |
| lyche-numerical-linear-algebra_FO2340 | 173 | 1.000 | [0,0,1]^{T} | | |
| lyche-numerical-linear-algebra_FO2341 | 173 | 1.000 | \boldsymbol{A}=\boldsymbol{U} \boldsymbol{\Sigma} \boldsymbol{V}^{T} | | |
| lyche-numerical-linear-algebra_FO2342 | 174 | 1.000 | \mathcal{R}(\boldsymbol{A}), \mathcal{N}(\boldsymbol{A}), \mathcal{R}\left(\boldsymbol{A}^{*}\right) | | |
| lyche-numerical-linear-algebra_FO2343 | 174 | 1.000 | m, n | | |
| lyche-numerical-linear-algebra_FO2344 | 174 | 0.985 | \left[\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{m}\right] \boldsymbol{\Sigma}\left[\boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{n}\right]^{*}=\boldsymbol{U} \boldsymbol{\Sigma} \boldsymbol{V}^{*} | | |
| lyche-numerical-linear-algebra_FO2345 | 174 | 1.000 | \left\{\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{r}\right\} | | |
| lyche-numerical-linear-algebra_FO2346 | 174 | 1.000 | \left\{\boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{r}\right\} | | |
| lyche-numerical-linear-algebra_FO2347 | 174 | 1.000 | \boldsymbol{A} \boldsymbol{V}=\boldsymbol{U} \boldsymbol{\Sigma} | | |
| lyche-numerical-linear-algebra_FO2348 | 174 | 0.980 | \boldsymbol{A}\left[\boldsymbol{V}_{1}, \boldsymbol{V}_{2}\right]=\left[\boldsymbol{U}_{1}, \boldsymbol{U}_{2}\right]\left[\begin{array}{cc}\Sigma_{1} & \mathbf{0} \\ \mathbf{0} & \mathbf{0}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2349 | 174 | 0.980 | \boldsymbol{A} \boldsymbol{V}_{1}=\boldsymbol{U}_{1} \boldsymbol{\Sigma}_{1}, \boldsymbol{A} \boldsymbol{V}_{2}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO2350 | 174 | 1.000 | \boldsymbol{A}^{*}=\boldsymbol{V} \boldsymbol{\Sigma}^{*} \boldsymbol{U}^{*} | | |
| lyche-numerical-linear-algebra_FO2351 | 174 | 1.000 | \boldsymbol{A}^{*} \boldsymbol{U}=\boldsymbol{V} \boldsymbol{\Sigma}^{*} | | |
| lyche-numerical-linear-algebra_FO2352 | 174 | 1.000 | \mathcal{R}(\boldsymbol{A}),\left\{\boldsymbol{A}^{*} \boldsymbol{u}_{1}, \ldots, \boldsymbol{A}^{*} \boldsymbol{u}_{r}\right\} | | |
| lyche-numerical-linear-algebra_FO2353 | 174 | 1.000 | \mathcal{R}\left(\boldsymbol{A}^{*}\right),\left\{\boldsymbol{v}_{r+1}, \ldots, \boldsymbol{v}_{m}\right\} | | |
| lyche-numerical-linear-algebra_FO2354 | 174 | 1.000 | \boldsymbol{T}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m} | | |
| lyche-numerical-linear-algebra_FO2355 | 174 | 0.825 | \boldsymbol{T} \boldsymbol{z}:=\boldsymbol{A} \boldsymbol{z} | | |
| lyche-numerical-linear-algebra_FO2356 | 174 | 0.825 | (\boldsymbol{A})=n | | |
| lyche-numerical-linear-algebra_FO2357 | 174 | 0.672 | \mathcal{S}:=\left\{z \in \mathbb{R}^{n}\right. | | |
| lyche-numerical-linear-algebra_FO2358 | 174 | 0.943 | \left.\|\boldsymbol{z}\|_{2}=1\right\} | | |
| lyche-numerical-linear-algebra_FO2359 | 174 | 0.943 | \mathcal{E}:=\boldsymbol{A} \mathcal{S}=\{\boldsymbol{A} \boldsymbol{z}: \boldsymbol{z} \in \mathcal{S}\} | | |
| lyche-numerical-linear-algebra_FO2360 | 174 | 1.000 | r=n | | |
| lyche-numerical-linear-algebra_FO2361 | 174 | 0.977 | \boldsymbol{U}_{1} \Sigma_{1} \boldsymbol{V}_{1}^{T} | | |
| lyche-numerical-linear-algebra_FO2362 | 175 | 0.407 | z \in \mathcal{S} | | |
| lyche-numerical-linear-algebra_FO2363 | 175 | 0.407 | \boldsymbol{A} z=\boldsymbol{U}_{1} \Sigma_{1} \boldsymbol{V}_{1}^{T} z=\boldsymbol{U}_{1} \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO2364 | 175 | 0.407 | \boldsymbol{y}:=\Sigma_{1} \boldsymbol{V}_{1}^{T} \boldsymbol{z} | | |
| lyche-numerical-linear-algebra_FO2365 | 175 | 0.997 | \operatorname{rank}(\boldsymbol{A})=n | | |
| lyche-numerical-linear-algebra_FO2366 | 175 | 0.997 | \boldsymbol{V}_{1}=\boldsymbol{V} | | |
| lyche-numerical-linear-algebra_FO2367 | 175 | 0.997 | \boldsymbol{V}_{1} \boldsymbol{V}_{1}^{T}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO2368 | 175 | 0.984 | \boldsymbol{V}_{1} \boldsymbol{\Sigma}_{1}^{-1} \boldsymbol{y}=\boldsymbol{z} | | |
| lyche-numerical-linear-algebra_FO2369 | 175 | 1.000 | \boldsymbol{y} \in \tilde{\mathcal{E}} | | |
| lyche-numerical-linear-algebra_FO2370 | 175 | 1.000 | \boldsymbol{x}=\boldsymbol{A} \boldsymbol{z}=\boldsymbol{U}_{1} \Sigma_{1} \boldsymbol{V}_{1}^{T} \boldsymbol{z}=\boldsymbol{U}_{1} \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO2371 | 175 | 1.000 | \mathcal{E}=\boldsymbol{U}_{1} \tilde{\mathcal{E}} | | |
| lyche-numerical-linear-algebra_FO2372 | 175 | 1.000 | 1=\frac{y_{1}^{2}}{\sigma_{1}^{2}}+\cdots+\frac{y_{n}^{2}}{\sigma_{n}^{2}} | | |
| lyche-numerical-linear-algebra_FO2373 | 175 | 1.000 | \sigma_{j} | | |
| lyche-numerical-linear-algebra_FO2374 | 175 | 1.000 | \boldsymbol{e}_{j} | | |
| lyche-numerical-linear-algebra_FO2375 | 175 | 0.992 | \boldsymbol{U}_{1} \boldsymbol{y} \rightarrow \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO2376 | 175 | 0.992 | \mathcal{E}=\boldsymbol{A} \mathcal{S} | | |
| lyche-numerical-linear-algebra_FO2377 | 175 | 1.000 | \boldsymbol{u}_{j}=\boldsymbol{U} \boldsymbol{e}_{j} | | |
| lyche-numerical-linear-algebra_FO2378 | 175 | 1.000 | \sigma_{j}, j=1, \ldots, n | | |
| lyche-numerical-linear-algebra_FO2379 | 175 | 1.000 | \boldsymbol{A} \boldsymbol{v}_{j}=\sigma_{j} \boldsymbol{u}_{j} | | |
| lyche-numerical-linear-algebra_FO2380 | 175 | 1.000 | \mathcal{E} | | |
| lyche-numerical-linear-algebra_FO2381 | 175 | 1.000 | \boldsymbol{A}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} | | |
| lyche-numerical-linear-algebra_FO2382 | 175 | 1.000 | \sigma_{1}=3, \sigma_{2}=1, \boldsymbol{u}_{1}=[3,4]^{T} / 5 | | |
| lyche-numerical-linear-algebra_FO2383 | 175 | 1.000 | \boldsymbol{u}_{2}=[-4,3]^{T} / 5 | | |
| lyche-numerical-linear-algebra_FO2384 | 175 | 1.000 | y_{1}^{2} / \sigma_{1}^{2}+y_{2}^{2} / \sigma_{2}^{2}=1 | | |
| lyche-numerical-linear-algebra_FO2385 | 175 | 1.000 | \mathcal{E}=\boldsymbol{A} \mathcal{S}=\boldsymbol{U}_{1} \tilde{\mathcal{E}} | | |
| lyche-numerical-linear-algebra_FO2387 | 176 | 0.997 | \boldsymbol{y}=\boldsymbol{U}_{1}^{T} \boldsymbol{x}=\left[3 / 5 x_{1}+4 / 5 x_{2},-4 / 5 x_{1}+3 / 5 x_{2}\right]^{T} | | |
| lyche-numerical-linear-algebra_FO2388 | 176 | 1.000 | \left\|\boldsymbol{A}^{*}\right\|_{F}=\|\boldsymbol{A}\|_{F} | | |
| lyche-numerical-linear-algebra_FO2389 | 176 | 1.000 | \|\boldsymbol{A}\|_{F}^{2}=\sum_{j=1}^{n}\left\|\boldsymbol{a}_{: j}\right\|_{2}^{2} | | |
| lyche-numerical-linear-algebra_FO2390 | 177 | 1.000 | \|\boldsymbol{U} \boldsymbol{A}\|_{F}=\|\boldsymbol{A} \boldsymbol{V}\|_{F}=\|\boldsymbol{A}\|_{F} | | |
| lyche-numerical-linear-algebra_FO2391 | 177 | 1.000 | \boldsymbol{V} \in | | |
| lyche-numerical-linear-algebra_FO2392 | 177 | 0.999 | \|\boldsymbol{A} \boldsymbol{B}\|_{F} \leq\|\boldsymbol{A}\|_{F}\|\boldsymbol{B}\|_{F} | | |
| lyche-numerical-linear-algebra_FO2393 | 177 | 0.999 | \boldsymbol{B} \in \mathbb{C}^{n, k}, \quad k \in \mathbb{N} | | |
| lyche-numerical-linear-algebra_FO2394 | 177 | 0.980 | \|\boldsymbol{A} \boldsymbol{x}\|_{2} \leq\|\boldsymbol{A}\|_{F}\|\boldsymbol{x}\|_{2} | | |
| lyche-numerical-linear-algebra_FO2395 | 177 | 1.000 | \left\|\boldsymbol{A}^{*}\right\|_{F}^{2}=\sum_{j=1}^{n} \sum_{i=1}^{m}\left|\bar{a}_{i j}\right|^{2}=\sum_{i=1}^{m} \sum_{j=1}^{n}\left|a_{i j}\right|^{2}=\|\boldsymbol{A}\|_{F}^{2} | | |
| lyche-numerical-linear-algebra_FO2396 | 177 | 1.000 | \|\boldsymbol{A}\|_{F}:=\|\operatorname{vec}(\boldsymbol{A})\|_{2} | | |
| lyche-numerical-linear-algebra_FO2397 | 177 | 1.000 | \operatorname{vec}(\boldsymbol{A}) \in \mathbb{C}^{m n} | | |
| lyche-numerical-linear-algebra_FO2398 | 177 | 1.000 | \boldsymbol{U}^{*} \boldsymbol{U}=I | | |
| lyche-numerical-linear-algebra_FO2399 | 177 | 1.000 | \|\boldsymbol{U} \boldsymbol{x}\|_{2}=\|\boldsymbol{x}\|_{2} | | |
| lyche-numerical-linear-algebra_FO2400 | 177 | 0.997 | \|\boldsymbol{U} \boldsymbol{A}\|_{F}^{2} \stackrel{2 .}{=} \sum_{j=1}^{n}\left\|\boldsymbol{U} \boldsymbol{a}_{: j}\right\|_{2}^{2}=\sum_{j=1}^{n}\left\|\boldsymbol{a}_{: j}\right\|_{2}^{2} \stackrel{2 .}{=}\|\boldsymbol{A}\|_{F}^{2} | | |
| lyche-numerical-linear-algebra_FO2401 | 177 | 0.933 | \boldsymbol{V} \boldsymbol{V}^{*}=I | | |
| lyche-numerical-linear-algebra_FO2402 | 177 | 0.933 | \|\boldsymbol{A} \boldsymbol{V}\|_{F} \stackrel{1 .}{=}\left\|\boldsymbol{V}^{*} \boldsymbol{A}^{*}\right\|_{F}=\left\|\boldsymbol{A}^{*}\right\|_{F} \stackrel{1 .}{=}\|\boldsymbol{A}\|_{F} | | |
| lyche-numerical-linear-algebra_FO2403 | 177 | 0.986 | \|\boldsymbol{v}\|_{F}=\|\boldsymbol{v}\|_{2} | | |
| lyche-numerical-linear-algebra_FO2404 | 177 | 0.986 | \boldsymbol{B}=\boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO2405 | 177 | 0.996 | \|\boldsymbol{A}\|_{F}= | | |
| lyche-numerical-linear-algebra_FO2406 | 177 | 0.999 | \sqrt{\sigma_{1}^{2}+\cdots+\sigma_{n}^{2}} | | |
| lyche-numerical-linear-algebra_FO2407 | 177 | 0.999 | \sigma_{1}, \ldots, \sigma_{n} | | |
| lyche-numerical-linear-algebra_FO2408 | 177 | 1.000 | m \geq n \geq 1 | | |
| lyche-numerical-linear-algebra_FO2409 | 177 | 1.000 | \boldsymbol{U}\left[\begin{array}{c}\boldsymbol{D} \\ \mathbf{0}\end{array}\right] \boldsymbol{V}^{*} | | |
| lyche-numerical-linear-algebra_FO2410 | 177 | 1.000 | \boldsymbol{D}=\operatorname{diag}\left(\sigma_{1}, \ldots, \sigma_{n}\right) | | |
| lyche-numerical-linear-algebra_FO2411 | 177 | 1.000 | \epsilon>0 | | |
| lyche-numerical-linear-algebra_FO2412 | 177 | 1.000 | 1 \leq r \leq n | | |
| lyche-numerical-linear-algebra_FO2413 | 177 | 1.000 | \sigma_{r+1}^{2}+\cdots+\sigma_{n}^{2}<\epsilon^{2} | | |
| lyche-numerical-linear-algebra_FO2414 | 177 | 1.000 | \boldsymbol{A}^{\prime}:=\boldsymbol{U}\left[\begin{array}{c}\boldsymbol{D}^{\prime} \\ \mathbf{0}\end{array}\right] \boldsymbol{V}^{*} | | |
| lyche-numerical-linear-algebra_FO2415 | 177 | 1.000 | \boldsymbol{D}^{\prime}:=\operatorname{diag}\left(\sigma_{1}, \ldots, \sigma_{r}, 0, \ldots, 0\right) \in \mathbb{R}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO2416 | 178 | 1.000 | \boldsymbol{A}^{\prime} | | |
| lyche-numerical-linear-algebra_FO2417 | 178 | 1.000 | \sqrt{\sigma_{r+1}^{2}+\cdots+\sigma_{n}^{2}} | | |
| lyche-numerical-linear-algebra_FO2418 | 178 | 0.972 | \operatorname{rank}(\boldsymbol{A})=r | | |
| lyche-numerical-linear-algebra_FO2419 | 178 | 1.000 | r, \boldsymbol{A}^{\prime} | | |
| lyche-numerical-linear-algebra_FO2420 | 178 | 1.000 | \sigma_{1} \geq \cdots \geq \sigma_{n} \geq 0 | | |
| lyche-numerical-linear-algebra_FO2421 | 178 | 1.000 | r \leq \operatorname{rank}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO2422 | 178 | 1.000 | \boldsymbol{A}=\left[\begin{array}{l}3 \\ 4\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2423 | 178 | 1.000 | \boldsymbol{A}=\left[\begin{array}{ll}1 & 1 \\ 2 & 2 \\ 2 & 2\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2424 | 179 | 1.000 | \boldsymbol{A}=\boldsymbol{e}_{1} | | |
| lyche-numerical-linear-algebra_FO2425 | 179 | 1.000 | \boldsymbol{A}=\boldsymbol{e}_{n}^{T} | | |
| lyche-numerical-linear-algebra_FO2426 | 179 | 1.000 | \boldsymbol{A}=\left[\begin{array}{rr}-1 & 0 \\ 0 & 3\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2427 | 179 | 1.000 | \boldsymbol{U} \boldsymbol{\Sigma} \boldsymbol{V} | | |
| lyche-numerical-linear-algebra_FO2428 | 179 | 1.000 | \boldsymbol{A}^{*} \boldsymbol{A}=\boldsymbol{V} \operatorname{diag}\left(\sigma_{1}^{2}, \ldots, \sigma_{n}^{2}\right) \boldsymbol{V}^{*} | | |
| lyche-numerical-linear-algebra_FO2429 | 179 | 1.000 | \boldsymbol{A} \boldsymbol{A}^{*}=\boldsymbol{U} \operatorname{diag}\left(\sigma_{1}^{2}, \ldots, \sigma_{m}^{2}\right) \boldsymbol{U}^{*} | | |
| lyche-numerical-linear-algebra_FO2430 | 179 | 0.803 | \left(\sigma_{1}, \ldots, \sigma_{n}\right) | | |
| lyche-numerical-linear-algebra_FO2431 | 179 | 0.803 | \boldsymbol{A}^{2} | | |
| lyche-numerical-linear-algebra_FO2432 | 179 | 1.000 | \left(\sigma_{1}^{2}, \ldots, \sigma_{n}^{2}\right) | | |
| lyche-numerical-linear-algebra_FO2433 | 179 | 0.884 | \boldsymbol{A}=\frac{1}{25}\left[\begin{array}{ll}11 & 48 \\ 48 & 39\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2434 | 179 | 0.998 | \boldsymbol{U} \boldsymbol{D} \boldsymbol{U}^{T} | | |
| lyche-numerical-linear-algebra_FO2435 | 179 | 1.000 | \boldsymbol{A}=\frac{1}{5}\left[\begin{array}{cc}3 & -4 \\ 4 & 3\end{array}\right]\left[\begin{array}{ll}3 & 0 \\ 0 & 1\end{array}\right] \frac{1}{5}\left[\begin{array}{cc}3 & 4 \\ 4 & -3\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2436 | 180 | 1.000 | \operatorname{rank}(\boldsymbol{A})=\operatorname{rank}\left(\boldsymbol{A}^{*}\right) | | |
| lyche-numerical-linear-algebra_FO2437 | 180 | 1.000 | \operatorname{rank}(\boldsymbol{A})+\operatorname{null}(\boldsymbol{A})=n | | |
| lyche-numerical-linear-algebra_FO2438 | 180 | 1.000 | \operatorname{rank}(\boldsymbol{A})+\operatorname{null}\left(\boldsymbol{A}^{*}\right)=m | | |
| lyche-numerical-linear-algebra_FO2439 | 180 | 1.000 | \operatorname{null}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO2440 | 180 | 1.000 | \operatorname{rank} \boldsymbol{A}=\operatorname{rank}\left(\boldsymbol{A}^{*} \boldsymbol{A}\right)=\operatorname{rank}\left(\boldsymbol{A} \boldsymbol{A}^{*}\right) | | |
| lyche-numerical-linear-algebra_FO2441 | 180 | 0.997 | \operatorname{null}\left(\boldsymbol{A}^{*} \boldsymbol{A}\right)=\operatorname{null} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO2442 | 180 | 0.997 | \operatorname{null}\left(\boldsymbol{A} \boldsymbol{A}^{*}\right)=\operatorname{null}\left(\boldsymbol{A}^{*}\right) | | |
| lyche-numerical-linear-algebra_FO2443 | 180 | 1.000 | \mathcal{R}(\boldsymbol{B}) | | |
| lyche-numerical-linear-algebra_FO2444 | 180 | 1.000 | \mathcal{N}(\boldsymbol{B}) | | |
| lyche-numerical-linear-algebra_FO2445 | 180 | 1.000 | \mathcal{R}\left(\boldsymbol{A}^{*} \boldsymbol{A}\right)= | | |
| lyche-numerical-linear-algebra_FO2446 | 180 | 1.000 | \mathcal{R}\left(\boldsymbol{V}_{1}\right)=\mathcal{R}\left(\boldsymbol{A}^{*}\right) | | |
| lyche-numerical-linear-algebra_FO2447 | 180 | 1.000 | \boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{m} \in \mathbb{C}^{m} | | |
| lyche-numerical-linear-algebra_FO2448 | 180 | 1.000 | \boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{n} \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO2449 | 181 | 1.000 | \boldsymbol{B}:= | | |
| lyche-numerical-linear-algebra_FO2450 | 181 | 0.981 | \sqrt{\boldsymbol{A}^{T} \boldsymbol{A}} | | |
| lyche-numerical-linear-algebra_FO2451 | 181 | 0.981 | \boldsymbol{P}=\boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO2452 | 181 | 1.000 | \boldsymbol{X}_{k}=\boldsymbol{U} \boldsymbol{\Sigma}_{k} \boldsymbol{V}^{T} | | |
| lyche-numerical-linear-algebra_FO2453 | 181 | 0.985 | \Sigma_{k} | | |
| lyche-numerical-linear-algebra_FO2454 | 181 | 0.985 | k=0,1,2, \ldots | | |
| lyche-numerical-linear-algebra_FO2455 | 181 | 0.749 | [\mathrm{Q}, \mathrm{P}, \mathrm{k}]= | | |
| lyche-numerical-linear-algebra_FO2456 | 181 | 0.749 | (\mathrm{A}, \mathrm{tol}, \mathrm{K}) | | |
| lyche-numerical-linear-algebra_FO2457 | 181 | 1.000 | \boldsymbol{P}=\boldsymbol{Q}^{T} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO2458 | 181 | 0.977 | \boldsymbol{A}=\boldsymbol{Q} \boldsymbol{P} | | |
| lyche-numerical-linear-algebra_FO2459 | 181 | 0.919 | \left\|\boldsymbol{X}_{k+1}-\boldsymbol{X}_{k}\right\|_{F}< | | |
| lyche-numerical-linear-algebra_FO2460 | 181 | 0.919 | *\left\|\boldsymbol{X}_{k+1}\right\|_{F} | | |
| lyche-numerical-linear-algebra_FO2461 | 181 | 0.919 | k=K+1 | | |
| lyche-numerical-linear-algebra_FO2462 | 181 | 1.000 | \|\boldsymbol{A}\|_{1} | | |
| lyche-numerical-linear-algebra_FO2463 | 181 | 1.000 | \|\boldsymbol{A}\|_{\infty} | | |
| lyche-numerical-linear-algebra_FO2464 | 181 | 0.972 | \left(\boldsymbol{B}^{T}\right) | | |
| lyche-numerical-linear-algebra_FO2465 | 181 | 0.972 | \operatorname{ker}(\boldsymbol{B}) | | |
| lyche-numerical-linear-algebra_FO2466 | 182 | 1.000 | \boldsymbol{x} \in \mathbb{R}^{3} | | |
| lyche-numerical-linear-algebra_FO2467 | 182 | 1.000 | \boldsymbol{b} \in \mathbb{R}^{m} | | |
| lyche-numerical-linear-algebra_FO2468 | 182 | 1.000 | \boldsymbol{A}^{T} \boldsymbol{A} \boldsymbol{x}=\boldsymbol{A}^{T} \boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO2469 | 182 | 1.000 | \mathcal{R}(\boldsymbol{A}), \mathcal{R}\left(\boldsymbol{A}^{T}\right), \mathcal{N}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO2470 | 182 | 1.000 | \mathcal{N}\left(\boldsymbol{A}^{T}\right) | | |
| lyche-numerical-linear-algebra_FO2471 | 182 | 0.999 | \boldsymbol{B} \in \mathbb{R}^{4,3} | | |
| lyche-numerical-linear-algebra_FO2472 | 182 | 0.999 | \|\boldsymbol{A}-\boldsymbol{B}\|_{F} \geq 6 | | |
| lyche-numerical-linear-algebra_FO2473 | 182 | 1.000 | \left\|\boldsymbol{A}-\boldsymbol{A}_{1}\right\|_{F}=6 | | |
| lyche-numerical-linear-algebra_FO2474 | 182 | 0.980 | (n, 1) | | |
| lyche-numerical-linear-algebra_FO2475 | 182 | 1.000 | -2^{2-n} | | |
| lyche-numerical-linear-algebra_FO2476 | 182 | 1.000 | \boldsymbol{x}:=\left[2^{n-2}, 2^{n-3}, \ldots, 2^{0}, 1\right]^{T} | | |
| lyche-numerical-linear-algebra_FO2477 | 182 | 1.000 | \operatorname{det}(\boldsymbol{A})=1 | | |
| lyche-numerical-linear-algebra_FO2478 | 182 | 1.000 | \|\boldsymbol{A}-\boldsymbol{B}\|_{F}=2^{2-n} | | |
| lyche-numerical-linear-algebra_FO2479 | 182 | 1.000 | \boldsymbol{A}-\boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO2480 | 182 | 1.000 | \sigma_{n} | | |
| lyche-numerical-linear-algebra_FO2481 | 182 | 1.000 | 2^{2-n} | | |
| lyche-numerical-linear-algebra_FO2482 | 183 | 1.000 | \|\boldsymbol{A}\|_{1},\|\boldsymbol{A}\|_{\infty} | | |
| lyche-numerical-linear-algebra_FO2483 | 183 | 1.000 | \|\boldsymbol{A}\|_{F} | | |
| lyche-numerical-linear-algebra_FO2484 | 183 | 1.000 | \boldsymbol{T}=\boldsymbol{L} \boldsymbol{L}^{T} | | |
| lyche-numerical-linear-algebra_FO2485 | 183 | 1.000 | \boldsymbol{A}^{\prime}=\sum_{i=1}^{r} \sigma_{i} \boldsymbol{u}_{i} \boldsymbol{v}_{i}^{*} | | |
| lyche-numerical-linear-algebra_FO2486 | 183 | 1.000 | 1 \leq r \leq n, \sigma_{i} | | |
| lyche-numerical-linear-algebra_FO2487 | 183 | 1.000 | \boldsymbol{u}_{i}, \boldsymbol{v}_{i} | | |
| lyche-numerical-linear-algebra_FO2488 | 185 | 1.000 | \|\cdot\|: \mathcal{V} \rightarrow \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO2489 | 185 | 0.728 | (\mathcal{V}, \mathbb{R},\|\cdot\|) | | |
| lyche-numerical-linear-algebra_FO2490 | 185 | 0.728 | (\mathcal{V}, \mathbb{C},\|\cdot\|) | | |
| lyche-numerical-linear-algebra_FO2491 | 185 | 1.000 | \mathbb{R}^{n}, \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO2492 | 185 | 1.000 | \mathcal{V}=\mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO2493 | 185 | 1.000 | \mathcal{V}=\mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO2494 | 186 | 0.987 | p \geq 1 | | |
| lyche-numerical-linear-algebra_FO2495 | 186 | 1.000 | p=1,2, \infty | | |
| lyche-numerical-linear-algebra_FO2496 | 186 | 1.000 | \|\boldsymbol{x}\|_{1}:=\sum_{j=1}^{n}\left|x_{j}\right| | | |
| lyche-numerical-linear-algebra_FO2497 | 186 | 1.000 | \|\boldsymbol{x}\|_{2}:=\sqrt{\sum_{j=1}^{n}\left|x_{j}\right|^{2}} | | |
| lyche-numerical-linear-algebra_FO2498 | 186 | 1.000 | l_{2} | | |
| lyche-numerical-linear-algebra_FO2499 | 186 | 1.000 | \|\boldsymbol{x}\|_{\infty}:=\max _{1 \leq j \leq n}\left|x_{j}\right| | | |
| lyche-numerical-linear-algebra_FO2500 | 186 | 1.000 | l_{\infty} | | |
| lyche-numerical-linear-algebra_FO2501 | 186 | 1.000 | 1 \leq p \leq \infty | | |
| lyche-numerical-linear-algebra_FO2502 | 186 | 1.000 | \|\boldsymbol{x}+\boldsymbol{y}\|_{p} \leq\|\boldsymbol{x}\|_{p}+\|\boldsymbol{y}\|_{p} | | |
| lyche-numerical-linear-algebra_FO2503 | 186 | 1.000 | \frac{1}{p}+\frac{1}{q}=1 | | |
| lyche-numerical-linear-algebra_FO2504 | 186 | 1.000 | p=1 | | |
| lyche-numerical-linear-algebra_FO2505 | 186 | 1.000 | q=\infty | | |
| lyche-numerical-linear-algebra_FO2506 | 186 | 1.000 | p=2 | | |
| lyche-numerical-linear-algebra_FO2507 | 186 | 1.000 | \lim _{p \rightarrow \infty} n^{1 / p}=1 | | |
| lyche-numerical-linear-algebra_FO2508 | 187 | 0.987 | \|\cdot\| | | |
| lyche-numerical-linear-algebra_FO2509 | 187 | 0.987 | \|\cdot\|^{\prime} | | |
| lyche-numerical-linear-algebra_FO2510 | 187 | 1.000 | M | | |
| lyche-numerical-linear-algebra_FO2511 | 187 | 1.000 | \boldsymbol{x} \in \mathcal{V} | | |
| lyche-numerical-linear-algebra_FO2512 | 187 | 1.000 | \infty | | |
| lyche-numerical-linear-algebra_FO2513 | 187 | 1.000 | \|\boldsymbol{x}-\boldsymbol{y}\| \geq|\|\boldsymbol{x}\|-\|\boldsymbol{y}\|| | | |
| lyche-numerical-linear-algebra_FO2514 | 187 | 1.000 | \boldsymbol{x}, \boldsymbol{y} \in \mathbb{C}^{n} \quad | | |
| lyche-numerical-linear-algebra_FO2515 | 187 | 1.000 | \mathcal{V} \rightarrow \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO2516 | 187 | 1.000 | \|\boldsymbol{x}\|=\|\boldsymbol{x}-\boldsymbol{y}+\boldsymbol{y}\| \leq\|\boldsymbol{x}-\boldsymbol{y}\|+\|\boldsymbol{y}\| | | |
| lyche-numerical-linear-algebra_FO2517 | 187 | 1.000 | \|\boldsymbol{x}-\boldsymbol{y}\| \geq\|\boldsymbol{x}\|-\|\boldsymbol{y}\| | | |
| lyche-numerical-linear-algebra_FO2518 | 187 | 1.000 | \|\boldsymbol{x}-\boldsymbol{y}\|=\|\boldsymbol{y}-\boldsymbol{x}\| \geq\|\boldsymbol{y}\|-\|\boldsymbol{x}\| | | |
| lyche-numerical-linear-algebra_FO2519 | 187 | 1.000 | f: \mathcal{S} \rightarrow \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO2520 | 187 | 1.000 | f(\boldsymbol{y})=\|\boldsymbol{y}\| | | |
| lyche-numerical-linear-algebra_FO2521 | 187 | 1.000 | \boldsymbol{y}:=\boldsymbol{x} /\|\boldsymbol{x}\|^{\prime} \in \mathcal{S} | | |
| lyche-numerical-linear-algebra_FO2522 | 188 | 0.997 | \left(\mathbb{C}^{m \times n}, \mathbb{C}\right) | | |
| lyche-numerical-linear-algebra_FO2523 | 188 | 1.000 | \left(\mathbb{R}^{m \times n}, \mathbb{R}\right) | | |
| lyche-numerical-linear-algebra_FO2524 | 188 | 1.000 | \left\|\|: \mathbb{C}^{m \times n}, \rightarrow \mathbb{R}\right. | | |
| lyche-numerical-linear-algebra_FO2525 | 188 | 1.000 | m n | | |
| lyche-numerical-linear-algebra_FO2526 | 188 | 1.000 | \mathbb{C}^{m n} | | |
| lyche-numerical-linear-algebra_FO2527 | 188 | 1.000 | \|\cdot\|_{V} | | |
| lyche-numerical-linear-algebra_FO2528 | 188 | 1.000 | \|\boldsymbol{A}\|:= | | |
| lyche-numerical-linear-algebra_FO2529 | 188 | 1.000 | \|\operatorname{vec}(\boldsymbol{A})\|_{V} | | |
| lyche-numerical-linear-algebra_FO2530 | 189 | 1.000 | \|\boldsymbol{A} \boldsymbol{B}\| \leq\|\boldsymbol{A}\|\|\boldsymbol{B}\| | | |
| lyche-numerical-linear-algebra_FO2531 | 189 | 0.998 | \|\| | | |
| lyche-numerical-linear-algebra_FO2532 | 189 | 1.000 | \left\|\|_{\beta}\right. | | |
| lyche-numerical-linear-algebra_FO2533 | 189 | 1.000 | \| \| | | |
| lyche-numerical-linear-algebra_FO2534 | 189 | 0.999 | \|\boldsymbol{A} \boldsymbol{x}\| \leq\|\boldsymbol{A}\|\|\boldsymbol{x}\|_{\beta} | | |
| lyche-numerical-linear-algebra_FO2535 | 189 | 1.000 | m, n \in \mathbb{N}, \boldsymbol{A} \in \mathbb{C}^{m \times n} | | |
| lyche-numerical-linear-algebra_FO2536 | 189 | 1.000 | \boldsymbol{B}:=\boldsymbol{x} \in \mathbb{C}^{n \times 1} | | |
| lyche-numerical-linear-algebra_FO2537 | 190 | 0.969 | n \in \mathbb{N} .^{1} | | |
| lyche-numerical-linear-algebra_FO2538 | 190 | 1.000 | \left\|\|\right. | | |
| lyche-numerical-linear-algebra_FO2539 | 190 | 1.000 | \|\|: \mathcal{S} \rightarrow \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO2540 | 190 | 0.995 | f(\boldsymbol{x})=\|\boldsymbol{A} \boldsymbol{x}\| | | |
| lyche-numerical-linear-algebra_FO2541 | 190 | 1.000 | \| \|_{\beta} | | |
| lyche-numerical-linear-algebra_FO2542 | 190 | 0.587 | \|\boldsymbol{A}\|:=\max _{\boldsymbol{x} \neq 0} \frac{\|\boldsymbol{A} \boldsymbol{x}\|}{\|\boldsymbol{x}\| \beta} | | |
| lyche-numerical-linear-algebra_FO2543 | 191 | 1.000 | \|\boldsymbol{I}\|=1 | | |
| lyche-numerical-linear-algebra_FO2544 | 191 | 1.000 | \|\boldsymbol{A}\|=0 | | |
| lyche-numerical-linear-algebra_FO2545 | 191 | 1.000 | \|\boldsymbol{A} \boldsymbol{y}\|=0 | | |
| lyche-numerical-linear-algebra_FO2546 | 191 | 1.000 | \boldsymbol{A}=0 | | |
| lyche-numerical-linear-algebra_FO2547 | 191 | 1.000 | \|c \boldsymbol{A}\|=\max _{\boldsymbol{x}}\|c \boldsymbol{A} \boldsymbol{x}\|=\max _{\boldsymbol{x}}|c|\|\boldsymbol{A} \boldsymbol{x}\|=|c|\|\boldsymbol{A}\| | | |
| lyche-numerical-linear-algebra_FO2548 | 191 | 1.000 | \|\boldsymbol{A}+\boldsymbol{B}\|=\max _{\boldsymbol{x}}\|(\boldsymbol{A}+\boldsymbol{B}) \boldsymbol{x}\| \leq \max _{\boldsymbol{x}}\|\boldsymbol{A} \boldsymbol{x}\|+\max _{\boldsymbol{x}}\|\boldsymbol{B} \boldsymbol{x}\|=\|\boldsymbol{A}\|+\|\boldsymbol{B}\| | | |
| lyche-numerical-linear-algebra_FO2549 | 191 | 1.000 | \|\boldsymbol{I}\|_{F}=\sqrt{n} | | |
| lyche-numerical-linear-algebra_FO2550 | 191 | 1.000 | n>1 | | |
| lyche-numerical-linear-algebra_FO2552 | 191 | 1.000 | \ell_{p} | | |
| lyche-numerical-linear-algebra_FO2553 | 191 | 1.000 | \left\|\|_{p}\right. | | |
| lyche-numerical-linear-algebra_FO2554 | 192 | 1.000 | \|\boldsymbol{A}\|_{2} | | |
| lyche-numerical-linear-algebra_FO2555 | 192 | 0.998 | K_{p} | | |
| lyche-numerical-linear-algebra_FO2556 | 192 | 0.998 | \|\boldsymbol{A} \boldsymbol{x}\|_{p} \leq K_{p} | | |
| lyche-numerical-linear-algebra_FO2557 | 192 | 0.998 | \|\boldsymbol{x}\|_{p}=1 | | |
| lyche-numerical-linear-algebra_FO2558 | 192 | 1.000 | \boldsymbol{y}_{e} \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO2559 | 192 | 1.000 | \left\|\boldsymbol{y}_{e}\right\|_{p}=1 | | |
| lyche-numerical-linear-algebra_FO2560 | 192 | 1.000 | \left\|\boldsymbol{A} \boldsymbol{y}_{e}\right\|_{p}=K_{p} | | |
| lyche-numerical-linear-algebra_FO2561 | 192 | 1.000 | \|\boldsymbol{A}\|_{p}=\left\|\boldsymbol{A} \boldsymbol{y}_{e}\right\|_{p}=K_{p} | | |
| lyche-numerical-linear-algebra_FO2562 | 192 | 1.000 | K_{2}= | | |
| lyche-numerical-linear-algebra_FO2563 | 192 | 0.998 | \sigma_{1}, \boldsymbol{c}:=\boldsymbol{V}^{*} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO2564 | 192 | 0.998 | \boldsymbol{y}_{e}=\boldsymbol{v}_{1} | | |
| lyche-numerical-linear-algebra_FO2565 | 192 | 0.998 | \sigma_{1} | | |
| lyche-numerical-linear-algebra_FO2566 | 192 | 1.000 | \boldsymbol{V} \boldsymbol{c},\|\boldsymbol{c}\|_{2}=\|\boldsymbol{x}\|_{2}=1 | | |
| lyche-numerical-linear-algebra_FO2567 | 192 | 0.998 | \|\boldsymbol{A} \boldsymbol{x}\|_{2}^{2}=\left\|\boldsymbol{U} \boldsymbol{\Sigma} \boldsymbol{V}^{*} \boldsymbol{x}\right\|_{2}^{2}=\|\boldsymbol{\Sigma} \boldsymbol{c}\|_{2}^{2}=\sum_{j=1}^{n} \sigma_{j}^{2}\left|c_{j}\right|^{2} \leq \sigma_{1}^{2} \sum_{j=1}^{n}\left|c_{j}\right|^{2}=\sigma_{1}^{2} | | |
| lyche-numerical-linear-algebra_FO2568 | 192 | 1.000 | \left\|\boldsymbol{A} \boldsymbol{v}_{1}\right\|_{2}=\left\|\sigma_{1} \boldsymbol{u}_{1}\right\|_{2}=\sigma_{1} | | |
| lyche-numerical-linear-algebra_FO2569 | 192 | 1.000 | K_{1}, c | | |
| lyche-numerical-linear-algebra_FO2570 | 192 | 1.000 | \boldsymbol{y}_{e} | | |
| lyche-numerical-linear-algebra_FO2571 | 192 | 1.000 | K_{1}:=\left\|\boldsymbol{A} \boldsymbol{e}_{c}\right\|_{1}=\max _{1 \leq j \leq n}\left\|\boldsymbol{A} \boldsymbol{e}_{j}\right\|_{1} | | |
| lyche-numerical-linear-algebra_FO2572 | 192 | 1.000 | \boldsymbol{y}_{e}:= | | |
| lyche-numerical-linear-algebra_FO2573 | 192 | 1.000 | \boldsymbol{e}_{c} | | |
| lyche-numerical-linear-algebra_FO2574 | 192 | 1.000 | \left\|\boldsymbol{y}_{e}\right\|_{1}=1 | | |
| lyche-numerical-linear-algebra_FO2575 | 192 | 1.000 | \left\|\boldsymbol{A} \boldsymbol{y}_{e}\right\|_{1}=K_{1} | | |
| lyche-numerical-linear-algebra_FO2576 | 192 | 0.997 | K_{\infty}, r | | |
| lyche-numerical-linear-algebra_FO2577 | 192 | 0.997 | K_{\infty}:=\left\|\boldsymbol{e}_{r}^{T} \boldsymbol{A}\right\|_{1}=\max _{1 \leq k \leq m}\left\|\boldsymbol{e}_{k}^{T} \boldsymbol{A}\right\|_{1} | | |
| lyche-numerical-linear-algebra_FO2578 | 192 | 0.998 | \boldsymbol{y}_{e}:=\left[e^{-i \theta_{1}}, \ldots, e^{-i \theta_{n}}\right]^{T} | | |
| lyche-numerical-linear-algebra_FO2579 | 192 | 0.998 | a_{r j}=\left|a_{r j}\right| e^{i \theta_{j}} | | |
| lyche-numerical-linear-algebra_FO2580 | 192 | 1.000 | \left\|\boldsymbol{A} \boldsymbol{y}^{*}\right\|_{\infty}=\max _{1 \leq k \leq m}\left|\sum_{j=1}^{n} a_{k j} e^{-i \theta_{j}}\right|=K_{\infty} | | |
| lyche-numerical-linear-algebra_FO2581 | 192 | 1.000 | \left|\sum_{j=1}^{n} a_{k j} e^{-i \theta_{j}}\right| \leq \sum_{j=1}^{n}\left|a_{k j}\right| \leq K_{\infty} | | |
| lyche-numerical-linear-algebra_FO2582 | 192 | 1.000 | k=r | | |
| lyche-numerical-linear-algebra_FO2583 | 192 | 0.999 | \boldsymbol{A}:=\frac{1}{15}\left[\begin{array}{ccc}14 & 4 & 16 \\ 2 & 22 & 13\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2584 | 193 | 1.000 | \sigma_{1} \geq \sigma_{2} \geq | | |
| lyche-numerical-linear-algebra_FO2585 | 193 | 1.000 | \cdots \geq \sigma_{n} | | |
| lyche-numerical-linear-algebra_FO2586 | 193 | 1.000 | \left|\lambda_{1}\right| \geq\left|\lambda_{2}\right| \geq \cdots \geq\left|\lambda_{n}\right| | | |
| lyche-numerical-linear-algebra_FO2587 | 193 | 1.000 | 1 / \sigma_{n} | | |
| lyche-numerical-linear-algebra_FO2588 | 193 | 1.000 | \|\boldsymbol{A}\|_{2}^{2} \leq | | |
| lyche-numerical-linear-algebra_FO2589 | 193 | 0.994 | \|\boldsymbol{A}\|_{1}\|\boldsymbol{A}\|_{\infty} | | |
| lyche-numerical-linear-algebra_FO2590 | 193 | 0.994 | \left(\sigma^{2}, \boldsymbol{v}\right) | | |
| lyche-numerical-linear-algebra_FO2591 | 193 | 1.000 | \left\|\boldsymbol{A}^{*}\right\|_{1}=\|\boldsymbol{A}\|_{\infty} | | |
| lyche-numerical-linear-algebra_FO2592 | 193 | 1.000 | \|\boldsymbol{v}\|_{1} | | |
| lyche-numerical-linear-algebra_FO2593 | 193 | 1.000 | \|\boldsymbol{U} \boldsymbol{A} \boldsymbol{V}\|=\|\boldsymbol{A}\| | | |
| lyche-numerical-linear-algebra_FO2594 | 193 | 1.000 | \boldsymbol{U}(\boldsymbol{A}+ | | |
| lyche-numerical-linear-algebra_FO2595 | 193 | 1.000 | \boldsymbol{E}) \boldsymbol{V}=\boldsymbol{U} \boldsymbol{A} \boldsymbol{V}+\boldsymbol{F} | | |
| lyche-numerical-linear-algebra_FO2596 | 193 | 1.000 | \|\boldsymbol{F}\|=\|\boldsymbol{E}\| | | |
| lyche-numerical-linear-algebra_FO2597 | 194 | 1.000 | \left\|\boldsymbol{A}^{*}\right\|_{2}=\|\boldsymbol{A}\|_{2} | | |
| lyche-numerical-linear-algebra_FO2598 | 194 | 1.000 | \boldsymbol{V}^{*} | | |
| lyche-numerical-linear-algebra_FO2599 | 194 | 1.000 | \|\boldsymbol{x}\|=\||\boldsymbol{x}|\| | | |
| lyche-numerical-linear-algebra_FO2600 | 194 | 1.000 | |\boldsymbol{x}|:=\left[\left|x_{1}\right|, \ldots,\left|x_{n}\right|\right]^{T} | | |
| lyche-numerical-linear-algebra_FO2601 | 195 | 1.000 | x_{1}=x_{2}=10 | | |
| lyche-numerical-linear-algebra_FO2602 | 195 | 1.000 | x_{1}=30, x_{2}=-10 | | |
| lyche-numerical-linear-algebra_FO2603 | 195 | 1.000 | 1-10^{-16} | | |
| lyche-numerical-linear-algebra_FO2604 | 195 | 1.000 | 1+10^{-16} | | |
| lyche-numerical-linear-algebra_FO2605 | 195 | 0.994 | \boldsymbol{A}, \boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO2606 | 195 | 1.000 | \boldsymbol{I} \in \mathbb{R}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO2607 | 195 | 0.996 | (\boldsymbol{A}+\boldsymbol{E}) \boldsymbol{y}=\boldsymbol{b}+\boldsymbol{e} | | |
| lyche-numerical-linear-algebra_FO2608 | 195 | 0.996 | \boldsymbol{A}, \boldsymbol{A}+\boldsymbol{E} \in | | |
| lyche-numerical-linear-algebra_FO2609 | 195 | 1.000 | \boldsymbol{b}, \boldsymbol{e} \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO2610 | 195 | 1.000 | \boldsymbol{y}-\boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO2611 | 195 | 1.000 | \|\boldsymbol{y}-\boldsymbol{x}\| | | |
| lyche-numerical-linear-algebra_FO2612 | 195 | 1.000 | \|\boldsymbol{y}-\boldsymbol{x}\| /\|\boldsymbol{x}\| | | |
| lyche-numerical-linear-algebra_FO2613 | 195 | 1.000 | \|\boldsymbol{y}-\boldsymbol{x}\| /\|\boldsymbol{y}\| | | |
| lyche-numerical-linear-algebra_FO2614 | 195 | 0.546 | \boldsymbol{b}, \boldsymbol{e} \in \mathbb{C}^{n}, \boldsymbol{b} \neq \mathbf{0} | | |
| lyche-numerical-linear-algebra_FO2615 | 195 | 0.546 | \boldsymbol{A} \boldsymbol{x}=\boldsymbol{b}, \boldsymbol{A} \boldsymbol{y}=\boldsymbol{b}+\boldsymbol{e} | | |
| lyche-numerical-linear-algebra_FO2616 | 195 | 1.000 | K(\boldsymbol{A})=\|\boldsymbol{A}\|\left\|\boldsymbol{A}^{-1}\right\| | | |
| lyche-numerical-linear-algebra_FO2617 | 195 | 1.000 | \boldsymbol{A} \boldsymbol{y}=\boldsymbol{b}+\boldsymbol{e} | | |
| lyche-numerical-linear-algebra_FO2618 | 195 | 1.000 | \boldsymbol{A}(\boldsymbol{y}-\boldsymbol{x})=\boldsymbol{e} | | |
| lyche-numerical-linear-algebra_FO2619 | 195 | 1.000 | \boldsymbol{y}-\boldsymbol{x}=\boldsymbol{A}^{-1} \boldsymbol{e} | | |
| lyche-numerical-linear-algebra_FO2620 | 195 | 1.000 | \|\boldsymbol{y}-\boldsymbol{x}\|=\left\|\boldsymbol{A}^{-1} \boldsymbol{e}\right\| \leq\left\|\boldsymbol{A}^{-1}\right\|\|\boldsymbol{e}\| | | |
| lyche-numerical-linear-algebra_FO2621 | 195 | 1.000 | \|\boldsymbol{b}\|=\|\boldsymbol{A} \boldsymbol{x}\| \leq\|\boldsymbol{A}\|\|\boldsymbol{x}\| | | |
| lyche-numerical-linear-algebra_FO2622 | 195 | 1.000 | \|\boldsymbol{e}\| \leq\|\boldsymbol{A}\|\|\boldsymbol{y}-\boldsymbol{x}\| | | |
| lyche-numerical-linear-algebra_FO2623 | 195 | 1.000 | \|\boldsymbol{x}\| \leq | | |
| lyche-numerical-linear-algebra_FO2624 | 195 | 1.000 | \left\|\boldsymbol{A}^{-1}\right\|\|\boldsymbol{b}\| | | |
| lyche-numerical-linear-algebra_FO2625 | 196 | 1.000 | \|\boldsymbol{e}\| /\|\boldsymbol{b}\| | | |
| lyche-numerical-linear-algebra_FO2626 | 196 | 1.000 | \boldsymbol{e} | | |
| lyche-numerical-linear-algebra_FO2627 | 196 | 0.999 | K(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO2628 | 196 | 1.000 | K(\boldsymbol{A}) \geq 1 | | |
| lyche-numerical-linear-algebra_FO2629 | 196 | 1.000 | \|\boldsymbol{x}\|=\|\boldsymbol{I} \boldsymbol{x}\| \leq\|\boldsymbol{I}\|\|\boldsymbol{x}\| | | |
| lyche-numerical-linear-algebra_FO2630 | 196 | 1.000 | \|\boldsymbol{I}\| \geq 1 | | |
| lyche-numerical-linear-algebra_FO2631 | 196 | 1.000 | \|\boldsymbol{A}\|\left\|\boldsymbol{A}^{-1}\right\| \geq | | |
| lyche-numerical-linear-algebra_FO2632 | 196 | 1.000 | \left\|\boldsymbol{A} \boldsymbol{A}^{-1}\right\|=\|\boldsymbol{I}\| \geq 1 | | |
| lyche-numerical-linear-algebra_FO2633 | 196 | 1.000 | \ell_{1}, \ell_{\infty} | | |
| lyche-numerical-linear-algebra_FO2634 | 196 | 0.874 | \boldsymbol{A x}=\boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO2635 | 196 | 1.000 | \boldsymbol{r}(\boldsymbol{y}):=\boldsymbol{A} \boldsymbol{y}-\boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO2636 | 196 | 1.000 | \boldsymbol{r} | | |
| lyche-numerical-linear-algebra_FO2637 | 196 | 0.987 | \boldsymbol{b} \neq \mathbf{0} | | |
| lyche-numerical-linear-algebra_FO2638 | 196 | 0.987 | \boldsymbol{r}(\boldsymbol{y})=\boldsymbol{A} \boldsymbol{y}-\boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO2639 | 196 | 1.000 | \boldsymbol{e}=\boldsymbol{r}(\boldsymbol{y}) | | |
| lyche-numerical-linear-algebra_FO2640 | 196 | 0.999 | \boldsymbol{A}, \boldsymbol{E} \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO2641 | 196 | 0.999 | \boldsymbol{A}, \boldsymbol{A}+\boldsymbol{E} | | |
| lyche-numerical-linear-algebra_FO2642 | 196 | 0.999 | (\boldsymbol{A}+\boldsymbol{E}) \boldsymbol{y}=\boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO2643 | 196 | 0.615 | \boldsymbol{A}, \boldsymbol{E} \in \mathbb{C}^{n \times n}, \boldsymbol{b} \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO2644 | 196 | 0.999 | r:=\left\|\boldsymbol{A}^{-1} \boldsymbol{E}\right\|<1 | | |
| lyche-numerical-linear-algebra_FO2645 | 196 | 0.999 | \boldsymbol{A}+\boldsymbol{E} | | |
| lyche-numerical-linear-algebra_FO2646 | 196 | 1.000 | r \geq 1 | | |
| lyche-numerical-linear-algebra_FO2647 | 196 | 1.000 | (\boldsymbol{A}+\boldsymbol{E}) \boldsymbol{x}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO2648 | 196 | 1.000 | \left(\boldsymbol{I}+\boldsymbol{A}^{-1} \boldsymbol{E}\right) \boldsymbol{x}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO2649 | 196 | 1.000 | \|\boldsymbol{x}\|=\left\|\boldsymbol{A}^{-1} \boldsymbol{E} \boldsymbol{x}\right\| \leq r\|\boldsymbol{x}\| | | |
| lyche-numerical-linear-algebra_FO2650 | 196 | 0.967 | \boldsymbol{A}(\boldsymbol{x}-\boldsymbol{y})=\boldsymbol{E} \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO2651 | 196 | 1.000 | \boldsymbol{x}-\boldsymbol{y}=\boldsymbol{A}^{-1} \boldsymbol{E} \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO2652 | 196 | 1.000 | \|\boldsymbol{y}\| | | |
| lyche-numerical-linear-algebra_FO2653 | 197 | 1.000 | \boldsymbol{y}=\left(\boldsymbol{I}+\boldsymbol{A}^{-1} \boldsymbol{E}\right)^{-1} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO2654 | 197 | 0.995 | \|\boldsymbol{y}\| \leq\left\|\left(\boldsymbol{I}+\boldsymbol{A}^{-1} \boldsymbol{E}\right)^{-1}\right\|\|\boldsymbol{x}\| \leq \frac{\|\boldsymbol{x}\|}{1-r} | | |
| lyche-numerical-linear-algebra_FO2655 | 197 | 0.995 | \|\boldsymbol{y}-\boldsymbol{x}\| \leq r\|\boldsymbol{y}\| \leq | | |
| lyche-numerical-linear-algebra_FO2656 | 197 | 0.669 | \frac{r}{1-r}\|\boldsymbol{x}\| \leq \frac{K(\boldsymbol{A})}{1-r}\|\boldsymbol{E}\|\|\boldsymbol{A}\| | | |
| lyche-numerical-linear-algebra_FO2657 | 197 | 0.669 | \|\boldsymbol{x}\| | | |
| lyche-numerical-linear-algebra_FO2658 | 197 | 0.986 | \boldsymbol{x} .\|\boldsymbol{E}\| /\|\boldsymbol{A}\| | | |
| lyche-numerical-linear-algebra_FO2659 | 197 | 1.000 | \|\boldsymbol{E}\| /\|\boldsymbol{A}\| | | |
| lyche-numerical-linear-algebra_FO2660 | 197 | 1.000 | \left\|\boldsymbol{A}^{-1} \boldsymbol{E} \boldsymbol{y}\right\| \leq\left\|\boldsymbol{A}^{-1}\right\|\|\boldsymbol{E}\|\|\boldsymbol{y}\| | | |
| lyche-numerical-linear-algebra_FO2661 | 197 | 1.000 | \left\|\boldsymbol{A}^{-1} \boldsymbol{E} \boldsymbol{y}\right\| | | |
| lyche-numerical-linear-algebra_FO2662 | 197 | 1.000 | \sigma_{1} \geq \sigma_{2} \geq \cdots \geq \sigma_{n}>0 | | |
| lyche-numerical-linear-algebra_FO2663 | 197 | 1.000 | \left|\lambda_{1}\right| \geq\left|\lambda_{2}\right| \geq \cdots \geq | | |
| lyche-numerical-linear-algebra_FO2664 | 197 | 1.000 | \left|\lambda_{n}\right|>0 | | |
| lyche-numerical-linear-algebra_FO2665 | 197 | 0.999 | \sigma_{1} / \sigma_{n} | | |
| lyche-numerical-linear-algebra_FO2666 | 197 | 1.000 | \lambda_{1} / \lambda_{n} | | |
| lyche-numerical-linear-algebra_FO2667 | 197 | 1.000 | \|\boldsymbol{y}-\boldsymbol{x}\| /\|\boldsymbol{x}\| \approx\|\boldsymbol{r}(\boldsymbol{y})\| /\|\boldsymbol{b}\| | | |
| lyche-numerical-linear-algebra_FO2668 | 197 | 1.000 | \|\boldsymbol{b}\| \approx 1 | | |
| lyche-numerical-linear-algebra_FO2669 | 197 | 1.000 | \boldsymbol{B}:=\boldsymbol{A}+\boldsymbol{E} | | |
| lyche-numerical-linear-algebra_FO2670 | 197 | 1.000 | \left\|\boldsymbol{B}^{-1}\right\| | | |
| lyche-numerical-linear-algebra_FO2671 | 197 | 1.000 | \left\|\boldsymbol{A}^{-1}\right\| | | |
| lyche-numerical-linear-algebra_FO2672 | 197 | 1.000 | \left\|\boldsymbol{B}^{-1}-\boldsymbol{A}^{-1}\right\| | | |
| lyche-numerical-linear-algebra_FO2673 | 197 | 1.000 | \left\|\boldsymbol{B}^{-1}\right\| /\left\|\boldsymbol{A}^{-1}\right\|,\left\|\boldsymbol{B}^{-1}-\boldsymbol{A}^{-1}\right\| /\left\|\boldsymbol{B}^{-1}\right\| | | |
| lyche-numerical-linear-algebra_FO2674 | 197 | 1.000 | \| \boldsymbol{B}^{-1}- | | |
| lyche-numerical-linear-algebra_FO2675 | 197 | 1.000 | \boldsymbol{A}^{-1}\|/\| \boldsymbol{A}^{-1} \| | | |
| lyche-numerical-linear-algebra_FO2676 | 198 | 1.000 | \boldsymbol{B}:=\boldsymbol{A}+\boldsymbol{E} \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO2677 | 198 | 0.998 | K(\boldsymbol{A}):=\|\boldsymbol{A}\|\left\|\boldsymbol{A}^{-1}\right\| | | |
| lyche-numerical-linear-algebra_FO2678 | 198 | 0.922 | \left\|\boldsymbol{A}^{-1} \boldsymbol{E}\right\| | | |
| lyche-numerical-linear-algebra_FO2679 | 198 | 0.922 | \left\|\boldsymbol{E} \boldsymbol{A}^{-1}\right\| | | |
| lyche-numerical-linear-algebra_FO2680 | 198 | 0.999 | r<1 | | |
| lyche-numerical-linear-algebra_FO2681 | 198 | 0.999 | -\boldsymbol{E}= | | |
| lyche-numerical-linear-algebra_FO2682 | 198 | 1.000 | \boldsymbol{A}-\boldsymbol{B}=\boldsymbol{A}\left(\boldsymbol{B}^{-1}-\boldsymbol{A}^{-1}\right) \boldsymbol{B}=\boldsymbol{B}\left(\boldsymbol{B}^{-1}-\boldsymbol{A}^{-1}\right) \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO2683 | 198 | 1.000 | \left\|\boldsymbol{A}^{-1}\right\| \leq\left\|\boldsymbol{B}^{-1}\right\|+r\left\|\boldsymbol{B}^{-1}\right\| | | |
| lyche-numerical-linear-algebra_FO2684 | 198 | 1.000 | \left\|\boldsymbol{B}^{-1}\right\| /\left\|\boldsymbol{A}^{-1}\right\| | | |
| lyche-numerical-linear-algebra_FO2685 | 199 | 1.000 | 1 \leq p \leq \infty, \boldsymbol{x}, \boldsymbol{y} \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO2686 | 199 | 1.000 | \|\boldsymbol{x}\|_{p} \geq 0 | | |
| lyche-numerical-linear-algebra_FO2687 | 199 | 1.000 | \|a \boldsymbol{x}\|_{p}=|a|\|\boldsymbol{x}\|_{p} | | |
| lyche-numerical-linear-algebra_FO2688 | 199 | 1.000 | I \subset \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO2689 | 199 | 1.000 | f: I \rightarrow \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO2690 | 199 | 1.000 | x_{1}, x_{2} \in I | | |
| lyche-numerical-linear-algebra_FO2691 | 199 | 1.000 | x_{1}<x_{2} | | |
| lyche-numerical-linear-algebra_FO2692 | 199 | 1.000 | \lambda \in[0,1] | | |
| lyche-numerical-linear-algebra_FO2693 | 199 | 1.000 | \sum_{j=1}^{n} \lambda_{j} x_{j} | | |
| lyche-numerical-linear-algebra_FO2694 | 199 | 1.000 | x_{1}, \ldots, x_{n} | | |
| lyche-numerical-linear-algebra_FO2695 | 199 | 1.000 | \lambda_{j} \geq 0 | | |
| lyche-numerical-linear-algebra_FO2696 | 199 | 1.000 | \sum_{j=1}^{n} \lambda_{j}=1 | | |
| lyche-numerical-linear-algebra_FO2697 | 200 | 1.000 | f \in C^{2}[a, b] | | |
| lyche-numerical-linear-algebra_FO2698 | 200 | 1.000 | f^{\prime \prime}(x) \geq | | |
| lyche-numerical-linear-algebra_FO2699 | 200 | 1.000 | a \leq x_{1} \leq x \leq x_{2} \leq b | | |
| lyche-numerical-linear-algebra_FO2700 | 200 | 1.000 | c \in\left[x_{1}, x_{2}\right] | | |
| lyche-numerical-linear-algebra_FO2701 | 200 | 1.000 | f(x) \leq(1-\lambda) f\left(x_{1}\right)+\lambda f\left(x_{2}\right) | | |
| lyche-numerical-linear-algebra_FO2702 | 200 | 0.999 | I \in \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO2703 | 200 | 1.000 | z_{1}, \ldots, z_{n} \in I | | |
| lyche-numerical-linear-algebra_FO2704 | 200 | 1.000 | n \geq 2 | | |
| lyche-numerical-linear-algebra_FO2705 | 200 | 0.999 | \lambda_{j}, z_{j} | | |
| lyche-numerical-linear-algebra_FO2706 | 200 | 1.000 | \lambda_{i}<1 | | |
| lyche-numerical-linear-algebra_FO2707 | 200 | 1.000 | \lambda_{1}<1 | | |
| lyche-numerical-linear-algebra_FO2708 | 200 | 1.000 | u:=\sum_{j=2}^{n} \frac{\lambda_{j}}{1-\lambda_{1}} z_{j} | | |
| lyche-numerical-linear-algebra_FO2709 | 200 | 1.000 | \sum_{j=2}^{n} \lambda_{j}=1-\lambda_{1} | | |
| lyche-numerical-linear-algebra_FO2710 | 200 | 1.000 | f(u) \leq \sum_{j=2}^{n} \frac{\lambda_{j}}{1-\lambda_{1}} f\left(z_{j}\right) | | |
| lyche-numerical-linear-algebra_FO2711 | 201 | 1.000 | \sum_{j=1}^{n} \lambda_{j} a_{j} | | |
| lyche-numerical-linear-algebra_FO2712 | 201 | 1.000 | a_{1}, \ldots, a_{n} | | |
| lyche-numerical-linear-algebra_FO2713 | 201 | 1.000 | 0^{0}:=0 | | |
| lyche-numerical-linear-algebra_FO2714 | 201 | 1.000 | a_{j} | | |
| lyche-numerical-linear-algebra_FO2715 | 201 | 1.000 | a_{j}>0 | | |
| lyche-numerical-linear-algebra_FO2716 | 201 | 1.000 | f:(0, \infty) \rightarrow \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO2717 | 201 | 1.000 | f(x)=-\log x | | |
| lyche-numerical-linear-algebra_FO2718 | 201 | 1.000 | f^{\prime \prime}(x)=1 / x^{2}>0 | | |
| lyche-numerical-linear-algebra_FO2719 | 201 | 1.000 | x \in(0, \infty) | | |
| lyche-numerical-linear-algebra_FO2720 | 201 | 1.000 | \log \left(a_{1}^{\lambda_{1}} \cdots a_{n}^{\lambda_{n}}\right) \leq \log \left(\sum_{j=1}^{n} \lambda_{j} a_{j}\right) | | |
| lyche-numerical-linear-algebra_FO2721 | 201 | 1.000 | \exp (\log x) | | |
| lyche-numerical-linear-algebra_FO2722 | 201 | 1.000 | =x | | |
| lyche-numerical-linear-algebra_FO2723 | 201 | 1.000 | x>0 | | |
| lyche-numerical-linear-algebra_FO2724 | 201 | 1.000 | \lambda_{j}=\frac{1}{n} | | |
| lyche-numerical-linear-algebra_FO2725 | 201 | 1.000 | p=\infty | | |
| lyche-numerical-linear-algebra_FO2726 | 201 | 1.000 | 1<p< | | |
| lyche-numerical-linear-algebra_FO2727 | 201 | 1.000 | a, b \geq 0 | | |
| lyche-numerical-linear-algebra_FO2728 | 202 | 1.000 | (p-1) q=p | | |
| lyche-numerical-linear-algebra_FO2729 | 202 | 1.000 | a= \pm 1 | | |
| lyche-numerical-linear-algebra_FO2730 | 202 | 1.000 | \langle\boldsymbol{x}, \boldsymbol{x}\rangle=\|\boldsymbol{x}\|^{2} | | |
| lyche-numerical-linear-algebra_FO2731 | 202 | 1.000 | \boldsymbol{x}, \boldsymbol{y} \in \mathcal{V} | | |
| lyche-numerical-linear-algebra_FO2732 | 203 | 0.670 | \langle\boldsymbol{y}, \boldsymbol{z}\rangle=0 | | |
| lyche-numerical-linear-algebra_FO2733 | 203 | 0.670 | \langle\boldsymbol{x}, \boldsymbol{z}\rangle=2\left\langle\frac{\boldsymbol{x}}{2}, \boldsymbol{z}\right\rangle | | |
| lyche-numerical-linear-algebra_FO2734 | 203 | 1.000 | \boldsymbol{x}, \boldsymbol{z} \in \mathcal{V} | | |
| lyche-numerical-linear-algebra_FO2735 | 203 | 1.000 | 2\left\langle\frac{\boldsymbol{x}+\boldsymbol{y}}{2}, \boldsymbol{z}\right\rangle=\langle\boldsymbol{x}+\boldsymbol{y}, \boldsymbol{z}\rangle | | |
| lyche-numerical-linear-algebra_FO2736 | 203 | 1.000 | a=n | | |
| lyche-numerical-linear-algebra_FO2737 | 203 | 1.000 | a>0 | | |
| lyche-numerical-linear-algebra_FO2738 | 203 | 1.000 | \left\{a_{n}\right\} | | |
| lyche-numerical-linear-algebra_FO2739 | 204 | 1.000 | a\langle\boldsymbol{x}, \boldsymbol{y}\rangle=\langle a \boldsymbol{x}, \boldsymbol{y}\rangle | | |
| lyche-numerical-linear-algebra_FO2740 | 204 | 0.651 | a<0 | | |
| lyche-numerical-linear-algebra_FO2741 | 204 | 0.651 | (-a)>0 | | |
| lyche-numerical-linear-algebra_FO2742 | 204 | 1.000 | \mathcal{V}=\mathbb{C}^{n}, 1 \leq p \leq \infty, n \geq 2 | | |
| lyche-numerical-linear-algebra_FO2743 | 204 | 1.000 | \langle\boldsymbol{x}, \boldsymbol{x}\rangle=\|\boldsymbol{x}\|_{p}^{2} | | |
| lyche-numerical-linear-algebra_FO2744 | 204 | 1.000 | p \neq 2 | | |
| lyche-numerical-linear-algebra_FO2745 | 204 | 1.000 | \boldsymbol{x}:=\boldsymbol{e}_{1} | | |
| lyche-numerical-linear-algebra_FO2746 | 204 | 1.000 | \boldsymbol{y}:=\boldsymbol{e}_{2} | | |
| lyche-numerical-linear-algebra_FO2747 | 204 | 1.000 | 0<\lambda_{n} \leq \cdots \leq \lambda_{1} | | |
| lyche-numerical-linear-algebra_FO2748 | 204 | 1.000 | \boldsymbol{b}_{1}, \ldots, \boldsymbol{b}_{n} | | |
| lyche-numerical-linear-algebra_FO2749 | 204 | 1.000 | \boldsymbol{B}_{k}:=\left[\boldsymbol{b}_{1}, \ldots, \boldsymbol{b}_{k}\right] \in \mathbb{R}^{n \times k} | | |
| lyche-numerical-linear-algebra_FO2750 | 204 | 0.973 | \langle\boldsymbol{x}, \boldsymbol{y}\rangle:=\boldsymbol{x}^{T} \boldsymbol{A} \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO2751 | 204 | 1.000 | \|\boldsymbol{x}\|_{\boldsymbol{A}}:=\langle\boldsymbol{x}, \boldsymbol{x}\rangle^{1 / 2} | | |
| lyche-numerical-linear-algebra_FO2752 | 204 | 1.000 | \tilde{\boldsymbol{b}}_{1}:=\boldsymbol{b}_{1} | | |
| lyche-numerical-linear-algebra_FO2753 | 204 | 1.000 | \boldsymbol{B}_{k}^{T} \boldsymbol{A} \boldsymbol{B}_{k} | | |
| lyche-numerical-linear-algebra_FO2754 | 204 | 0.999 | \left\langle\tilde{\boldsymbol{b}}_{k}, \boldsymbol{b}_{j}\right\rangle=0 | | |
| lyche-numerical-linear-algebra_FO2755 | 204 | 0.999 | j=1, \ldots, k-1 | | |
| lyche-numerical-linear-algebra_FO2756 | 204 | 1.000 | \tilde{\boldsymbol{b}}_{k}-\boldsymbol{b}_{k} \in \operatorname{span}\left(\boldsymbol{b}_{1}, \ldots, \boldsymbol{b}_{k-1}\right) | | |
| lyche-numerical-linear-algebra_FO2757 | 205 | 0.985 | \tilde{\boldsymbol{b}}_{1}, \ldots, \tilde{\boldsymbol{b}}_{n} | | |
| lyche-numerical-linear-algebra_FO2758 | 205 | 0.984 | \left\langle\tilde{\boldsymbol{b}}_{i}, \tilde{\boldsymbol{b}}_{j}\right\rangle=0 | | |
| lyche-numerical-linear-algebra_FO2759 | 205 | 0.984 | i, j \leq n, i \neq j | | |
| lyche-numerical-linear-algebra_FO2760 | 205 | 1.000 | \tilde{\boldsymbol{B}}_{n}:=\left[\tilde{\boldsymbol{b}}_{1}, \ldots, \tilde{\boldsymbol{b}}_{n}\right] | | |
| lyche-numerical-linear-algebra_FO2761 | 205 | 1.000 | \boldsymbol{T} \in | | |
| lyche-numerical-linear-algebra_FO2762 | 205 | 1.000 | \boldsymbol{B}_{n}=\tilde{\boldsymbol{B}}_{n} \boldsymbol{T} | | |
| lyche-numerical-linear-algebra_FO2763 | 205 | 1.000 | \left|t_{i j}\right| \leq \frac{1}{2} | | |
| lyche-numerical-linear-algebra_FO2764 | 205 | 0.947 | \left\|\tilde{\boldsymbol{b}}_{k}\right\|_{\boldsymbol{A}}^{2} \leq 2\left\|\tilde{\boldsymbol{b}}_{k+1}\right\|_{\boldsymbol{A}}^{2} | | |
| lyche-numerical-linear-algebra_FO2765 | 205 | 0.947 | \operatorname{det}\left(\boldsymbol{B}_{n}\right)=1 | | |
| lyche-numerical-linear-algebra_FO2766 | 205 | 0.948 | \left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2767 | 205 | 1.000 | \sqrt{m n} | | |
| lyche-numerical-linear-algebra_FO2768 | 205 | 0.985 | \|\boldsymbol{A} \boldsymbol{x}\|_{\infty} \leq | | |
| lyche-numerical-linear-algebra_FO2769 | 205 | 1.000 | \|\boldsymbol{A}\|_{M}\|\boldsymbol{x}\|_{1} | | |
| lyche-numerical-linear-algebra_FO2770 | 205 | 1.000 | \|\boldsymbol{A}\|_{M}=\left|a_{k l}\right| | | |
| lyche-numerical-linear-algebra_FO2771 | 205 | 1.000 | \left\|\boldsymbol{A} \boldsymbol{e}_{l}\right\|_{\infty}=\|\boldsymbol{A}\|_{M}\left\|\boldsymbol{e}_{l}\right\|_{1} | | |
| lyche-numerical-linear-algebra_FO2772 | 205 | 1.000 | \|\boldsymbol{A}\|_{M}=\max _{\boldsymbol{x} \neq \mathbf{0}} \frac{\|\boldsymbol{A} \boldsymbol{x}\|_{\infty}}{\|\boldsymbol{x}\|_{1}} | | |
| lyche-numerical-linear-algebra_FO2773 | 205 | 0.480 | \left\|\tilde{\boldsymbol{b}}_{1}\right\|_{\boldsymbol{A}}^{2} \cdots\left\|\tilde{\boldsymbol{b}}_{n}\right\|_{\boldsymbol{A}}^{2}=\operatorname{det}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO2774 | 206 | 0.991 | \|\boldsymbol{A} \boldsymbol{x}\|_{2} \geq \sigma_{n} | | |
| lyche-numerical-linear-algebra_FO2775 | 206 | 1.000 | \|\boldsymbol{A}\|_{p} | | |
| lyche-numerical-linear-algebra_FO2776 | 206 | 1.000 | \left\|\boldsymbol{A}^{-1}\right\|_{p} | | |
| lyche-numerical-linear-algebra_FO2777 | 206 | 1.000 | \|\boldsymbol{V} \boldsymbol{A}\|_{2}= | | |
| lyche-numerical-linear-algebra_FO2778 | 206 | 1.000 | \boldsymbol{V}^{*} \boldsymbol{V}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO2779 | 206 | 1.000 | 8.12\left(\|\boldsymbol{A} \boldsymbol{U}\|_{2}\right. | | |
| lyche-numerical-linear-algebra_FO2780 | 206 | 1.000 | \left.\boldsymbol{A}\right) | | |
| lyche-numerical-linear-algebra_FO2781 | 206 | 1.000 | \boldsymbol{A} \in \mathbb{R}^{2 \times 2} | | |
| lyche-numerical-linear-algebra_FO2782 | 206 | 1.000 | \boldsymbol{U} \in \mathbb{R}^{2 \times 1} | | |
| lyche-numerical-linear-algebra_FO2783 | 206 | 1.000 | \|\boldsymbol{A} \boldsymbol{U}\|_{2}<\|\boldsymbol{A}\|_{2} | | |
| lyche-numerical-linear-algebra_FO2784 | 206 | 1.000 | \|\boldsymbol{A} \boldsymbol{U}\|_{2}=\|\boldsymbol{A}\|_{2} | | |
| lyche-numerical-linear-algebra_FO2785 | 206 | 0.999 | \|\boldsymbol{A}\|_{p}=\rho(\boldsymbol{A}):= | | |
| lyche-numerical-linear-algebra_FO2786 | 206 | 0.972 | \max \left|\lambda_{i}\right| | | |
| lyche-numerical-linear-algebra_FO2787 | 206 | 1.000 | \boldsymbol{a} \in \mathbb{C}^{m} | | |
| lyche-numerical-linear-algebra_FO2788 | 206 | 1.000 | \boldsymbol{A} \in \mathbb{C}^{m, 1} | | |
| lyche-numerical-linear-algebra_FO2789 | 206 | 1.000 | \|\boldsymbol{A}\|_{p}=\|\boldsymbol{a}\|_{p} | | |
| lyche-numerical-linear-algebra_FO2790 | 206 | 1.000 | |\boldsymbol{A}| \in \mathbb{R}^{m \times n} | | |
| lyche-numerical-linear-algebra_FO2791 | 206 | 1.000 | \left|a_{i j}\right| | | |
| lyche-numerical-linear-algebra_FO2792 | 206 | 0.999 | |\boldsymbol{A}| | | |
| lyche-numerical-linear-algebra_FO2793 | 206 | 0.999 | \boldsymbol{A}=\left[\begin{array}{cc}1+i & -2 \\ 1 & 1-i\end{array}\right], \quad i=\sqrt{-1} | | |
| lyche-numerical-linear-algebra_FO2794 | 206 | 1.000 | \boldsymbol{A} \in \mathbb{C}^{m \times n}\|\boldsymbol{A}\|_{F}=\||\boldsymbol{A}|\|_{F},\|\boldsymbol{A}\|_{p}=\||\boldsymbol{A}|\|_{p} | | |
| lyche-numerical-linear-algebra_FO2795 | 206 | 1.000 | p=1, \infty | | |
| lyche-numerical-linear-algebra_FO2796 | 206 | 1.000 | \boldsymbol{A} \in \mathbb{C}^{m \times n}\|\boldsymbol{A}\|_{2} \leq\||\boldsymbol{A}|\|_{2} | | |
| lyche-numerical-linear-algebra_FO2797 | 206 | 0.806 | \|\boldsymbol{A}\|_{2}<\||\boldsymbol{A}|\|_{2} | | |
| lyche-numerical-linear-algebra_FO2798 | 206 | 1.000 | \left\{\lambda_{1}, \lambda_{2}, \ldots, \lambda_{m}\right\} | | |
| lyche-numerical-linear-algebra_FO2799 | 206 | 1.000 | \boldsymbol{x}_{0} \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO2800 | 207 | 1.000 | \left\{\boldsymbol{x}_{k}\right\}_{k=0}^{m-1} | | |
| lyche-numerical-linear-algebra_FO2801 | 207 | 1.000 | c_{i k} | | |
| lyche-numerical-linear-algebra_FO2802 | 207 | 1.000 | \left\{\left(\sigma_{i}, \boldsymbol{u}_{i}\right)\right\}_{i=1}^{n} | | |
| lyche-numerical-linear-algebra_FO2803 | 207 | 1.000 | l \leq m | | |
| lyche-numerical-linear-algebra_FO2804 | 207 | 1.000 | \boldsymbol{x}_{l}=\boldsymbol{x}_{l+1}=\cdots=\boldsymbol{x}_{m}=\boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO2805 | 207 | 1.000 | \boldsymbol{T}=\operatorname{tridiag}(c, d, c) | | |
| lyche-numerical-linear-algebra_FO2806 | 207 | 1.000 | d>2 c | | |
| lyche-numerical-linear-algebra_FO2807 | 207 | 1.000 | \boldsymbol{T} \boldsymbol{x}=\boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO2808 | 207 | 1.000 | b_{i j}=0 | | |
| lyche-numerical-linear-algebra_FO2809 | 207 | 1.000 | |i-j| \leq 1 | | |
| lyche-numerical-linear-algebra_FO2810 | 207 | 1.000 | \boldsymbol{A}=\boldsymbol{T}+\boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO2811 | 207 | 0.991 | \rho(\boldsymbol{E}):=\max _{i}\left|\lambda_{i}\right| | | |
| lyche-numerical-linear-algebra_FO2812 | 207 | 1.000 | \rho\left(\boldsymbol{T}^{-1} \boldsymbol{B}\right) \leq \rho\left(\boldsymbol{T}^{-1}\right) \rho(\boldsymbol{B}) | | |
| lyche-numerical-linear-algebra_FO2813 | 208 | 1.000 | \delta:=\|\boldsymbol{x}-\boldsymbol{y}\|_{2} /\|\boldsymbol{x}\|_{2} | | |
| lyche-numerical-linear-algebra_FO2814 | 208 | 1.000 | K_{2}(\boldsymbol{A})=\|\boldsymbol{A}\|_{2}\left\|\boldsymbol{A}^{-1}\right\|_{2} | | |
| lyche-numerical-linear-algebra_FO2815 | 208 | 1.000 | \delta | | |
| lyche-numerical-linear-algebra_FO2816 | 208 | 1.000 | b_{2}=2.0 | | |
| lyche-numerical-linear-algebra_FO2817 | 208 | 1.000 | \|\boldsymbol{e}\|_{2}=0.1 | | |
| lyche-numerical-linear-algebra_FO2818 | 208 | 1.000 | b_{1} | | |
| lyche-numerical-linear-algebra_FO2819 | 208 | 1.000 | \boldsymbol{y}_{\boldsymbol{A}} | | |
| lyche-numerical-linear-algebra_FO2820 | 208 | 1.000 | \boldsymbol{y}_{\boldsymbol{A}^{-1}} | | |
| lyche-numerical-linear-algebra_FO2821 | 208 | 1.000 | \left\|\boldsymbol{y}_{\boldsymbol{A}}\right\|=\left\|\boldsymbol{y}_{\boldsymbol{A}^{-1}}\right\|=1 | | |
| lyche-numerical-linear-algebra_FO2822 | 208 | 0.968 | \|\boldsymbol{A}\|=\left\|\boldsymbol{A} \boldsymbol{y}_{\boldsymbol{A}}\right\| | | |
| lyche-numerical-linear-algebra_FO2823 | 208 | 0.968 | \left\|\boldsymbol{A}^{-1}\right\|=\left\|\boldsymbol{A}^{-1} \boldsymbol{y}_{\boldsymbol{A}^{-1}}\right\| | | |
| lyche-numerical-linear-algebra_FO2824 | 208 | 1.000 | \boldsymbol{b}=\boldsymbol{A} \boldsymbol{y}_{\boldsymbol{A}} | | |
| lyche-numerical-linear-algebra_FO2825 | 208 | 1.000 | \boldsymbol{e}=\boldsymbol{y}_{\boldsymbol{A}^{-1}} | | |
| lyche-numerical-linear-algebra_FO2826 | 208 | 0.980 | \boldsymbol{b}=\boldsymbol{y}_{\boldsymbol{A}^{-1}} | | |
| lyche-numerical-linear-algebra_FO2827 | 208 | 0.980 | \boldsymbol{e}=\boldsymbol{A} \boldsymbol{y}_{\boldsymbol{A}} | | |
| lyche-numerical-linear-algebra_FO2828 | 208 | 1.000 | m \geq 1 | | |
| lyche-numerical-linear-algebra_FO2829 | 208 | 1.000 | \boldsymbol{T}:=\operatorname{tridiag}(-1,2,-1) \in \mathbb{R}^{m \times m} | | |
| lyche-numerical-linear-algebra_FO2830 | 208 | 1.000 | \operatorname{cond}_{p}(\boldsymbol{T}):=\|\boldsymbol{T}\|_{p}\left\|\boldsymbol{T}^{-1}\right\|_{p} | | |
| lyche-numerical-linear-algebra_FO2831 | 208 | 1.000 | m \geq 3 | | |
| lyche-numerical-linear-algebra_FO2832 | 208 | 1.000 | \operatorname{cond}_{1}(\boldsymbol{T})=\operatorname{cond}_{\infty}(\boldsymbol{T})=3 | | |
| lyche-numerical-linear-algebra_FO2833 | 208 | 1.000 | m=2 | | |
| lyche-numerical-linear-algebra_FO2834 | 209 | 1.000 | \tan x>x | | |
| lyche-numerical-linear-algebra_FO2835 | 209 | 1.000 | 0<x<\pi / 2 | | |
| lyche-numerical-linear-algebra_FO2836 | 209 | 1.000 | \cot ^{2} x>\frac{1}{x^{2}}-\frac{2}{3} | | |
| lyche-numerical-linear-algebra_FO2837 | 209 | 1.000 | \boldsymbol{I}-\boldsymbol{E} | | |
| lyche-numerical-linear-algebra_FO2838 | 209 | 1.000 | \|\boldsymbol{E}\|<1 | | |
| lyche-numerical-linear-algebra_FO2839 | 209 | 1.000 | (\boldsymbol{I}-\boldsymbol{E})^{-1} | | |
| lyche-numerical-linear-algebra_FO2840 | 209 | 1.000 | \boldsymbol{x}=\left[x_{0}, \ldots, x_{n}\right]^{T} \in \mathbb{R}^{n+1} | | |
| lyche-numerical-linear-algebra_FO2841 | 210 | 0.999 | \boldsymbol{y}:=\left[y_{0}, \ldots, y_{n}\right]^{T} \in \mathbb{R}^{n+1} | | |
| lyche-numerical-linear-algebra_FO2842 | 210 | 0.999 | y_{0}=y_{n} | | |
| lyche-numerical-linear-algebra_FO2843 | 210 | 0.902 | \left(x_{i-1}, x_{i}\right) | | |
| lyche-numerical-linear-algebra_FO2844 | 210 | 1.000 | C^{2}[a, b] | | |
| lyche-numerical-linear-algebra_FO2845 | 210 | 0.664 | s_{i}:=g^{\prime}\left(x_{i}\right), i=0, \ldots, n | | |
| lyche-numerical-linear-algebra_FO2846 | 210 | 1.000 | \left[s_{1}, \ldots, s_{n}\right]^{T} | | |
| lyche-numerical-linear-algebra_FO2847 | 211 | 1.000 | \left\|\boldsymbol{A}^{-1}\right\|_{\infty} \leq 1 | | |
| lyche-numerical-linear-algebra_FO2848 | 211 | 1.000 | \left\|\boldsymbol{A}^{-1}\right\|_{1} \leq \frac{3}{2} | | |
| lyche-numerical-linear-algebra_FO2849 | 211 | 1.000 | \boldsymbol{e} \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO2850 | 211 | 1.000 | \|\boldsymbol{e}\|_{p} /\|\boldsymbol{b}\|_{p} \leq 0.01 | | |
| lyche-numerical-linear-algebra_FO2851 | 211 | 1.000 | \hat{\boldsymbol{s}} | | |
| lyche-numerical-linear-algebra_FO2852 | 211 | 1.000 | \boldsymbol{A} \in \mathbb{R}^{m \times n}, \boldsymbol{b} \in \mathbb{R}^{m} | | |
| lyche-numerical-linear-algebra_FO2853 | 211 | 1.000 | \boldsymbol{U} \in \mathbb{R}^{m \times n} | | |
| lyche-numerical-linear-algebra_FO2854 | 211 | 1.000 | \boldsymbol{\Sigma}, \boldsymbol{V} \in \mathbb{R}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO2855 | 211 | 0.932 | [\mathrm{x}, \mathrm{K}]=l \mathrm{sq}(\mathrm{A}, \mathrm{b}) | | |
| lyche-numerical-linear-algebra_FO2856 | 211 | 1.000 | \boldsymbol{x}=\boldsymbol{V} \boldsymbol{\Sigma}^{-1} \boldsymbol{U}^{T} \boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO2857 | 211 | 1.000 | \boldsymbol{A} \boldsymbol{x}= | | |
| lyche-numerical-linear-algebra_FO2858 | 211 | 0.909 | [\mathrm{U}, \mathrm{Sigma}, \mathrm{V}]=\operatorname{svd}(\mathrm{A}, 0) | | |
| lyche-numerical-linear-algebra_FO2859 | 211 | 0.998 | \|\cdot\| p | | |
| lyche-numerical-linear-algebra_FO2860 | 211 | 1.000 | p=1, p=\infty | | |
| lyche-numerical-linear-algebra_FO2861 | 211 | 0.992 | \langle\boldsymbol{x}, \boldsymbol{y}\rangle=s(\boldsymbol{x}, \boldsymbol{y})+i s(\boldsymbol{x}, i \boldsymbol{y}) | | |
| lyche-numerical-linear-algebra_FO2862 | 211 | 0.992 | s(\boldsymbol{x}, \boldsymbol{y}):=\frac{1}{4}\left(\|\boldsymbol{x}+\boldsymbol{y}\|^{2}-\|\boldsymbol{x}-\boldsymbol{y}\|^{2}\right) | | |
| lyche-numerical-linear-algebra_FO2863 | 212 | 1.000 | S_{p} | | |
| lyche-numerical-linear-algebra_FO2864 | 212 | 1.000 | \|\boldsymbol{x}\|_{\infty} \leq\|\boldsymbol{x}\|_{p} \leq n^{1 / p}\|\boldsymbol{x}\|_{\infty} | | |
| lyche-numerical-linear-algebra_FO2865 | 212 | 1.000 | \boldsymbol{x}_{l} | | |
| lyche-numerical-linear-algebra_FO2866 | 212 | 1.000 | \left\|\boldsymbol{x}_{l}\right\|_{\infty}=\left\|\boldsymbol{x}_{l}\right\|_{p} | | |
| lyche-numerical-linear-algebra_FO2867 | 212 | 1.000 | \boldsymbol{x}_{u} | | |
| lyche-numerical-linear-algebra_FO2868 | 212 | 1.000 | \left\|\boldsymbol{x}_{u}\right\|_{p}=n^{1 / p}\left\|\boldsymbol{x}_{u}\right\|_{\infty} | | |
| lyche-numerical-linear-algebra_FO2869 | 212 | 1.000 | 1 \leq q \leq p \leq \infty | | |
| lyche-numerical-linear-algebra_FO2870 | 212 | 1.000 | f(z)=z^{p / q} | | |
| lyche-numerical-linear-algebra_FO2871 | 212 | 1.000 | z_{i}=\left|x_{i}\right|^{q} | | |
| lyche-numerical-linear-algebra_FO2872 | 212 | 1.000 | y_{i}=x_{i} /\|\boldsymbol{x}\|_{\infty} | | |
| lyche-numerical-linear-algebra_FO2873 | 212 | 0.551 | \| \boldsymbol{A} \boldsymbol{x}\|\leq\| \boldsymbol{A}\left\|_{F}\right\| \boldsymbol{x} \| | | |
| lyche-numerical-linear-algebra_FO2874 | 212 | 0.980 | \mathbb{C}^{n \times n}, \boldsymbol{x} \in \mathbb{C}^{n}, m, n \in \mathbb{N} | | |
| lyche-numerical-linear-algebra_FO2875 | 212 | 1.000 | \|\boldsymbol{A}\|_{2} \leq\|\boldsymbol{A}\|_{F} | | |
| lyche-numerical-linear-algebra_FO2876 | 213 | 1.000 | \boldsymbol{b} \notin \mathcal{R}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO2877 | 213 | 0.999 | \|\boldsymbol{A} \boldsymbol{x}-\boldsymbol{b}\| | | |
| lyche-numerical-linear-algebra_FO2878 | 213 | 1.000 | \boldsymbol{b} \in \mathbb{C}^{m} | | |
| lyche-numerical-linear-algebra_FO2879 | 213 | 1.000 | E: \mathbb{C}^{n} \rightarrow \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO2880 | 213 | 1.000 | E(\boldsymbol{x}) | | |
| lyche-numerical-linear-algebra_FO2881 | 213 | 1.000 | \sqrt{E(\boldsymbol{x})} | | |
| lyche-numerical-linear-algebra_FO2882 | 214 | 1.000 | 6 x_{1}-8=0 | | |
| lyche-numerical-linear-algebra_FO2883 | 214 | 1.000 | x_{1}=4 / 3 | | |
| lyche-numerical-linear-algebra_FO2884 | 214 | 1.000 | b_{1}, b_{2}, b_{3} | | |
| lyche-numerical-linear-algebra_FO2885 | 214 | 1.000 | \boldsymbol{A}^{*} \boldsymbol{A} \boldsymbol{x}=\boldsymbol{A}^{*} \boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO2886 | 214 | 1.000 | p(t)=x_{1}+x_{2} t | | |
| lyche-numerical-linear-algebra_FO2887 | 214 | 1.000 | \left(t_{k}, y_{k}\right) \in \mathbb{R}^{2}, k=1, \ldots, m | | |
| lyche-numerical-linear-algebra_FO2888 | 214 | 1.000 | m>2 | | |
| lyche-numerical-linear-algebra_FO2889 | 214 | 1.000 | \left\{t_{1}, \ldots, t_{m}\right\} | | |
| lyche-numerical-linear-algebra_FO2890 | 214 | 1.000 | t_{i} \neq t_{j} | | |
| lyche-numerical-linear-algebra_FO2891 | 215 | 1.000 | t_{1}=\cdots=t_{m} | | |
| lyche-numerical-linear-algebra_FO2892 | 215 | 0.970 | t_{1} \neq t_{2} | | |
| lyche-numerical-linear-algebra_FO2893 | 215 | 1.000 | p\left(t_{k}\right)=y_{k}, k=1,2 | | |
| lyche-numerical-linear-algebra_FO2894 | 215 | 1.000 | \left[\begin{array}{cc}4 & 10 \\ 10 & 30\end{array}\right]\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]=\left[\begin{array}{c}6 \\ 10.1\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO2895 | 215 | 1.000 | p(t)=x_{1}+x_{2} t=3.95-0.98 t | | |
| lyche-numerical-linear-algebra_FO2896 | 215 | 1.000 | y \in \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO2897 | 216 | 0.994 | \min \|\boldsymbol{A} \boldsymbol{x}-\boldsymbol{b}\|_{2}^{2} | | |
| lyche-numerical-linear-algebra_FO2898 | 216 | 1.000 | 1 \leq n \leq m | | |
| lyche-numerical-linear-algebra_FO2899 | 216 | 1.000 | \mathcal{S}:=\left\{t_{1}, t_{2}, \ldots, t_{m}\right\} \subset[a, b] | | |
| lyche-numerical-linear-algebra_FO2900 | 216 | 1.000 | \boldsymbol{y}=\left[y_{1}, y_{2}, \ldots, y_{m}\right]^{*} \in \mathbb{R}^{m} | | |
| lyche-numerical-linear-algebra_FO2901 | 216 | 1.000 | \phi_{j}:[a, b] \rightarrow \mathbb{R}, j=1, \ldots, n | | |
| lyche-numerical-linear-algebra_FO2902 | 216 | 0.999 | p:[a, b] \rightarrow \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO2903 | 216 | 0.999 | p:=\sum_{j=1}^{n} x_{j} \phi_{j} | | |
| lyche-numerical-linear-algebra_FO2904 | 216 | 1.000 | p\left(t_{k}\right) \approx y_{k} | | |
| lyche-numerical-linear-algebra_FO2905 | 217 | 1.000 | \phi_{j} | | |
| lyche-numerical-linear-algebra_FO2906 | 217 | 1.000 | w_{k}>0 | | |
| lyche-numerical-linear-algebra_FO2907 | 217 | 1.000 | y_{k} | | |
| lyche-numerical-linear-algebra_FO2908 | 217 | 1.000 | w_{k} | | |
| lyche-numerical-linear-algebra_FO2909 | 217 | 1.000 | p\left(t_{k}\right)-y_{k} | | |
| lyche-numerical-linear-algebra_FO2910 | 217 | 1.000 | \delta y_{k} | | |
| lyche-numerical-linear-algebra_FO2911 | 217 | 1.000 | w_{k}=1 /\left(\delta y_{k}\right)^{2}, k=1,2, \ldots, m | | |
| lyche-numerical-linear-algebra_FO2912 | 217 | 1.000 | w_{k}=1 | | |
| lyche-numerical-linear-algebra_FO2913 | 217 | 1.000 | 4.2 \boldsymbol{A}^{*} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO2914 | 217 | 0.989 | (\boldsymbol{A} \boldsymbol{x})_{k}=\sum_{j=1}^{n} x_{j} \phi_{j}\left(t_{k}\right), k=1, \ldots, m | | |
| lyche-numerical-linear-algebra_FO2915 | 217 | 1.000 | \phi_{j}(t):=t^{j-1} | | |
| lyche-numerical-linear-algebra_FO2916 | 217 | 1.000 | t_{k}=(k-1) /(m-1) | | |
| lyche-numerical-linear-algebra_FO2917 | 217 | 1.000 | \boldsymbol{B}_{n} \boldsymbol{x}=\boldsymbol{c}_{n} | | |
| lyche-numerical-linear-algebra_FO2918 | 217 | 1.000 | \boldsymbol{B}_{n} | | |
| lyche-numerical-linear-algebra_FO2919 | 217 | 1.000 | t | | |
| lyche-numerical-linear-algebra_FO2920 | 217 | 1.000 | \frac{1}{m} \boldsymbol{B}_{n} \approx \boldsymbol{H}_{n} | | |
| lyche-numerical-linear-algebra_FO2921 | 217 | 1.000 | \boldsymbol{H}_{n} \in \mathbb{R}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO2922 | 217 | 1.000 | 1 /(i+j-1) | | |
| lyche-numerical-linear-algebra_FO2923 | 218 | 1.000 | \frac{1}{m} \boldsymbol{B}_{n} | | |
| lyche-numerical-linear-algebra_FO2924 | 218 | 1.000 | \boldsymbol{H}_{n} | | |
| lyche-numerical-linear-algebra_FO2925 | 218 | 1.000 | \boldsymbol{B}_{n}=\left[b_{i, j}\right]_{i, j=1}^{n} | | |
| lyche-numerical-linear-algebra_FO2926 | 218 | 1.000 | \boldsymbol{H}_{n}^{-1} | | |
| lyche-numerical-linear-algebra_FO2927 | 218 | 1.000 | K_{1}\left(\boldsymbol{H}_{6}\right):= | | |
| lyche-numerical-linear-algebra_FO2928 | 218 | 0.998 | \left\|\boldsymbol{H}_{6}\right\|_{1}\left\|\boldsymbol{H}_{6}^{-1}\right\|_{1} \approx 3 \cdot 10^{7} | | |
| lyche-numerical-linear-algebra_FO2929 | 218 | 1.000 | (t-\tilde{t})^{j-1}, j=1, \ldots, n | | |
| lyche-numerical-linear-algebra_FO2930 | 218 | 1.000 | \tilde{t} | | |
| lyche-numerical-linear-algebra_FO2931 | 218 | 1.000 | \langle\boldsymbol{x}, \boldsymbol{y}\rangle=\boldsymbol{y}^{*} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO2932 | 218 | 1.000 | \mathcal{S}:=\mathcal{R}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO2933 | 218 | 1.000 | \mathcal{T}:=\mathcal{N}\left(\boldsymbol{A}^{*}\right) | | |
| lyche-numerical-linear-algebra_FO2934 | 218 | 1.000 | \boldsymbol{b}=\boldsymbol{b}_{1}+\boldsymbol{b}_{2} | | |
| lyche-numerical-linear-algebra_FO2935 | 218 | 1.000 | \boldsymbol{b}_{1} | | |
| lyche-numerical-linear-algebra_FO2936 | 218 | 1.000 | \boldsymbol{b}_{2} | | |
| lyche-numerical-linear-algebra_FO2937 | 218 | 1.000 | \boldsymbol{A} \boldsymbol{x}-\boldsymbol{b}_{1} \in \mathcal{R}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO2938 | 218 | 1.000 | \boldsymbol{b}_{2} \in \mathcal{N}\left(\boldsymbol{A}^{*}\right) | | |
| lyche-numerical-linear-algebra_FO2939 | 218 | 1.000 | \left\langle\boldsymbol{A} \boldsymbol{x}-\boldsymbol{b}_{1}, \boldsymbol{b}_{2}\right\rangle=0 | | |
| lyche-numerical-linear-algebra_FO2940 | 219 | 1.000 | \boldsymbol{A} \boldsymbol{x}=\boldsymbol{b}_{1} | | |
| lyche-numerical-linear-algebra_FO2941 | 219 | 1.000 | \boldsymbol{b}_{1} \in \mathcal{R}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO2942 | 219 | 1.000 | \boldsymbol{A} \boldsymbol{x}-\boldsymbol{b}_{1}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO2943 | 219 | 0.699 | \boldsymbol{A}^{*}(\boldsymbol{A x}-\boldsymbol{b})=\boldsymbol{A}^{*}\left(\boldsymbol{A x}-\boldsymbol{b}_{1}\right)=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO2944 | 219 | 1.000 | \boldsymbol{A}^{*} \boldsymbol{b}_{2}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO2945 | 219 | 0.995 | \boldsymbol{A}^{*}\left(\boldsymbol{A} \boldsymbol{x}-\boldsymbol{b}_{1}\right)=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO2946 | 219 | 0.995 | \boldsymbol{A} \boldsymbol{x}-\boldsymbol{b}_{1} \in \mathcal{R}(\boldsymbol{A}) \cap \mathcal{N}\left(\boldsymbol{A}^{*}\right) | | |
| lyche-numerical-linear-algebra_FO2947 | 219 | 1.000 | m \geq n, \boldsymbol{A} \in \mathbb{C}^{m \times n}, \boldsymbol{b} \in \mathbb{C}^{m} | | |
| lyche-numerical-linear-algebra_FO2948 | 219 | 1.000 | \boldsymbol{B}:=\boldsymbol{A}^{*} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO2949 | 219 | 0.998 | \left(\boldsymbol{A}^{*} \boldsymbol{A}\right)_{i, j}=\sum_{k=1}^{m} \bar{a}_{k, i} a_{k, j}, i, j=1, \ldots, n | | |
| lyche-numerical-linear-algebra_FO2950 | 219 | 1.000 | \boldsymbol{A}^{*} \boldsymbol{A}=\sum_{k=1}^{m}\left[\begin{array}{c}\bar{a}_{k, 1} \\ \vdots \\ \bar{a}_{k, n}\end{array}\right]\left[a_{k 1} \cdots a_{k n}\right], \boldsymbol{A}^{*} \boldsymbol{b}=\sum_{k=1}^{m}\left[\begin{array}{c}\bar{a}_{k, 1} \\ \vdots \\ \bar{a}_{k, n}\end{array}\right] b_{k} | | |
| lyche-numerical-linear-algebra_FO2951 | 219 | 1.000 | n(n+1) / 2 | | |
| lyche-numerical-linear-algebra_FO2952 | 219 | 1.000 | m n^{2} | | |
| lyche-numerical-linear-algebra_FO2953 | 220 | 1.000 | \boldsymbol{B}, 2 m n | | |
| lyche-numerical-linear-algebra_FO2954 | 220 | 1.000 | \boldsymbol{c}:=\boldsymbol{A}^{*} \boldsymbol{b}, n^{3} / 3 | | |
| lyche-numerical-linear-algebra_FO2955 | 220 | 1.000 | \boldsymbol{B}=\boldsymbol{L} \boldsymbol{L}^{*}, n^{2} | | |
| lyche-numerical-linear-algebra_FO2956 | 220 | 1.000 | \boldsymbol{L} \boldsymbol{y}=\boldsymbol{c} | | |
| lyche-numerical-linear-algebra_FO2957 | 220 | 1.000 | \boldsymbol{L}^{*} \boldsymbol{x}=\boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO2958 | 220 | 1.000 | m \approx n | | |
| lyche-numerical-linear-algebra_FO2959 | 220 | 1.000 | \frac{4}{3} n^{3}= | | |
| lyche-numerical-linear-algebra_FO2960 | 220 | 1.000 | \lambda_{1} \geq \cdots \geq \lambda_{n}>0 | | |
| lyche-numerical-linear-algebra_FO2961 | 220 | 1.000 | \sigma_{1} \geq \cdots \geq \sigma_{n}>0 | | |
| lyche-numerical-linear-algebra_FO2962 | 220 | 1.000 | K_{2}(\boldsymbol{A})=\frac{\sigma_{1}}{\sigma_{n}} | | |
| lyche-numerical-linear-algebra_FO2963 | 220 | 0.999 | K_{2}\left(\boldsymbol{A}^{*} \boldsymbol{A}\right)=\frac{\lambda_{1}}{\lambda_{n}} | | |
| lyche-numerical-linear-algebra_FO2964 | 220 | 0.999 | \lambda_{i}=\sigma_{i}^{2} | | |
| lyche-numerical-linear-algebra_FO2965 | 220 | 1.000 | \boldsymbol{R}_{1}^{*} | | |
| lyche-numerical-linear-algebra_FO2966 | 220 | 1.000 | \boldsymbol{c}_{1} | | |
| lyche-numerical-linear-algebra_FO2967 | 220 | 0.997 | \boldsymbol{R}=\boldsymbol{Q}^{*} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO2968 | 220 | 0.997 | \boldsymbol{c}=\boldsymbol{Q}^{*} \boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO2969 | 220 | 1.000 | \boldsymbol{c} | | |
| lyche-numerical-linear-algebra_FO2970 | 220 | 0.769 | [\mathrm{R}, \mathrm{c}]= | | |
| lyche-numerical-linear-algebra_FO2971 | 221 | 0.958 | \boldsymbol{x}=[1,0,0]^{*} | | |
| lyche-numerical-linear-algebra_FO2972 | 221 | 1.000 | \boldsymbol{R}_{1} \boldsymbol{x}=\boldsymbol{c}_{1} | | |
| lyche-numerical-linear-algebra_FO2973 | 221 | 1.000 | K_{2}\left(\boldsymbol{R}_{1}\right)=K_{2}\left(\boldsymbol{Q}_{1} \boldsymbol{R}_{1}\right)=K_{2}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO2974 | 221 | 1.000 | K_{2}\left(\boldsymbol{A}^{*} \boldsymbol{A}\right) | | |
| lyche-numerical-linear-algebra_FO2975 | 221 | 1.000 | 2 m n^{2}- | | |
| lyche-numerical-linear-algebra_FO2976 | 221 | 1.000 | \mathrm{x}=\mathrm{A} \backslash \mathrm{b} | | |
| lyche-numerical-linear-algebra_FO2977 | 221 | 0.996 | \mathrm{x}=\operatorname{lscov}(\mathrm{A}, \mathrm{b}) | | |
| lyche-numerical-linear-algebra_FO2978 | 222 | 1.000 | \boldsymbol{U}_{2} | | |
| lyche-numerical-linear-algebra_FO2979 | 222 | 1.000 | \boldsymbol{V}_{2} | | |
| lyche-numerical-linear-algebra_FO2980 | 222 | 1.000 | \boldsymbol{A}^{\dagger} \in \mathbb{C}^{n \times m} | | |
| lyche-numerical-linear-algebra_FO2981 | 222 | 1.000 | \boldsymbol{U}_{1} \boldsymbol{\Sigma}_{1} \boldsymbol{V}_{1}^{*} | | |
| lyche-numerical-linear-algebra_FO2982 | 222 | 1.000 | \left(\boldsymbol{A}^{\dagger} \boldsymbol{A}\right)^{*}=\boldsymbol{V}_{1} \boldsymbol{V}_{1}^{*} | | |
| lyche-numerical-linear-algebra_FO2983 | 222 | 1.000 | \left(\boldsymbol{A} \boldsymbol{A}^{\dagger}\right)^{*}=\boldsymbol{U}_{1} \boldsymbol{U}_{1}^{*} | | |
| lyche-numerical-linear-algebra_FO2984 | 222 | 1.000 | \boldsymbol{A}^{\dagger} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO2985 | 222 | 1.000 | \boldsymbol{A} \boldsymbol{A}^{\dagger} | | |
| lyche-numerical-linear-algebra_FO2986 | 223 | 1.000 | \boldsymbol{A}^{\dagger} | | |
| lyche-numerical-linear-algebra_FO2987 | 223 | 0.999 | \boldsymbol{A}^{-1}=\boldsymbol{A}^{\dagger} | | |
| lyche-numerical-linear-algebra_FO2988 | 223 | 1.000 | \boldsymbol{A}^{-1} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO2989 | 223 | 1.000 | \boldsymbol{A} \boldsymbol{A}^{-1} | | |
| lyche-numerical-linear-algebra_FO2990 | 223 | 1.000 | \boldsymbol{A} \boldsymbol{A}^{-1} \boldsymbol{A}=\boldsymbol{A}, \boldsymbol{A}^{-1} \boldsymbol{A} \boldsymbol{A}^{-1}=\boldsymbol{A}^{-1} | | |
| lyche-numerical-linear-algebra_FO2991 | 223 | 1.000 | \mathcal{R}(\boldsymbol{A}), \mathcal{N}\left(\boldsymbol{A}^{*}\right) | | |
| lyche-numerical-linear-algebra_FO2992 | 223 | 1.000 | \boldsymbol{v} \in \mathbb{C}^{m} | | |
| lyche-numerical-linear-algebra_FO2993 | 223 | 1.000 | \boldsymbol{P}_{\mathcal{S}} | | |
| lyche-numerical-linear-algebra_FO2994 | 223 | 1.000 | \boldsymbol{P}_{\mathcal{S}} \boldsymbol{v} | | |
| lyche-numerical-linear-algebra_FO2995 | 223 | 0.998 | \boldsymbol{A}=\boldsymbol{U} \boldsymbol{\Sigma} \boldsymbol{V}^{*} \in \mathbb{C}^{m \times n} | | |
| lyche-numerical-linear-algebra_FO2996 | 223 | 1.000 | \boldsymbol{s}=\boldsymbol{U}_{1} \boldsymbol{U}_{1}^{*} \boldsymbol{v} \in \mathcal{R}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO2997 | 223 | 1.000 | \boldsymbol{t}=\boldsymbol{U}_{2} \boldsymbol{U}_{2}^{*} \boldsymbol{v} \in \mathcal{N}\left(\boldsymbol{A}^{*}\right) | | |
| lyche-numerical-linear-algebra_FO2998 | 223 | 1.000 | \boldsymbol{v}=\left(\boldsymbol{U}_{1} \boldsymbol{U}_{1}^{*}+\boldsymbol{U}_{2} \boldsymbol{U}_{2}^{*}\right) \boldsymbol{v} | | |
| lyche-numerical-linear-algebra_FO2999 | 223 | 0.999 | \boldsymbol{U}_{1} \boldsymbol{U}_{1}^{*}+\boldsymbol{U}_{2} \boldsymbol{U}_{2}^{*}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO3000 | 223 | 0.999 | \boldsymbol{U}_{2} \boldsymbol{U}_{2}^{*}=\boldsymbol{I}-\boldsymbol{U}_{1} \boldsymbol{U}_{1}^{*}=\boldsymbol{I}-\boldsymbol{A} \boldsymbol{A}^{\dagger} | | |
| lyche-numerical-linear-algebra_FO3001 | 223 | 0.848 | \min _{x}\|\boldsymbol{A} \boldsymbol{x}-b\|_{2}^{2} | | |
| lyche-numerical-linear-algebra_FO3002 | 223 | 0.848 | \boldsymbol{x}=\boldsymbol{A}^{\dagger} \boldsymbol{b}+\boldsymbol{z} | | |
| lyche-numerical-linear-algebra_FO3003 | 223 | 0.893 | \boldsymbol{z} \in \mathcal{N}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO3004 | 224 | 1.000 | \boldsymbol{b}_{1}:=\boldsymbol{A} \boldsymbol{A}^{\dagger} \boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO3005 | 224 | 0.998 | \boldsymbol{z}:=\boldsymbol{x}-\boldsymbol{A}^{\dagger} \boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO3006 | 224 | 0.998 | \boldsymbol{A} \boldsymbol{z}= | | |
| lyche-numerical-linear-algebra_FO3007 | 224 | 1.000 | \boldsymbol{A} \boldsymbol{x}-\boldsymbol{A} \boldsymbol{A}^{\dagger} \boldsymbol{b}=\boldsymbol{b}_{1}-\boldsymbol{b}_{1}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO3008 | 224 | 0.997 | \boldsymbol{A} \boldsymbol{z}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO3009 | 224 | 0.997 | \boldsymbol{A} \boldsymbol{x}=\boldsymbol{A}\left(\boldsymbol{A}^{\dagger} \boldsymbol{b}+\boldsymbol{z}\right)=\boldsymbol{b}_{1} | | |
| lyche-numerical-linear-algebra_FO3010 | 224 | 1.000 | \boldsymbol{A}^{\dagger} \boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO3011 | 224 | 1.000 | \boldsymbol{x}=\boldsymbol{A}^{\dagger} \boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO3012 | 224 | 0.999 | \boldsymbol{z}=\boldsymbol{V}_{2} \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO3013 | 224 | 0.999 | \boldsymbol{V}_{2}^{*} \boldsymbol{V}_{1}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO3014 | 224 | 1.000 | \boldsymbol{z}^{*} \boldsymbol{A}^{\dagger} \boldsymbol{b}=\boldsymbol{y}^{*} \boldsymbol{V}_{2}^{*} \boldsymbol{V}_{1} \boldsymbol{\Sigma}^{-1} \boldsymbol{U}_{1}^{*} \boldsymbol{b}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO3015 | 224 | 1.000 | \|\boldsymbol{x}\|_{2}^{2}=\left\|\boldsymbol{A}^{\dagger} \boldsymbol{b}+\boldsymbol{z}\right\|_{2}^{2}=\left\|\boldsymbol{A}^{\dagger} \boldsymbol{b}\right\|_{2}^{2}+\|\boldsymbol{z}\|_{2}^{2} \geq\left\|\boldsymbol{A}^{\dagger} \boldsymbol{b}\right\|_{2}^{2} | | |
| lyche-numerical-linear-algebra_FO3016 | 224 | 0.997 | \boldsymbol{A}=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO3017 | 224 | 0.997 | \boldsymbol{b}=[1,1]^{*} | | |
| lyche-numerical-linear-algebra_FO3018 | 224 | 1.000 | [1 / 2,1 / 2]+[a,-a] | | |
| lyche-numerical-linear-algebra_FO3019 | 224 | 0.653 | a=1 / 2 | | |
| lyche-numerical-linear-algebra_FO3020 | 224 | 0.761 | [1 / 2,1 / 2] | | |
| lyche-numerical-linear-algebra_FO3021 | 224 | 0.933 | \min \|\boldsymbol{A} \boldsymbol{x}-\boldsymbol{b}\|_{2} | | |
| lyche-numerical-linear-algebra_FO3022 | 225 | 1.000 | \boldsymbol{b}, \boldsymbol{e} \in \mathbb{C}^{m} | | |
| lyche-numerical-linear-algebra_FO3023 | 225 | 0.994 | \min \|\boldsymbol{A} \boldsymbol{y}-\boldsymbol{b}-\boldsymbol{e}\|_{2} | | |
| lyche-numerical-linear-algebra_FO3024 | 225 | 0.994 | \boldsymbol{b}_{1}, \boldsymbol{e}_{1} | | |
| lyche-numerical-linear-algebra_FO3025 | 225 | 1.000 | \boldsymbol{b}_{1} \neq \mathbf{0} | | |
| lyche-numerical-linear-algebra_FO3026 | 225 | 1.000 | \boldsymbol{x}=\boldsymbol{A}^{\dagger} \boldsymbol{b}_{1} | | |
| lyche-numerical-linear-algebra_FO3027 | 225 | 1.000 | \boldsymbol{y}=\boldsymbol{A}^{\dagger} \boldsymbol{b}_{1}+\boldsymbol{A}^{\dagger} \boldsymbol{e}_{1} | | |
| lyche-numerical-linear-algebra_FO3028 | 225 | 1.000 | \boldsymbol{y}-\boldsymbol{x}=\boldsymbol{A}^{\dagger} \boldsymbol{e}_{1} | | |
| lyche-numerical-linear-algebra_FO3029 | 225 | 1.000 | \|\boldsymbol{y}-\boldsymbol{x}\|=\left\|\boldsymbol{A}^{\dagger} \boldsymbol{e}_{1}\right\| \leq\left\|\boldsymbol{A}^{\dagger}\right\|\left\|\boldsymbol{e}_{1}\right\| | | |
| lyche-numerical-linear-algebra_FO3030 | 225 | 1.000 | \left\|\boldsymbol{b}_{1}\right\|=\|\boldsymbol{A} \boldsymbol{x}\| \leq\|\boldsymbol{A}\|\|\boldsymbol{x}\| | | |
| lyche-numerical-linear-algebra_FO3031 | 225 | 1.000 | \|\boldsymbol{y}-\boldsymbol{x}\| /\|\boldsymbol{x}\| \leq\|\boldsymbol{A}\|\left\|\boldsymbol{A}^{\dagger}\right\|\left\|\boldsymbol{e}_{1}\right\| /\left\|\boldsymbol{b}_{1}\right\| | | |
| lyche-numerical-linear-algebra_FO3032 | 225 | 1.000 | \boldsymbol{A}(\boldsymbol{x}-\boldsymbol{y})=\boldsymbol{e}_{1} | | |
| lyche-numerical-linear-algebra_FO3033 | 225 | 1.000 | K(\boldsymbol{A})=\|\boldsymbol{A}\|\left\|\boldsymbol{A}^{\dagger}\right\| | | |
| lyche-numerical-linear-algebra_FO3034 | 225 | 1.000 | \|\boldsymbol{A}\|\left\|\boldsymbol{A}^{-1}\right\| | | |
| lyche-numerical-linear-algebra_FO3035 | 225 | 0.997 | \left\|\boldsymbol{e}_{1}\right\| /\left\|\boldsymbol{b}_{1}\right\| | | |
| lyche-numerical-linear-algebra_FO3036 | 225 | 1.000 | \|\boldsymbol{b}\| /\left\|\boldsymbol{b}_{1}\right\| | | |
| lyche-numerical-linear-algebra_FO3037 | 226 | 1.000 | K_{\infty}(\boldsymbol{A})=\|\boldsymbol{A}\|_{\infty}\left\|\boldsymbol{A}^{\dagger}\right\|_{\infty}=2 \cdot 2=4 | | |
| lyche-numerical-linear-algebra_FO3038 | 226 | 0.999 | \boldsymbol{A} \boldsymbol{A}^{\dagger}=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO3039 | 226 | 1.000 | \left\|\boldsymbol{e}_{1}\right\|_{\infty} /\left\|\boldsymbol{b}_{1}\right\|_{\infty}=10^{-2} | | |
| lyche-numerical-linear-algebra_FO3040 | 226 | 1.000 | \boldsymbol{x}=\boldsymbol{A}^{\dagger} \boldsymbol{b}=\left[10^{-4}, 0\right]^{*} | | |
| lyche-numerical-linear-algebra_FO3041 | 226 | 0.999 | \boldsymbol{y}=\boldsymbol{A}^{\dagger}(\boldsymbol{b}+\boldsymbol{e})=\left[10^{-4}+10^{-6}, 0\right]^{*} | | |
| lyche-numerical-linear-algebra_FO3042 | 226 | 1.000 | \boldsymbol{A}, \boldsymbol{E} \in \mathbb{C}^{m \times n}, m>n | | |
| lyche-numerical-linear-algebra_FO3043 | 226 | 1.000 | \alpha:=1-\|\boldsymbol{E}\|_{2}\left\|\boldsymbol{A}^{\dagger}\right\|_{2}>0 | | |
| lyche-numerical-linear-algebra_FO3044 | 226 | 1.000 | \boldsymbol{b}=\boldsymbol{b}_{1}+\boldsymbol{b}_{2} \in \mathbb{C}^{m} | | |
| lyche-numerical-linear-algebra_FO3045 | 226 | 0.755 | \min \|(\boldsymbol{A}+\boldsymbol{E}) \boldsymbol{y}-\boldsymbol{b}\|_{2} | | |
| lyche-numerical-linear-algebra_FO3046 | 226 | 1.000 | K(1+\beta K) / \alpha | | |
| lyche-numerical-linear-algebra_FO3047 | 226 | 1.000 | \|\boldsymbol{E}\|_{2} /\|\boldsymbol{A}\|_{2} | | |
| lyche-numerical-linear-algebra_FO3048 | 227 | 0.993 | \boldsymbol{A} . \beta | | |
| lyche-numerical-linear-algebra_FO3049 | 227 | 1.000 | \rho \leq \frac{1}{\alpha} K\|\boldsymbol{E}\|_{2} /\|\boldsymbol{A}\|_{2} | | |
| lyche-numerical-linear-algebra_FO3050 | 227 | 1.000 | \beta | | |
| lyche-numerical-linear-algebra_FO3051 | 227 | 1.000 | \frac{1}{\alpha} K^{2} \beta\|\boldsymbol{E}\|_{2} /\|\boldsymbol{A}\|_{2} | | |
| lyche-numerical-linear-algebra_FO3052 | 227 | 1.000 | K(\boldsymbol{A})^{2} \beta | | |
| lyche-numerical-linear-algebra_FO3053 | 227 | 0.999 | \boldsymbol{b}_{2}=\boldsymbol{b}-\boldsymbol{A} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO3054 | 227 | 0.999 | \sigma_{1}, \sigma_{2}, \ldots, \sigma_{n} | | |
| lyche-numerical-linear-algebra_FO3055 | 227 | 0.999 | \sigma_{1} \geq \cdots \geq \sigma_{n} | | |
| lyche-numerical-linear-algebra_FO3056 | 228 | 1.000 | \boldsymbol{A}, \boldsymbol{B} \in \mathbb{R}^{m \times n} | | |
| lyche-numerical-linear-algebra_FO3057 | 228 | 1.000 | n-j+1 | | |
| lyche-numerical-linear-algebra_FO3058 | 228 | 1.000 | \beta_{j} \leq \alpha_{j}+\|\boldsymbol{A}-\boldsymbol{B}\|_{2} | | |
| lyche-numerical-linear-algebra_FO3059 | 228 | 1.000 | \boldsymbol{A}, \boldsymbol{E} \in | | |
| lyche-numerical-linear-algebra_FO3060 | 228 | 1.000 | \alpha_{1} \geq \cdots \geq \alpha_{n} | | |
| lyche-numerical-linear-algebra_FO3061 | 228 | 1.000 | \epsilon_{1} \geq \cdots \geq \epsilon_{n} | | |
| lyche-numerical-linear-algebra_FO3062 | 228 | 1.000 | \operatorname{rank}(\boldsymbol{A}+\boldsymbol{E}) \leq | | |
| lyche-numerical-linear-algebra_FO3063 | 228 | 0.893 | \left\|\boldsymbol{A}^{\dagger}\right\|_{2}\|\boldsymbol{E}\|_{2}<1 | | |
| lyche-numerical-linear-algebra_FO3064 | 228 | 1.000 | \operatorname{rank}(\boldsymbol{A}+\boldsymbol{E})=\operatorname{rank}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO3065 | 228 | 1.000 | \left\|(\boldsymbol{A}+\boldsymbol{E})^{\dagger}\right\|_{2} \leq \frac{\left\|\boldsymbol{A}^{\dagger}\right\|_{2}}{1-\left\|\boldsymbol{A}^{\dagger}\right\|_{2}\|\boldsymbol{E}\|_{2}}=\frac{1}{\alpha_{r}-\epsilon_{1}} | | |
| lyche-numerical-linear-algebra_FO3066 | 228 | 1.000 | \beta_{1} \geq | | |
| lyche-numerical-linear-algebra_FO3067 | 228 | 1.000 | \cdots \geq \beta_{n} | | |
| lyche-numerical-linear-algebra_FO3068 | 228 | 1.000 | \epsilon_{1} / \alpha_{r}<1 | | |
| lyche-numerical-linear-algebra_FO3069 | 228 | 1.000 | \alpha_{r}>\epsilon_{1} | | |
| lyche-numerical-linear-algebra_FO3070 | 228 | 1.000 | \alpha_{r}-\beta_{r} \leq\|\boldsymbol{E}\|_{2}=\epsilon_{1} | | |
| lyche-numerical-linear-algebra_FO3071 | 228 | 0.703 | \beta_{r} \geq \alpha_{r}-\epsilon_{1}>0 | | |
| lyche-numerical-linear-algebra_FO3072 | 228 | 0.703 | \operatorname{rank}(\boldsymbol{A}+\boldsymbol{E}) \geq r | | |
| lyche-numerical-linear-algebra_FO3073 | 228 | 0.960 | \beta_{r} \geq \alpha_{r}-\epsilon_{1} | | |
| lyche-numerical-linear-algebra_FO3074 | 228 | 1.000 | \mu_{1} \geq \cdots \geq \mu_{n} | | |
| lyche-numerical-linear-algebra_FO3075 | 229 | 1.000 | \boldsymbol{A}, \boldsymbol{B} \in \mathbb{C}^{m \times n} | | |
| lyche-numerical-linear-algebra_FO3076 | 229 | 1.000 | \beta_{1} \geq \cdots \geq \beta_{n} | | |
| lyche-numerical-linear-algebra_FO3077 | 229 | 1.000 | \lambda_{1} \geq \cdots \geq \lambda_{m+n} | | |
| lyche-numerical-linear-algebra_FO3078 | 229 | 1.000 | \mu_{1} \geq \cdots \geq \mu_{m+n} | | |
| lyche-numerical-linear-algebra_FO3079 | 229 | 0.651 | \operatorname{SVD}\left[\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{m}\right] \boldsymbol{\Sigma}\left[\boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{n}\right]^{*} | | |
| lyche-numerical-linear-algebra_FO3080 | 229 | 1.000 | 2 r | | |
| lyche-numerical-linear-algebra_FO3081 | 229 | 1.000 | \alpha_{1},-\alpha_{1}, \ldots, \alpha_{r},-\alpha_{r} | | |
| lyche-numerical-linear-algebra_FO3082 | 229 | 1.000 | m+n-2 r | | |
| lyche-numerical-linear-algebra_FO3083 | 229 | 1.000 | 2 s | | |
| lyche-numerical-linear-algebra_FO3084 | 229 | 0.999 | \beta_{1},-\beta_{1}, \ldots, \beta_{s},-\beta_{s} | | |
| lyche-numerical-linear-algebra_FO3085 | 229 | 0.999 | m+n-2 s | | |
| lyche-numerical-linear-algebra_FO3086 | 230 | 1.000 | \sum_{i=1}^{t}\left|\alpha_{i}-\beta_{i}\right|^{2} \leq\|\boldsymbol{B}-\boldsymbol{A}\|_{F}^{2} | | |
| lyche-numerical-linear-algebra_FO3087 | 230 | 1.000 | t \leq n | | |
| lyche-numerical-linear-algebra_FO3088 | 230 | 1.000 | \alpha_{i}=\beta_{i}=0 | | |
| lyche-numerical-linear-algebra_FO3089 | 230 | 1.000 | i=t+1, \ldots, n | | |
| lyche-numerical-linear-algebra_FO3090 | 230 | 1.000 | \left(t-c_{1}\right)^{2}+\left(y-c_{2}\right)^{2}=r^{2} | | |
| lyche-numerical-linear-algebra_FO3091 | 230 | 1.000 | \left(t_{i}, y_{i}\right)_{i=1}^{m} | | |
| lyche-numerical-linear-algebra_FO3092 | 230 | 0.999 | (t, y) | | |
| lyche-numerical-linear-algebra_FO3093 | 230 | 1.000 | c_{1}, c_{2} | | |
| lyche-numerical-linear-algebra_FO3094 | 230 | 1.000 | c_{1}=x_{1} / 2, c_{2}=x_{2} / 2 | | |
| lyche-numerical-linear-algebra_FO3095 | 230 | 1.000 | r^{2}=c_{1}^{2}+c_{2}^{2}+x_{3} | | |
| lyche-numerical-linear-algebra_FO3096 | 230 | 1.000 | c_{1}, c_{2}, r | | |
| lyche-numerical-linear-algebra_FO3097 | 230 | 1.000 | \left[x_{1}, x_{2}, x_{3}\right] | | |
| lyche-numerical-linear-algebra_FO3098 | 230 | 0.593 | (1,4),(3,2),(1,0) | | |
| lyche-numerical-linear-algebra_FO3099 | 231 | 1.000 | \left[\begin{array}{cc}\sqrt{2} & \sqrt{2} \\ 0 & \sqrt{3}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO3100 | 231 | 0.993 | \boldsymbol{A}=\left[\begin{array}{cc}3 & \alpha \\ \alpha & 1\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO3101 | 231 | 1.000 | \boldsymbol{p}_{1}=(0,1), \boldsymbol{p}_{2}=(1,0), \boldsymbol{p}_{3}=(2,1) | | |
| lyche-numerical-linear-algebra_FO3102 | 231 | 1.000 | p(x)=m x+b | | |
| lyche-numerical-linear-algebra_FO3103 | 231 | 1.000 | \boldsymbol{p}_{i}=\left(x_{i}, y_{i}\right) | | |
| lyche-numerical-linear-algebra_FO3104 | 231 | 1.000 | F: \mathbb{R}^{n} \rightarrow \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO3105 | 231 | 1.000 | \boldsymbol{B}:=\boldsymbol{I}+\boldsymbol{A}^{T} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO3106 | 231 | 1.000 | \left(\boldsymbol{I}+\boldsymbol{A}^{T} \boldsymbol{A}\right) \boldsymbol{x}=\boldsymbol{A}^{T} \boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO3107 | 231 | 1.000 | \operatorname{diag}\left(d_{1}, d_{2}, \ldots, d_{m}\right) \in \mathbb{R}^{m \times m} | | |
| lyche-numerical-linear-algebra_FO3108 | 231 | 1.000 | d_{i}>0, i=1,2, \ldots, m | | |
| lyche-numerical-linear-algebra_FO3109 | 232 | 1.000 | r_{i}=r_{i}(\boldsymbol{x}), i=1,2, \ldots, m | | |
| lyche-numerical-linear-algebra_FO3110 | 232 | 1.000 | \|\boldsymbol{r}(\boldsymbol{x})\|_{D}^{2} | | |
| lyche-numerical-linear-algebra_FO3111 | 232 | 1.000 | \boldsymbol{x}=\boldsymbol{x}_{\text {min }} | | |
| lyche-numerical-linear-algebra_FO3112 | 232 | 1.000 | K_{2}(\boldsymbol{B}):=\|\boldsymbol{B}\|_{2}\left\|\boldsymbol{B}^{-1}\right\|_{2} | | |
| lyche-numerical-linear-algebra_FO3113 | 232 | 0.999 | \boldsymbol{B}, \boldsymbol{C} \in \mathbb{C}^{n \times m} | | |
| lyche-numerical-linear-algebra_FO3114 | 232 | 1.000 | \boldsymbol{B}=\boldsymbol{C} | | |
| lyche-numerical-linear-algebra_FO3115 | 232 | 1.000 | \boldsymbol{A}=\left[\begin{array}{ll}1 & 1 \\ 1 & 1 \\ 0 & 0\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO3116 | 232 | 1.000 | \boldsymbol{A}^{\dagger}=\frac{1}{4}\left[\begin{array}{lll}1 & 1 & 0 \\ 1 & 1 & 0\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO3117 | 232 | 1.000 | \boldsymbol{A}^{\dagger}=\left(\boldsymbol{A}^{*} \boldsymbol{A}\right)^{-1} \boldsymbol{A}^{*} | | |
| lyche-numerical-linear-algebra_FO3118 | 232 | 1.000 | \boldsymbol{A}^{\dagger}=\boldsymbol{A}^{*}\left(\boldsymbol{A} \boldsymbol{A}^{*}\right)^{-1} | | |
| lyche-numerical-linear-algebra_FO3119 | 233 | 1.000 | \mathcal{R}\left(\boldsymbol{A}^{*}\right), \mathcal{N}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO3120 | 233 | 0.996 | \quad \mathbb{C}^{n}=\mathcal{R}\left(\boldsymbol{A}^{*}\right) \stackrel{\perp}{\oplus} \mathcal{N}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO3121 | 233 | 1.000 | \boldsymbol{u}^{\dagger}=\left(\boldsymbol{u}^{*} \boldsymbol{u}\right)^{-1} \boldsymbol{u}^{*} | | |
| lyche-numerical-linear-algebra_FO3122 | 233 | 1.000 | \boldsymbol{u} \in \mathbb{C}^{n, 1} | | |
| lyche-numerical-linear-algebra_FO3123 | 233 | 1.000 | \boldsymbol{A}=\boldsymbol{u} \boldsymbol{v}^{*} | | |
| lyche-numerical-linear-algebra_FO3124 | 233 | 1.000 | \boldsymbol{u} \in \mathbb{C}^{m}, \boldsymbol{v} \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO3125 | 233 | 0.990 | \operatorname{diag}\left(\lambda_{1}, \ldots, \lambda_{n}\right)^{\dagger}=\operatorname{diag}\left(\lambda_{1}^{\dagger}, \ldots, \lambda_{n}^{\dagger}\right) | | |
| lyche-numerical-linear-algebra_FO3126 | 233 | 1.000 | \left(\boldsymbol{A}^{*}\right)^{\dagger}=\left(\boldsymbol{A}^{\dagger}\right)^{*} | | |
| lyche-numerical-linear-algebra_FO3127 | 233 | 0.999 | \left(\boldsymbol{A}^{\dagger}\right)^{\dagger}=\boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO3128 | 233 | 0.995 | (\alpha \boldsymbol{A})^{\dagger}=\frac{1}{\alpha} \boldsymbol{A}^{\dagger}, \quad \alpha \neq 0 | | |
| lyche-numerical-linear-algebra_FO3129 | 233 | 1.000 | k, m, n \in \mathbb{N}, \boldsymbol{A} \in | | |
| lyche-numerical-linear-algebra_FO3130 | 233 | 0.623 | \mathbb{C}^{m \times n}, \boldsymbol{B} \in \mathbb{C}^{n \times k} | | |
| lyche-numerical-linear-algebra_FO3131 | 233 | 0.993 | (\boldsymbol{A} \boldsymbol{B})^{\dagger}=\boldsymbol{B}^{\dagger} \boldsymbol{A}^{\dagger} | | |
| lyche-numerical-linear-algebra_FO3132 | 233 | 1.000 | \boldsymbol{E}=\boldsymbol{A} \boldsymbol{B}, \boldsymbol{F}=\boldsymbol{B}^{\dagger} \boldsymbol{A}^{\dagger} | | |
| lyche-numerical-linear-algebra_FO3133 | 233 | 1.000 | \boldsymbol{A}^{\dagger} \boldsymbol{A}=\boldsymbol{B} \boldsymbol{B}^{\dagger}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO3134 | 233 | 0.998 | \boldsymbol{A} \in \mathbb{R}^{1,2}, \boldsymbol{B} \in \mathbb{R}^{2,1} | | |
| lyche-numerical-linear-algebra_FO3135 | 233 | 0.998 | (\boldsymbol{A} \boldsymbol{B})^{\dagger} \neq \boldsymbol{B}^{\dagger} \boldsymbol{A}^{\dagger} | | |
| lyche-numerical-linear-algebra_FO3136 | 234 | 1.000 | \boldsymbol{A}^{*}=\boldsymbol{A}^{\dagger} | | |
| lyche-numerical-linear-algebra_FO3137 | 234 | 0.998 | \boldsymbol{A}\left(\boldsymbol{A}^{*} \boldsymbol{A}\right)^{-1} \boldsymbol{A}^{*} \boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO3138 | 234 | 1.000 | r>0 | | |
| lyche-numerical-linear-algebra_FO3139 | 234 | 1.000 | \boldsymbol{c}=\left[c_{1}, \ldots, c_{n}\right]^{*}=\boldsymbol{U}^{*} \boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO3140 | 234 | 1.000 | \boldsymbol{y}=\left[y_{1}, \ldots, y_{n}\right]^{*}=\boldsymbol{V}^{*} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO3141 | 234 | 0.999 | c_{r+1}=\cdots=c_{n}=0 | | |
| lyche-numerical-linear-algebra_FO3142 | 234 | 1.000 | \boldsymbol{A} \in \mathbb{C}^{m \times n}, \boldsymbol{b} \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO3143 | 234 | 1.000 | \boldsymbol{b} \in \mathcal{R}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO3144 | 234 | 1.000 | \boldsymbol{y} \in \mathcal{N}\left(\boldsymbol{A}^{*}\right) | | |
| lyche-numerical-linear-algebra_FO3145 | 234 | 1.000 | \boldsymbol{y}^{*} \boldsymbol{b} \neq 0 | | |
| lyche-numerical-linear-algebra_FO3146 | 234 | 1.000 | \mathbb{C}^{(m-n) \times n} | | |
| lyche-numerical-linear-algebra_FO3147 | 234 | 1.000 | \left\|\boldsymbol{A}^{\dagger}\right\|_{2} \leq\left\|\boldsymbol{B}^{-1}\right\|_{2} | | |
| lyche-numerical-linear-algebra_FO3148 | 235 | 1.000 | K(\boldsymbol{A})=\|\boldsymbol{A}\|_{2}\left\|\boldsymbol{A}^{\dagger}\right\|_{2} | | |
| lyche-numerical-linear-algebra_FO3149 | 235 | 0.999 | \boldsymbol{y}_{A^{\dagger}} | | |
| lyche-numerical-linear-algebra_FO3150 | 235 | 0.999 | \left\|\boldsymbol{y}_{\boldsymbol{A}}\right\|=\left\|\boldsymbol{y}_{A^{\dagger}}\right\|=1 | | |
| lyche-numerical-linear-algebra_FO3151 | 235 | 0.999 | \left\|\boldsymbol{A}^{\dagger}\right\|= | | |
| lyche-numerical-linear-algebra_FO3152 | 235 | 0.999 | \left\|\boldsymbol{A}^{\dagger} \boldsymbol{y}_{A^{\dagger}}\right\| | | |
| lyche-numerical-linear-algebra_FO3153 | 235 | 1.000 | \boldsymbol{b}=\boldsymbol{A} \boldsymbol{y}_{A}, \boldsymbol{e}_{1}=\boldsymbol{y}_{A^{\dagger}} | | |
| lyche-numerical-linear-algebra_FO3154 | 235 | 1.000 | \boldsymbol{A}^{\dagger}=\boldsymbol{A}^{-1} | | |
| lyche-numerical-linear-algebra_FO3155 | 235 | 1.000 | \boldsymbol{e}_{1}=\boldsymbol{e} | | |
| lyche-numerical-linear-algebra_FO3156 | 235 | 1.000 | (2,2) | | |
| lyche-numerical-linear-algebra_FO3157 | 235 | 1.000 | 3+2 \epsilon+\epsilon^{2} | | |
| lyche-numerical-linear-algebra_FO3158 | 235 | 1.000 | 3+2 \epsilon | | |
| lyche-numerical-linear-algebra_FO3159 | 235 | 1.000 | \epsilon<\sqrt{u}, u | | |
| lyche-numerical-linear-algebra_FO3160 | 235 | 1.000 | u=10^{-16} | | |
| lyche-numerical-linear-algebra_FO3161 | 235 | 1.000 | \sqrt{u}=10^{-8} | | |
| lyche-numerical-linear-algebra_FO3162 | 235 | 1.000 | \boldsymbol{A}(\epsilon) \in \mathbb{R}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO3163 | 235 | 1.000 | j \neq i, i+1 | | |
| lyche-numerical-linear-algebra_FO3164 | 235 | 1.000 | a_{k, k+1}=\epsilon \in \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO3165 | 235 | 1.000 | \sigma_{i}(\epsilon), i=1, \ldots, n | | |
| lyche-numerical-linear-algebra_FO3166 | 235 | 1.000 | \boldsymbol{A}(\epsilon) | | |
| lyche-numerical-linear-algebra_FO3167 | 238 | 1.000 | \Omega:=(0,1)^{2}=\{(x, y): 0<x, y<1\} | | |
| lyche-numerical-linear-algebra_FO3168 | 238 | 1.000 | \partial \Omega | | |
| lyche-numerical-linear-algebra_FO3169 | 238 | 1.000 | u= | | |
| lyche-numerical-linear-algebra_FO3170 | 238 | 1.000 | u(x, y) | | |
| lyche-numerical-linear-algebra_FO3171 | 238 | 1.000 | v_{j, k} \approx u(j h, k h) | | |
| lyche-numerical-linear-algebra_FO3172 | 238 | 1.000 | \Omega_{h}:=\{(j h, k h): j, k=1, \ldots, m\} | | |
| lyche-numerical-linear-algebra_FO3173 | 238 | 1.000 | \bar{\Omega}_{h} \backslash \Omega_{h} | | |
| lyche-numerical-linear-algebra_FO3174 | 239 | 1.000 | f_{j, k}:=f(j h, k h) | | |
| lyche-numerical-linear-algebra_FO3175 | 239 | 1.000 | n:=m^{2} | | |
| lyche-numerical-linear-algebra_FO3176 | 239 | 1.000 | v_{j, k} | | |
| lyche-numerical-linear-algebra_FO3177 | 239 | 1.000 | \boldsymbol{T}=\operatorname{tridiag}(-1,2,-1) \in \mathbb{R}^{m \times m} | | |
| lyche-numerical-linear-algebra_FO3178 | 239 | 0.913 | j, k | | |
| lyche-numerical-linear-algebra_FO3179 | 239 | 0.913 | \boldsymbol{T} \boldsymbol{V}+\boldsymbol{V T} | | |
| lyche-numerical-linear-algebra_FO3180 | 239 | 1.000 | \boldsymbol{B} \in \mathbb{R}^{m \times n} | | |
| lyche-numerical-linear-algebra_FO3181 | 240 | 1.000 | \boldsymbol{x}:=\operatorname{vec}(\boldsymbol{V}) \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO3182 | 240 | 1.000 | n=m^{2} | | |
| lyche-numerical-linear-algebra_FO3183 | 240 | 1.000 | x_{i}=v_{j, k} | | |
| lyche-numerical-linear-algebra_FO3184 | 240 | 1.000 | b_{i}=h^{2} f_{j, k} | | |
| lyche-numerical-linear-algebra_FO3185 | 240 | 1.000 | \boldsymbol{x}=\operatorname{vec}(\boldsymbol{V}), \boldsymbol{b}=h^{2} \operatorname{vec}(\boldsymbol{F}) | | |
| lyche-numerical-linear-algebra_FO3186 | 241 | 0.905 | \boldsymbol{T}_{1} \in \mathbb{R}^{m \times m} | | |
| lyche-numerical-linear-algebra_FO3187 | 241 | 1.000 | a, d | | |
| lyche-numerical-linear-algebra_FO3188 | 241 | 1.000 | \boldsymbol{T}_{2}=\left[a_{i j}\right] \in | | |
| lyche-numerical-linear-algebra_FO3189 | 242 | 0.956 | \boldsymbol{T}_{2} | | |
| lyche-numerical-linear-algebra_FO3190 | 242 | 1.000 | p, q, r, s | | |
| lyche-numerical-linear-algebra_FO3191 | 242 | 1.000 | \boldsymbol{A} \in \mathbb{R}^{p \times q} | | |
| lyche-numerical-linear-algebra_FO3192 | 242 | 1.000 | \boldsymbol{B} \in \mathbb{R}^{r \times s} | | |
| lyche-numerical-linear-algebra_FO3193 | 242 | 1.000 | \boldsymbol{C} \in \mathbb{R}^{p r \times q s} | | |
| lyche-numerical-linear-algebra_FO3194 | 242 | 1.000 | \boldsymbol{C}=\boldsymbol{A} \otimes \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO3195 | 242 | 1.000 | \boldsymbol{B} \otimes \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO3196 | 242 | 1.000 | \boldsymbol{u} \otimes \boldsymbol{v}=\left[\boldsymbol{u}^{T} v_{1}, \ldots, \boldsymbol{u}^{T} v_{r}\right]^{T} | | |
| lyche-numerical-linear-algebra_FO3197 | 242 | 1.000 | \boldsymbol{u} \in \mathbb{R}^{p} | | |
| lyche-numerical-linear-algebra_FO3198 | 242 | 1.000 | \boldsymbol{v} \in \mathbb{R}^{r} | | |
| lyche-numerical-linear-algebra_FO3199 | 242 | 1.000 | p \cdot r | | |
| lyche-numerical-linear-algebra_FO3200 | 243 | 1.000 | r, s, k | | |
| lyche-numerical-linear-algebra_FO3201 | 243 | 1.000 | \boldsymbol{A} \in \mathbb{R}^{r \times r}, \boldsymbol{B} \in | | |
| lyche-numerical-linear-algebra_FO3202 | 243 | 1.000 | \mathbb{R}^{s \times s} | | |
| lyche-numerical-linear-algebra_FO3203 | 243 | 1.000 | \boldsymbol{I}_{k} | | |
| lyche-numerical-linear-algebra_FO3204 | 243 | 1.000 | \boldsymbol{A} \otimes \boldsymbol{I}_{s}+\boldsymbol{I}_{r} \otimes \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO3205 | 243 | 1.000 | \lambda, \mu | | |
| lyche-numerical-linear-algebra_FO3206 | 243 | 1.000 | \boldsymbol{A}, \boldsymbol{A}_{1}, \boldsymbol{A}_{2}, \boldsymbol{B}, \boldsymbol{B}_{1}, \boldsymbol{B}_{2}, \boldsymbol{C} | | |
| lyche-numerical-linear-algebra_FO3207 | 243 | 1.000 | \boldsymbol{A} \otimes \boldsymbol{B} \neq \boldsymbol{B} \otimes \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO3208 | 243 | 1.000 | \boldsymbol{P}, \boldsymbol{Q} | | |
| lyche-numerical-linear-algebra_FO3209 | 243 | 1.000 | \boldsymbol{B} \otimes \boldsymbol{A}=\boldsymbol{P}(\boldsymbol{A} \otimes \boldsymbol{B}) \boldsymbol{Q} | | |
| lyche-numerical-linear-algebra_FO3210 | 243 | 0.989 | \boldsymbol{B} \boldsymbol{D} | | |
| lyche-numerical-linear-algebra_FO3211 | 243 | 1.000 | (\boldsymbol{A} \otimes \boldsymbol{B})(\boldsymbol{C} \otimes \boldsymbol{D}) | | |
| lyche-numerical-linear-algebra_FO3212 | 243 | 1.000 | \boldsymbol{B} \in \mathbb{R}^{r, t} | | |
| lyche-numerical-linear-algebra_FO3213 | 243 | 1.000 | \boldsymbol{D} \in \mathbb{R}^{t, s} | | |
| lyche-numerical-linear-algebra_FO3214 | 243 | 1.000 | r, s, t | | |
| lyche-numerical-linear-algebra_FO3215 | 244 | 1.000 | r, s \in \mathbb{N} | | |
| lyche-numerical-linear-algebra_FO3216 | 244 | 1.000 | \boldsymbol{A} \in \mathbb{R}^{r \times r} | | |
| lyche-numerical-linear-algebra_FO3217 | 244 | 1.000 | \boldsymbol{B} \in \mathbb{R}^{s \times s} | | |
| lyche-numerical-linear-algebra_FO3218 | 244 | 1.000 | \left(\lambda_{i}, \boldsymbol{u}_{i}\right) i=1, \ldots, r | | |
| lyche-numerical-linear-algebra_FO3219 | 244 | 1.000 | \left(\mu_{j}, \boldsymbol{v}_{j}\right), j=1, \ldots, s | | |
| lyche-numerical-linear-algebra_FO3220 | 244 | 1.000 | \boldsymbol{F}, \boldsymbol{V} \in \mathbb{R}^{r \times s} | | |
| lyche-numerical-linear-algebra_FO3221 | 244 | 0.999 | (\boldsymbol{A} \otimes \boldsymbol{B})^{T}=\boldsymbol{A}^{T} \otimes \boldsymbol{B}^{T} | | |
| lyche-numerical-linear-algebra_FO3222 | 244 | 0.999 | \boldsymbol{A} \otimes \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO3223 | 244 | 0.999 | (\boldsymbol{A} \otimes \boldsymbol{B})^{-1}= | | |
| lyche-numerical-linear-algebra_FO3224 | 244 | 1.000 | \boldsymbol{A}^{-1} \otimes \boldsymbol{B}^{-1} | | |
| lyche-numerical-linear-algebra_FO3225 | 244 | 1.000 | (\boldsymbol{A} \otimes \boldsymbol{B})\left(\boldsymbol{u}_{i} \otimes \boldsymbol{v}_{j}\right)=\lambda_{i} \mu_{j}\left(\boldsymbol{u}_{i} \otimes \boldsymbol{v}_{j}\right), \quad i=1, \ldots, r, \quad j=1, \ldots, s | | |
| lyche-numerical-linear-algebra_FO3226 | 244 | 0.999 | \left(\boldsymbol{A} \otimes \boldsymbol{I}_{s}+\boldsymbol{I}_{r} \otimes \boldsymbol{B}\right)\left(\boldsymbol{u}_{i} \otimes \boldsymbol{v}_{j}\right)=\left(\lambda_{i}+\mu_{j}\right)\left(\boldsymbol{u}_{i} \otimes \boldsymbol{v}_{j}\right), \quad i=1, \ldots, r, \quad j=1, \ldots, s | | |
| lyche-numerical-linear-algebra_FO3227 | 244 | 1.000 | \boldsymbol{A} \otimes \boldsymbol{I}+\boldsymbol{I} \otimes \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO3228 | 244 | 0.689 | \boldsymbol{A} \boldsymbol{V} \boldsymbol{B}^{T}=\boldsymbol{F} \quad \Leftrightarrow \quad(\boldsymbol{A} \otimes \boldsymbol{B}) \operatorname{vec}(\boldsymbol{V})=\operatorname{vec}(\boldsymbol{F}) | | |
| lyche-numerical-linear-algebra_FO3229 | 244 | 1.000 | \boldsymbol{A} \boldsymbol{V}+\boldsymbol{V} \boldsymbol{B}^{T}=\boldsymbol{F} \quad \Leftrightarrow \quad\left(\boldsymbol{A} \otimes \boldsymbol{I}_{s}+\boldsymbol{I}_{r} \otimes \boldsymbol{B}\right) \operatorname{vec}(\boldsymbol{V})=\operatorname{vec}(\boldsymbol{F}) | | |
| lyche-numerical-linear-algebra_FO3230 | 244 | 1.000 | \boldsymbol{A}=\boldsymbol{T} \otimes \boldsymbol{I}+\boldsymbol{I} \otimes \boldsymbol{T} | | |
| lyche-numerical-linear-algebra_FO3231 | 244 | 1.000 | \boldsymbol{A} \boldsymbol{V} \boldsymbol{B}^{T}=\boldsymbol{F} | | |
| lyche-numerical-linear-algebra_FO3232 | 244 | 1.000 | \boldsymbol{A} \boldsymbol{W}= | | |
| lyche-numerical-linear-algebra_FO3233 | 244 | 1.000 | \boldsymbol{B} \boldsymbol{V}^{T}=\boldsymbol{W}^{T} | | |
| lyche-numerical-linear-algebra_FO3234 | 244 | 0.544 | (\boldsymbol{A} \otimes \boldsymbol{B}) \operatorname{vec}(\boldsymbol{V})=\operatorname{vec}(\boldsymbol{F}) | | |
| lyche-numerical-linear-algebra_FO3235 | 244 | 0.965 | \boldsymbol{T} \boldsymbol{V}+\boldsymbol{V T}=\boldsymbol{F} | | |
| lyche-numerical-linear-algebra_FO3236 | 245 | 0.998 | (\boldsymbol{A} \otimes \boldsymbol{B})\left(\boldsymbol{A}^{-1} \otimes \boldsymbol{B}^{-1}\right)=\left(\boldsymbol{A} \boldsymbol{A}^{-1}\right) \otimes\left(\boldsymbol{B} \boldsymbol{B}^{-1}\right)= | | |
| lyche-numerical-linear-algebra_FO3237 | 245 | 0.997 | \boldsymbol{I}_{r} \otimes \boldsymbol{I}_{s}=\boldsymbol{I}_{r s} | | |
| lyche-numerical-linear-algebra_FO3238 | 245 | 0.997 | (\boldsymbol{A} \otimes \boldsymbol{B}) | | |
| lyche-numerical-linear-algebra_FO3239 | 245 | 1.000 | 1,(\boldsymbol{A} \otimes \boldsymbol{B})^{T}=\boldsymbol{A}^{T} \otimes \boldsymbol{B}^{T}=\boldsymbol{A} \otimes \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO3240 | 245 | 1.000 | \boldsymbol{A} \otimes \boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO3241 | 245 | 1.000 | \boldsymbol{I} \otimes \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO3242 | 245 | 1.000 | (\boldsymbol{A} \otimes \boldsymbol{B})\left(\boldsymbol{u}_{i} \otimes \boldsymbol{v}_{j}\right)=\left(\boldsymbol{A} \boldsymbol{u}_{i}\right) \otimes\left(\boldsymbol{B} \boldsymbol{v}_{j}\right)=\left(\lambda_{i} \boldsymbol{u}_{i}\right) \otimes\left(\mu_{j} \boldsymbol{v}_{j}\right)=\left(\lambda_{i} \mu_{j}\right)\left(\boldsymbol{u}_{i} \otimes \boldsymbol{v}_{j}\right) | | |
| lyche-numerical-linear-algebra_FO3243 | 245 | 0.475 | \left(\boldsymbol{A} \otimes \boldsymbol{I}_{s}\right)\left(\boldsymbol{u}_{i} \otimes \boldsymbol{v}_{j}\right)=\lambda_{i}\left(\boldsymbol{u}_{i} \otimes \boldsymbol{v}_{j}\right), \quad | | |
| lyche-numerical-linear-algebra_FO3244 | 245 | 0.475 | \quad\left(\boldsymbol{I}_{r} \otimes \boldsymbol{B}\right)\left(\boldsymbol{u}_{i} \otimes \boldsymbol{v}_{j}\right)=\mu_{j}\left(\boldsymbol{u}_{i} \otimes \boldsymbol{v}_{j}\right) | | |
| lyche-numerical-linear-algebra_FO3245 | 245 | 0.892 | 1, \boldsymbol{A} \otimes \boldsymbol{I}+\boldsymbol{I} \otimes \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO3246 | 245 | 0.892 | \lambda_{i}+\mu_{j} | | |
| lyche-numerical-linear-algebra_FO3247 | 245 | 0.500 | \mu_{j} | | |
| lyche-numerical-linear-algebra_FO3248 | 245 | 1.000 | \boldsymbol{V}, \boldsymbol{F} | | |
| lyche-numerical-linear-algebra_FO3249 | 245 | 1.000 | \boldsymbol{B}^{T} | | |
| lyche-numerical-linear-algebra_FO3250 | 245 | 1.000 | \boldsymbol{V}=\left[\boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{s}\right], \boldsymbol{F}=\left[\boldsymbol{f}_{1}, \ldots, \boldsymbol{f}_{s}\right] | | |
| lyche-numerical-linear-algebra_FO3251 | 245 | 1.000 | \boldsymbol{B}^{T}=\left[\boldsymbol{b}_{1}, \ldots, \boldsymbol{b}_{s}\right] | | |
| lyche-numerical-linear-algebra_FO3252 | 245 | 0.890 | \left(\boldsymbol{A} \otimes \boldsymbol{I}_{s}\right) \operatorname{vec}(\boldsymbol{V})=\boldsymbol{A} \boldsymbol{V} \boldsymbol{I}_{s}^{T} | | |
| lyche-numerical-linear-algebra_FO3253 | 245 | 0.890 | \left(\boldsymbol{I}_{r} \otimes \boldsymbol{B}\right) \operatorname{vec}(\boldsymbol{V})= | | |
| lyche-numerical-linear-algebra_FO3254 | 245 | 1.000 | \boldsymbol{I}_{r} \boldsymbol{V} \boldsymbol{B}^{T} | | |
| lyche-numerical-linear-algebra_FO3255 | 246 | 0.527 | \left(\lambda_{j}, \boldsymbol{s}_{j}\right) | | |
| lyche-numerical-linear-algebra_FO3256 | 246 | 0.975 | \boldsymbol{s}_{j} \otimes \boldsymbol{s}_{k} | | |
| lyche-numerical-linear-algebra_FO3257 | 246 | 1.000 | d>0 | | |
| lyche-numerical-linear-algebra_FO3258 | 246 | 1.000 | d \geq 2|a| | | |
| lyche-numerical-linear-algebra_FO3259 | 246 | 1.000 | j, k, p, q=1, \ldots, m | | |
| lyche-numerical-linear-algebra_FO3260 | 246 | 1.000 | \lambda_{j} | | |
| lyche-numerical-linear-algebra_FO3261 | 246 | 0.990 | \mathbb{R}^{m^{2} \times m^{2}} | | |
| lyche-numerical-linear-algebra_FO3262 | 246 | 1.000 | d=2, a=-1 | | |
| lyche-numerical-linear-algebra_FO3263 | 246 | 1.000 | \lambda_{j, k} | | |
| lyche-numerical-linear-algebra_FO3264 | 247 | 0.967 | \|\boldsymbol{A}\|_{2}\left\|\boldsymbol{A}^{-1}\right\|_{2}=\frac{\lambda_{\text {max }}}{\lambda_{\text {min }}} | | |
| lyche-numerical-linear-algebra_FO3265 | 247 | 1.000 | \lambda=2 a+2 d | | |
| lyche-numerical-linear-algebra_FO3266 | 247 | 1.000 | [1,1,1,1]^{T} | | |
| lyche-numerical-linear-algebra_FO3267 | 247 | 1.000 | j=k=1 | | |
| lyche-numerical-linear-algebra_FO3268 | 247 | 1.000 | -\Delta u=f, u=0 | | |
| lyche-numerical-linear-algebra_FO3269 | 248 | 1.000 | -\left(\square_{h} v\right)_{j, k}=(\mu f)_{j, k} | | |
| lyche-numerical-linear-algebra_FO3270 | 248 | 1.000 | j, k=1, \ldots, m | | |
| lyche-numerical-linear-algebra_FO3271 | 248 | 1.000 | 0=v_{0, k}=v_{m+1, k}=v_{j, 0}=v_{j, m+1}, j, k=0,1, \ldots, m+1 | | |
| lyche-numerical-linear-algebra_FO3272 | 248 | 1.000 | -\left(\square_{h} v\right)_{j, k}=\left[20 v_{j, k}-4 v_{j-1, k}-4 v_{j, k-1}-4 v_{j+1, k}-4 v_{j, k+1}\right. | | |
| lyche-numerical-linear-algebra_FO3273 | 248 | 1.000 | \left.-v_{j-1, k-1}-v_{j+1, k-1}-v_{j-1, k+1}-v_{j+1, k+1}\right] /\left(6 h^{2}\right) | | |
| lyche-numerical-linear-algebra_FO3274 | 248 | 0.877 | (\mu f)_{j, k}=\left[8 f_{j, k}+f_{j-1, k}+f_{j, k-1}+f_{j+1, k}+f_{j, k+1}\right] / 12 | | |
| lyche-numerical-linear-algebra_FO3275 | 248 | 1.000 | j, k=1,2 | | |
| lyche-numerical-linear-algebra_FO3276 | 248 | 1.000 | f(x, y)=2 \pi^{2} \sin (\pi x) \sin (\pi y) | | |
| lyche-numerical-linear-algebra_FO3277 | 248 | 1.000 | v_{j, k}=5 \pi^{2} / 66 | | |
| lyche-numerical-linear-algebra_FO3278 | 248 | 1.000 | O\left(h^{4}\right) | | |
| lyche-numerical-linear-algebra_FO3279 | 248 | 0.997 | \mu \boldsymbol{F} | | |
| lyche-numerical-linear-algebra_FO3280 | 248 | 0.997 | (\mu f)_{j, k} | | |
| lyche-numerical-linear-algebra_FO3281 | 248 | 1.000 | \boldsymbol{A}=\boldsymbol{T} \otimes \boldsymbol{I}+\boldsymbol{I} \otimes \boldsymbol{T}-\frac{1}{6} \boldsymbol{T} \otimes \boldsymbol{T}, \boldsymbol{x}=\operatorname{vec}(\boldsymbol{V}) | | |
| lyche-numerical-linear-algebra_FO3282 | 248 | 1.000 | \boldsymbol{b}=h^{2} \operatorname{vec}(\mu \boldsymbol{F}) | | |
| lyche-numerical-linear-algebra_FO3283 | 248 | 1.000 | \Delta u=0 | | |
| lyche-numerical-linear-algebra_FO3284 | 248 | 0.994 | \Delta^{2} u=u_{x x x x}+2 u_{x x y y}+u_{y y y y} | | |
| lyche-numerical-linear-algebra_FO3285 | 248 | 1.000 | v=-\Delta u | | |
| lyche-numerical-linear-algebra_FO3286 | 248 | 1.000 | \boldsymbol{T}=\operatorname{tridiag}(-1,2,-1) \in \mathbb{R}^{m \times m}, h=1 /(m+ | | |
| lyche-numerical-linear-algebra_FO3287 | 248 | 0.584 | \boldsymbol{F}=(f(j h, k h))_{j, k=1}^{m} | | |
| lyche-numerical-linear-algebra_FO3288 | 249 | 1.000 | \boldsymbol{A}=(\boldsymbol{T} \otimes \boldsymbol{I}+\boldsymbol{I} \otimes \boldsymbol{T})^{2} | | |
| lyche-numerical-linear-algebra_FO3289 | 249 | 0.999 | \boldsymbol{A}=\boldsymbol{T}^{2} \otimes \boldsymbol{I}+2 \boldsymbol{T} \otimes \boldsymbol{T}+\boldsymbol{I} \otimes \boldsymbol{T}^{2} \in \mathbb{R}^{n \times n}, \boldsymbol{x}=\operatorname{vec}(\boldsymbol{U}) | | |
| lyche-numerical-linear-algebra_FO3290 | 249 | 0.999 | \boldsymbol{b}=h^{4} \operatorname{vec}(\boldsymbol{F}) | | |
| lyche-numerical-linear-algebra_FO3291 | 249 | 0.831 | \mathbf{b} | | |
| lyche-numerical-linear-algebra_FO3292 | 249 | 0.831 | \mathbf{c} | | |
| lyche-numerical-linear-algebra_FO3293 | 249 | 1.000 | \boldsymbol{T}_{1}:=\operatorname{tridiagonal}(a, d, a) | | |
| lyche-numerical-linear-algebra_FO3294 | 249 | 1.000 | (\boldsymbol{A} \otimes | | |
| lyche-numerical-linear-algebra_FO3295 | 249 | 1.000 | \boldsymbol{B}) \operatorname{vec}(\boldsymbol{V})=\operatorname{vec}(\boldsymbol{F}) | | |
| lyche-numerical-linear-algebra_FO3296 | 249 | 0.999 | \left(\boldsymbol{A} \otimes \boldsymbol{I}_{s}+\boldsymbol{I}_{r} \otimes \boldsymbol{B}\right) \operatorname{vec}(\boldsymbol{V})=\operatorname{vec}(\boldsymbol{F}) | | |
| lyche-numerical-linear-algebra_FO3297 | 250 | 1.000 | n=9 | | |
| lyche-numerical-linear-algebra_FO3298 | 251 | 1.000 | d=\sqrt{n} | | |
| lyche-numerical-linear-algebra_FO3299 | 251 | 1.000 | O\left(n d^{2}\right)=O\left(n^{2}\right) | | |
| lyche-numerical-linear-algebra_FO3301 | 252 | 1.000 | \boldsymbol{D}_{1}, \ldots, \boldsymbol{D}_{m} | | |
| lyche-numerical-linear-algebra_FO3302 | 252 | 1.000 | \boldsymbol{U}_{1}, \ldots, \boldsymbol{U}_{m} | | |
| lyche-numerical-linear-algebra_FO3303 | 252 | 1.000 | \boldsymbol{A}_{1}, \ldots, \boldsymbol{A}_{m-1}, \boldsymbol{L}_{1} | | |
| lyche-numerical-linear-algebra_FO3304 | 252 | 1.000 | \ldots, \boldsymbol{L}_{m-1} | | |
| lyche-numerical-linear-algebra_FO3305 | 252 | 1.000 | \boldsymbol{C}_{1}, \ldots, \boldsymbol{C}_{m-1} | | |
| lyche-numerical-linear-algebra_FO3306 | 252 | 0.995 | \boldsymbol{b}^{T}= | | |
| lyche-numerical-linear-algebra_FO3307 | 252 | 1.000 | \left[\boldsymbol{b}_{1}^{T}, \ldots, \boldsymbol{b}_{m}^{T}\right] | | |
| lyche-numerical-linear-algebra_FO3308 | 252 | 1.000 | \boldsymbol{x}^{T}=\left[\boldsymbol{x}_{1}^{T}, \ldots, \boldsymbol{x}_{m}^{T}\right] | | |
| lyche-numerical-linear-algebra_FO3309 | 252 | 1.000 | \boldsymbol{L}_{k} | | |
| lyche-numerical-linear-algebra_FO3310 | 252 | 1.000 | \boldsymbol{L}_{k} \boldsymbol{U}_{k}=\boldsymbol{A}_{k} | | |
| lyche-numerical-linear-algebra_FO3311 | 252 | 1.000 | \boldsymbol{x}_{k} | | |
| lyche-numerical-linear-algebra_FO3312 | 252 | 1.000 | O\left(n^{3 / 2}\right) | | |
| lyche-numerical-linear-algebra_FO3313 | 252 | 1.000 | O(n \log n) | | |
| lyche-numerical-linear-algebra_FO3314 | 252 | 1.000 | \boldsymbol{V}= | | |
| lyche-numerical-linear-algebra_FO3315 | 252 | 1.000 | \left(v_{j, k}\right) \in \mathbb{R}^{m \times m} | | |
| lyche-numerical-linear-algebra_FO3316 | 252 | 1.000 | \boldsymbol{F}=\left(f_{j, k}\right)=(f(j h, k h)) \in \mathbb{R}^{m \times m} | | |
| lyche-numerical-linear-algebra_FO3317 | 253 | 0.999 | \boldsymbol{X} \in \mathbb{R}^{m \times m} | | |
| lyche-numerical-linear-algebra_FO3318 | 253 | 0.999 | \boldsymbol{V}=\boldsymbol{S} \boldsymbol{X} \boldsymbol{S} | | |
| lyche-numerical-linear-algebra_FO3319 | 253 | 0.999 | \boldsymbol{T} \boldsymbol{V}+\boldsymbol{V} \boldsymbol{T}=h^{2} \boldsymbol{F} | | |
| lyche-numerical-linear-algebra_FO3320 | 253 | 1.000 | \boldsymbol{D} \boldsymbol{X}+\boldsymbol{X} \boldsymbol{D}=4 h^{4} \boldsymbol{G} | | |
| lyche-numerical-linear-algebra_FO3321 | 254 | 1.000 | \boldsymbol{G}=\boldsymbol{S} \boldsymbol{F} \boldsymbol{S} | | |
| lyche-numerical-linear-algebra_FO3322 | 254 | 1.000 | x_{j, k}=h^{4} g_{j, k} /\left(\sigma_{j}+\sigma_{k}\right), \quad j, k=1, \ldots, m | | |
| lyche-numerical-linear-algebra_FO3323 | 254 | 1.000 | \boldsymbol{X}, \boldsymbol{S} | | |
| lyche-numerical-linear-algebra_FO3324 | 254 | 0.999 | -\Delta u=f | | |
| lyche-numerical-linear-algebra_FO3325 | 254 | 0.999 | \Omega=(0,1)^{2} | | |
| lyche-numerical-linear-algebra_FO3326 | 254 | 0.999 | m \in | | |
| lyche-numerical-linear-algebra_FO3327 | 254 | 1.000 | \mathbb{N}, h=1 /(m+1) | | |
| lyche-numerical-linear-algebra_FO3328 | 254 | 1.000 | \boldsymbol{F}=(f(j h, k h)) \in \mathbb{R}^{m \times m} | | |
| lyche-numerical-linear-algebra_FO3329 | 254 | 1.000 | \boldsymbol{V} \in \mathbb{R}^{(m+2) \times(m+2)} | | |
| lyche-numerical-linear-algebra_FO3330 | 254 | 1.000 | O\left(m^{3}\right) | | |
| lyche-numerical-linear-algebra_FO3331 | 254 | 1.000 | O\left(4 \times 2 m^{3}\right)= | | |
| lyche-numerical-linear-algebra_FO3332 | 254 | 0.943 | O\left(8 n^{3 / 2}\right) .{ }^{1} | | |
| lyche-numerical-linear-algebra_FO3333 | 254 | 1.000 | 10^{6} | | |
| lyche-numerical-linear-algebra_FO3334 | 254 | 1.000 | 1000 \times 1000 | | |
| lyche-numerical-linear-algebra_FO3335 | 254 | 1.000 | O\left(4 n^{3 / 2}\right) | | |
| lyche-numerical-linear-algebra_FO3336 | 255 | 1.000 | \boldsymbol{A} \in \mathbb{R}^{m \times m} | | |
| lyche-numerical-linear-algebra_FO3337 | 255 | 1.000 | \boldsymbol{S} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO3338 | 255 | 1.000 | \boldsymbol{A} \boldsymbol{S} | | |
| lyche-numerical-linear-algebra_FO3339 | 255 | 1.000 | O\left(m^{2} \log _{2} m\right) | | |
| lyche-numerical-linear-algebra_FO3340 | 255 | 1.000 | \boldsymbol{v}=\left[v_{1}, \ldots, v_{m}\right]^{T} \in \mathbb{R}^{m} | | |
| lyche-numerical-linear-algebra_FO3341 | 255 | 1.000 | \boldsymbol{w}=\left[w_{1}, \ldots, w_{m}\right]^{T} | | |
| lyche-numerical-linear-algebra_FO3342 | 255 | 1.000 | \boldsymbol{w}=\boldsymbol{S} \boldsymbol{v} | | |
| lyche-numerical-linear-algebra_FO3343 | 255 | 1.000 | \boldsymbol{B}=\boldsymbol{S} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO3344 | 255 | 1.000 | \boldsymbol{A} \in \mathbb{R}^{m, n} | | |
| lyche-numerical-linear-algebra_FO3345 | 255 | 1.000 | \boldsymbol{B}=\boldsymbol{A} \boldsymbol{S} | | |
| lyche-numerical-linear-algebra_FO3346 | 255 | 1.000 | \boldsymbol{B}=\left(\boldsymbol{S} \boldsymbol{A}^{T}\right)^{T} | | |
| lyche-numerical-linear-algebra_FO3347 | 255 | 1.000 | N \in \mathbb{N} | | |
| lyche-numerical-linear-algebra_FO3348 | 256 | 1.000 | \boldsymbol{y}=\left[y_{1}, \ldots, y_{N}\right]^{T} \in \mathbb{R}^{N} | | |
| lyche-numerical-linear-algebra_FO3349 | 256 | 1.000 | z=\left[z_{1}, \ldots, z_{N}\right]^{T} | | |
| lyche-numerical-linear-algebra_FO3350 | 256 | 0.991 | \boldsymbol{z}=\boldsymbol{F}_{N} \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO3351 | 256 | 0.991 | \boldsymbol{F}_{N} \in \mathbb{C}^{N \times N} | | |
| lyche-numerical-linear-algebra_FO3352 | 256 | 1.000 | \omega_{N}^{j k}, j, k=0,1, \ldots, N-1 | | |
| lyche-numerical-linear-algebra_FO3353 | 256 | 1.000 | \boldsymbol{B}=\boldsymbol{F}_{N} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO3354 | 256 | 1.000 | \omega_{4}^{2}=(-i)^{2}=-1, \omega_{4}^{3}=(-i)(-1)=i, \omega_{4}^{4}=(-1)^{2}=1, \omega_{4}^{6}=i^{2}=-1 | | |
| lyche-numerical-linear-algebra_FO3355 | 256 | 1.000 | \omega_{4}^{9}=i^{3}=-i | | |
| lyche-numerical-linear-algebra_FO3356 | 256 | 1.000 | 2 m+2 | | |
| lyche-numerical-linear-algebra_FO3357 | 256 | 1.000 | w | | |
| lyche-numerical-linear-algebra_FO3358 | 256 | 1.000 | \boldsymbol{S} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO3359 | 256 | 1.000 | i / 2 | | |
| lyche-numerical-linear-algebra_FO3360 | 256 | 1.000 | \boldsymbol{F}_{2 m+2} \boldsymbol{z} | | |
| lyche-numerical-linear-algebra_FO3361 | 256 | 1.000 | \omega=\omega_{2 m+2}=e^{-2 \pi i /(2 m+2)}=e^{-\pi i /(m+1)} | | |
| lyche-numerical-linear-algebra_FO3362 | 257 | 1.000 | -2 i | | |
| lyche-numerical-linear-algebra_FO3363 | 257 | 1.000 | -1 /(2 i)=-i /\left(2 i^{2}\right)=i / 2 | | |
| lyche-numerical-linear-algebra_FO3364 | 257 | 1.000 | N=2 m+2 | | |
| lyche-numerical-linear-algebra_FO3365 | 257 | 1.000 | \boldsymbol{F}_{N} \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO3366 | 257 | 1.000 | \boldsymbol{F}_{N} | | |
| lyche-numerical-linear-algebra_FO3367 | 257 | 1.000 | \boldsymbol{F}_{N / 2} | | |
| lyche-numerical-linear-algebra_FO3368 | 257 | 1.000 | N / 2 | | |
| lyche-numerical-linear-algebra_FO3369 | 257 | 1.000 | O\left(N^{2}\right) | | |
| lyche-numerical-linear-algebra_FO3370 | 257 | 1.000 | O\left(N \log _{2} N\right) | | |
| lyche-numerical-linear-algebra_FO3371 | 257 | 1.000 | \boldsymbol{P}_{N} \in \mathbb{R}^{N \times N} | | |
| lyche-numerical-linear-algebra_FO3372 | 257 | 0.998 | \boldsymbol{e}_{k}=\left(\delta_{j, k}\right) | | |
| lyche-numerical-linear-algebra_FO3373 | 257 | 0.998 | \left[\boldsymbol{a}_{1}, \ldots, \boldsymbol{a}_{N}\right] | | |
| lyche-numerical-linear-algebra_FO3374 | 257 | 0.965 | \boldsymbol{P}_{N} | | |
| lyche-numerical-linear-algebra_FO3375 | 258 | 1.000 | \boldsymbol{F}_{4} \boldsymbol{P}_{4} | | |
| lyche-numerical-linear-algebra_FO3376 | 258 | 1.000 | \boldsymbol{F}_{2} | | |
| lyche-numerical-linear-algebra_FO3377 | 258 | 1.000 | \omega_{2}=\exp ^{-2 \pi i / 2}=-1 | | |
| lyche-numerical-linear-algebra_FO3378 | 258 | 1.000 | N=2 m | | |
| lyche-numerical-linear-algebra_FO3379 | 258 | 1.000 | p, q | | |
| lyche-numerical-linear-algebra_FO3380 | 258 | 1.000 | 1 \leq p, q \leq m | | |
| lyche-numerical-linear-algebra_FO3381 | 258 | 1.000 | j:=p-1 | | |
| lyche-numerical-linear-algebra_FO3382 | 258 | 1.000 | k:=q-1 | | |
| lyche-numerical-linear-algebra_FO3383 | 258 | 1.000 | \boldsymbol{F}_{2 m} \boldsymbol{P}_{2 m} | | |
| lyche-numerical-linear-algebra_FO3384 | 258 | 1.000 | \boldsymbol{y} \in \mathbb{R}^{2 m} | | |
| lyche-numerical-linear-algebra_FO3385 | 258 | 1.000 | \boldsymbol{w}=\boldsymbol{P}_{2 m}^{T} \boldsymbol{y}=\left[\boldsymbol{w}_{1}^{T}, \boldsymbol{w}_{2}^{T}\right]^{T} | | |
| lyche-numerical-linear-algebra_FO3386 | 259 | 1.000 | \boldsymbol{F}_{2 m} \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO3387 | 259 | 1.000 | \boldsymbol{F}_{m} \boldsymbol{w}_{1} | | |
| lyche-numerical-linear-algebra_FO3388 | 259 | 1.000 | \boldsymbol{F}_{m} \boldsymbol{w}_{2} | | |
| lyche-numerical-linear-algebra_FO3389 | 259 | 1.000 | n=2^{k} | | |
| lyche-numerical-linear-algebra_FO3390 | 259 | 1.000 | \boldsymbol{y} \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO3391 | 259 | 1.000 | \gamma N \log _{2} N | | |
| lyche-numerical-linear-algebra_FO3392 | 259 | 1.000 | \gamma | | |
| lyche-numerical-linear-algebra_FO3393 | 259 | 1.000 | N=2^{k} | | |
| lyche-numerical-linear-algebra_FO3394 | 259 | 1.000 | N / 2=2^{k-1} | | |
| lyche-numerical-linear-algebra_FO3395 | 259 | 1.000 | \boldsymbol{D}_{N / 2} | | |
| lyche-numerical-linear-algebra_FO3396 | 259 | 1.000 | x_{k}=2 x_{k-1}+\gamma 2^{k} | | |
| lyche-numerical-linear-algebra_FO3397 | 259 | 1.000 | x_{0}=0 | | |
| lyche-numerical-linear-algebra_FO3398 | 259 | 1.000 | x_{k}=\gamma k 2^{k} | | |
| lyche-numerical-linear-algebra_FO3399 | 259 | 1.000 | x_{k-1}=\gamma(k-1) 2^{k-1} | | |
| lyche-numerical-linear-algebra_FO3400 | 259 | 1.000 | x_{k}=2 x_{k-1}+\gamma 2^{k}=2 \gamma(k-1) 2^{k-1}+\gamma 2^{k}=\gamma k 2^{k} | | |
| lyche-numerical-linear-algebra_FO3401 | 259 | 1.000 | \gamma \approx 5 | | |
| lyche-numerical-linear-algebra_FO3402 | 259 | 1.000 | O\left(8 n^{2}\right) | | |
| lyche-numerical-linear-algebra_FO3403 | 260 | 1.000 | 5 N \log _{2} N | | |
| lyche-numerical-linear-algebra_FO3404 | 260 | 1.000 | N=2^{20} \approx 10^{6} | | |
| lyche-numerical-linear-algebra_FO3405 | 260 | 1.000 | \boldsymbol{H}=\boldsymbol{S} \boldsymbol{F} | | |
| lyche-numerical-linear-algebra_FO3406 | 260 | 0.998 | \boldsymbol{G}=\boldsymbol{H} \boldsymbol{S} | | |
| lyche-numerical-linear-algebra_FO3407 | 260 | 1.000 | \boldsymbol{Z}=\boldsymbol{S} \boldsymbol{X} | | |
| lyche-numerical-linear-algebra_FO3408 | 260 | 1.000 | \boldsymbol{V}=\boldsymbol{Z} \boldsymbol{S} | | |
| lyche-numerical-linear-algebra_FO3409 | 260 | 1.000 | O\left(\gamma(2 m+2) \log _{2}(2 m+2)\right) | | |
| lyche-numerical-linear-algebra_FO3410 | 260 | 1.000 | O\left(8 n^{3 / 2}\right) | | |
| lyche-numerical-linear-algebra_FO3411 | 260 | 1.000 | \boldsymbol{F}_{4} | | |
| lyche-numerical-linear-algebra_FO3412 | 261 | 1.000 | \left(v_{i, j}\right)_{i, j=1}^{m} | | |
| lyche-numerical-linear-algebra_FO3413 | 261 | 0.608 | \boldsymbol{T} \boldsymbol{V}+\boldsymbol{V T}=h^{2} \boldsymbol{F} | | |
| lyche-numerical-linear-algebra_FO3414 | 261 | 0.608 | \boldsymbol{D}=\operatorname{diag}\left(\lambda_{1}, \ldots, \lambda_{m}\right) | | |
| lyche-numerical-linear-algebra_FO3415 | 261 | 0.608 | \lambda_{j}=4 \sin ^{2}(j \pi h / 2) | | |
| lyche-numerical-linear-algebra_FO3416 | 261 | 1.000 | \boldsymbol{X}=\left[\boldsymbol{x}_{1}, \ldots, \boldsymbol{x}_{m}\right] | | |
| lyche-numerical-linear-algebra_FO3417 | 261 | 1.000 | \boldsymbol{C}=\left[\boldsymbol{c}_{1}, \ldots, \boldsymbol{c}_{m}\right] | | |
| lyche-numerical-linear-algebra_FO3418 | 261 | 1.000 | 8 m-7 | | |
| lyche-numerical-linear-algebra_FO3419 | 261 | 1.000 | O\left(\delta m^{2}\right) | | |
| lyche-numerical-linear-algebra_FO3420 | 261 | 1.000 | O\left(4 m^{3}\right)=O\left(4 n^{3 / 2}\right) | | |
| lyche-numerical-linear-algebra_FO3421 | 261 | 1.000 | O\left(2 \gamma n \log _{2} n\right) | | |
| lyche-numerical-linear-algebra_FO3422 | 261 | 1.000 | \boldsymbol{V}=\boldsymbol{S} \boldsymbol{X} \boldsymbol{S}=\left(x_{j, k}\right) | | |
| lyche-numerical-linear-algebra_FO3423 | 262 | 1.000 | \lambda_{j}=4 \sin ^{2}(j \pi h / 2), j=1, \ldots, m | | |
| lyche-numerical-linear-algebra_FO3424 | 262 | 1.000 | \sigma_{j}+\sigma_{k}-\frac{2}{3} \sigma_{j} \sigma_{k}>0 | | |
| lyche-numerical-linear-algebra_FO3425 | 262 | 1.000 | j, k=1,2, \ldots, m | | |
| lyche-numerical-linear-algebra_FO3426 | 262 | 1.000 | \boldsymbol{X}=\left(x_{j, k}\right) | | |
| lyche-numerical-linear-algebra_FO3427 | 262 | 1.000 | \boldsymbol{U}=\boldsymbol{S} \boldsymbol{X} \boldsymbol{S} | | |
| lyche-numerical-linear-algebra_FO3428 | 262 | 1.000 | O\left(\delta n^{3 / 2}\right) | | |
| lyche-numerical-linear-algebra_FO3429 | 262 | 0.713 | \boldsymbol{F}= | | |
| lyche-numerical-linear-algebra_FO3430 | 262 | 1.000 | \boldsymbol{x}=\boldsymbol{A} \backslash \boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO3431 | 262 | 0.581 | \boldsymbol{x} \in \mathbb{R}^{m^{2}} | | |
| lyche-numerical-linear-algebra_FO3432 | 262 | 1.000 | U | | |
| lyche-numerical-linear-algebra_FO3433 | 262 | 1.000 | m=50 | | |
| lyche-numerical-linear-algebra_FO3434 | 264 | 1.000 | \boldsymbol{x}_{0} | | |
| lyche-numerical-linear-algebra_FO3435 | 264 | 1.000 | \left\{\boldsymbol{x}_{k}\right\} | | |
| lyche-numerical-linear-algebra_FO3436 | 264 | 1.000 | \boldsymbol{x}_{k} \rightarrow \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO3437 | 265 | 1.000 | \left[\begin{array}{cc}2 & -1 \\ -1 & 2\end{array}\right]\left[\begin{array}{l}y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 1\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO3438 | 265 | 1.000 | y=(z+1) / 2 | | |
| lyche-numerical-linear-algebra_FO3439 | 265 | 1.000 | z=(y+1) / 2 | | |
| lyche-numerical-linear-algebra_FO3440 | 265 | 1.000 | y_{0}, z_{0} | | |
| lyche-numerical-linear-algebra_FO3441 | 265 | 0.996 | \left\{y_{k}\right\} | | |
| lyche-numerical-linear-algebra_FO3442 | 265 | 0.996 | \left\{z_{k}\right\} | | |
| lyche-numerical-linear-algebra_FO3443 | 265 | 0.996 | y_{k+1}=\left(z_{k}+1\right) / 2 | | |
| lyche-numerical-linear-algebra_FO3444 | 265 | 1.000 | z_{k+1}=\left(y_{k}+1\right) / 2 | | |
| lyche-numerical-linear-algebra_FO3445 | 265 | 1.000 | y_{0}=z_{0}=0 | | |
| lyche-numerical-linear-algebra_FO3446 | 265 | 1.000 | y_{1}=z_{1}=1 / 2 | | |
| lyche-numerical-linear-algebra_FO3447 | 265 | 1.000 | y_{k}=z_{k}=1-2^{-k} | | |
| lyche-numerical-linear-algebra_FO3448 | 265 | 1.000 | k=0,1,2,3, \ldots | | |
| lyche-numerical-linear-algebra_FO3449 | 265 | 1.000 | [1,1]^{T} | | |
| lyche-numerical-linear-algebra_FO3450 | 266 | 1.000 | z_{k+1}=\left(y_{k+1}+1\right) / 2 | | |
| lyche-numerical-linear-algebra_FO3451 | 266 | 1.000 | y_{1}=1 / 2, z_{1}=3 / 4, y_{2}=7 / 8 | | |
| lyche-numerical-linear-algebra_FO3452 | 266 | 1.000 | z_{2}=15 / 16 | | |
| lyche-numerical-linear-algebra_FO3453 | 266 | 1.000 | y_{k}=1-2 \cdot 4^{-k} | | |
| lyche-numerical-linear-algebra_FO3454 | 266 | 1.000 | z_{k}=1-4^{-k} | | |
| lyche-numerical-linear-algebra_FO3455 | 266 | 1.000 | k=1,2,3, \ldots | | |
| lyche-numerical-linear-algebra_FO3456 | 266 | 1.000 | \boldsymbol{x}_{k}=\left[\boldsymbol{x}_{k}(1), \ldots, \boldsymbol{x}_{k}(n)\right]^{T} | | |
| lyche-numerical-linear-algebra_FO3457 | 266 | 1.000 | \boldsymbol{x}(i) | | |
| lyche-numerical-linear-algebra_FO3458 | 266 | 1.000 | \boldsymbol{x}_{k+1}(i) | | |
| lyche-numerical-linear-algebra_FO3459 | 266 | 1.000 | 0<\omega<2 | | |
| lyche-numerical-linear-algebra_FO3460 | 266 | 1.000 | \boldsymbol{x}(i)=\omega \boldsymbol{x}(i)+(1-\omega) \boldsymbol{x}(i) | | |
| lyche-numerical-linear-algebra_FO3461 | 266 | 1.000 | \omega=1 | | |
| lyche-numerical-linear-algebra_FO3462 | 266 | 1.000 | \boldsymbol{x}_{k+1}^{g s} | | |
| lyche-numerical-linear-algebra_FO3463 | 266 | 1.000 | \boldsymbol{x}_{k+1}=\omega \boldsymbol{x}_{k+1}^{g s}+(1- | | |
| lyche-numerical-linear-algebra_FO3464 | 266 | 1.000 | \omega) \boldsymbol{x}_{k} | | |
| lyche-numerical-linear-algebra_FO3465 | 266 | 1.000 | \boldsymbol{x}_{k+1} | | |
| lyche-numerical-linear-algebra_FO3466 | 267 | 1.000 | 1 \leq \omega<2 | | |
| lyche-numerical-linear-algebra_FO3467 | 267 | 1.000 | \boldsymbol{x}_{k+1 / 2} | | |
| lyche-numerical-linear-algebra_FO3468 | 267 | 1.000 | i=n, n-1, \ldots 1 | | |
| lyche-numerical-linear-algebra_FO3469 | 267 | 0.999 | \boldsymbol{v}(i, j) | | |
| lyche-numerical-linear-algebra_FO3470 | 267 | 0.570 | e_{i, j}:=f_{i, j} /(m+1)^{2} | | |
| lyche-numerical-linear-algebra_FO3476 | 268 | 0.995 | \left(i_{1}, j_{1}\right)< | | |
| lyche-numerical-linear-algebra_FO3477 | 268 | 0.960 | i_{2}, j_{2} | | |
| lyche-numerical-linear-algebra_FO3478 | 268 | 0.960 | j_{1} \leq j_{2} | | |
| lyche-numerical-linear-algebra_FO3479 | 268 | 0.960 | i_{1}<i_{2} | | |
| lyche-numerical-linear-algebra_FO3480 | 268 | 0.960 | j_{1}=j_{2} | | |
| lyche-numerical-linear-algebra_FO3481 | 268 | 1.000 | \boldsymbol{F}=\left(f_{i, j}\right) \in \mathbb{R}^{m \times m} | | |
| lyche-numerical-linear-algebra_FO3482 | 268 | 1.000 | \boldsymbol{V}_{0}=\mathbf{0} \in | | |
| lyche-numerical-linear-algebra_FO3483 | 268 | 1.000 | \mathbb{R}^{(m+2) \times(m+2)} | | |
| lyche-numerical-linear-algebra_FO3484 | 268 | 1.000 | \left\|\boldsymbol{V}^{(k)}-\boldsymbol{U}\right\|_{M}:= | | |
| lyche-numerical-linear-algebra_FO3485 | 268 | 0.911 | \max _{i, j}\left|v_{i j}-u_{i j}\right|< | | |
| lyche-numerical-linear-algebra_FO3486 | 268 | 0.911 | \left[u_{i j}\right] \in \mathbb{R}^{(m+2) \times(m+2)} | | |
| lyche-numerical-linear-algebra_FO3487 | 268 | 0.843 | k=k_{n} | | |
| lyche-numerical-linear-algebra_FO3488 | 268 | 0.843 | \boldsymbol{F}=\operatorname{ones}(m, m) | | |
| lyche-numerical-linear-algebra_FO3489 | 268 | 0.843 | m=10,50, K=10^{4} | | |
| lyche-numerical-linear-algebra_FO3490 | 268 | 0.825 | =10^{-8} | | |
| lyche-numerical-linear-algebra_FO3491 | 268 | 1.000 | \omega^{*}:=2 /\left(1+\sin \left(\frac{\pi}{m+1}\right)\right) | | |
| lyche-numerical-linear-algebra_FO3492 | 269 | 1.000 | 0<w<2 | | |
| lyche-numerical-linear-algebra_FO3493 | 269 | 1.000 | w=1 | | |
| lyche-numerical-linear-algebra_FO3494 | 269 | 1.000 | k_{n}=O(n) | | |
| lyche-numerical-linear-algebra_FO3495 | 269 | 1.000 | k_{n}=O(\sqrt{n}) | | |
| lyche-numerical-linear-algebra_FO3496 | 269 | 1.000 | \sqrt{n} | | |
| lyche-numerical-linear-algebra_FO3497 | 269 | 1.000 | O\left(k_{n} \times n\right) | | |
| lyche-numerical-linear-algebra_FO3498 | 270 | 1.000 | \boldsymbol{G} \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO3499 | 270 | 1.000 | \boldsymbol{x}=\boldsymbol{G} \boldsymbol{x}+\boldsymbol{c} | | |
| lyche-numerical-linear-algebra_FO3500 | 270 | 1.000 | \boldsymbol{I}-\boldsymbol{G} | | |
| lyche-numerical-linear-algebra_FO3501 | 270 | 1.000 | \lim _{k \rightarrow \infty} \boldsymbol{x}_{k}=\boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO3502 | 270 | 1.000 | \boldsymbol{M}^{-1} \boldsymbol{A} \boldsymbol{x}=\boldsymbol{M}^{-1} \boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO3503 | 270 | 1.000 | \boldsymbol{x}=\boldsymbol{x}-\boldsymbol{M}^{-1} \boldsymbol{A} \boldsymbol{x}+\boldsymbol{M}^{-1} \boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO3504 | 270 | 1.000 | \boldsymbol{A}=\boldsymbol{D}-\boldsymbol{A}_{\boldsymbol{L}}-\boldsymbol{A}_{\boldsymbol{R}} | | |
| lyche-numerical-linear-algebra_FO3505 | 270 | 1.000 | -\boldsymbol{A}_{\boldsymbol{L}}, \boldsymbol{D} | | |
| lyche-numerical-linear-algebra_FO3506 | 270 | 1.000 | -\boldsymbol{A}_{\boldsymbol{R}} | | |
| lyche-numerical-linear-algebra_FO3507 | 270 | 1.000 | \boldsymbol{D}:=\operatorname{diag}\left(a_{11}, \ldots, a_{n n}\right) | | |
| lyche-numerical-linear-algebra_FO3508 | 271 | 0.753 | \boldsymbol{M}_{J}, \boldsymbol{M}_{1} | | |
| lyche-numerical-linear-algebra_FO3509 | 271 | 0.753 | \boldsymbol{M}_{\omega} | | |
| lyche-numerical-linear-algebra_FO3510 | 271 | 0.753 | J, G S | | |
| lyche-numerical-linear-algebra_FO3511 | 271 | 0.955 | \boldsymbol{D} \boldsymbol{x}_{k+1}=(\boldsymbol{D}-\boldsymbol{A}) \boldsymbol{x}_{k}+\boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO3512 | 271 | 0.955 | \boldsymbol{M}_{J}=\boldsymbol{D} | | |
| lyche-numerical-linear-algebra_FO3513 | 271 | 1.000 | \boldsymbol{A}_{L} \boldsymbol{x}_{k+1} | | |
| lyche-numerical-linear-algebra_FO3514 | 271 | 1.000 | \left(\omega^{-1} \boldsymbol{D}-\boldsymbol{A}_{L}\right) \boldsymbol{x}_{k+1}=\boldsymbol{A}_{R} \boldsymbol{x}_{k}+\boldsymbol{b}+\left(\omega^{-1}-1\right) \boldsymbol{D} \boldsymbol{x}_{k} | | |
| lyche-numerical-linear-algebra_FO3515 | 271 | 1.000 | \boldsymbol{M}_{\omega}= | | |
| lyche-numerical-linear-algebra_FO3516 | 271 | 1.000 | \omega^{-1} \boldsymbol{D}-\boldsymbol{A}_{L} | | |
| lyche-numerical-linear-algebra_FO3517 | 271 | 1.000 | \boldsymbol{M}_{1} | | |
| lyche-numerical-linear-algebra_FO3518 | 271 | 1.000 | \boldsymbol{G}_{\omega}=\boldsymbol{I}-\boldsymbol{M}_{\omega}^{-1} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO3519 | 272 | 1.000 | \boldsymbol{x}_{k+1}(1) | | |
| lyche-numerical-linear-algebra_FO3520 | 272 | 1.000 | \boldsymbol{x}_{k+1}:=\boldsymbol{G} \boldsymbol{x}_{k}+\boldsymbol{c} | | |
| lyche-numerical-linear-algebra_FO3521 | 272 | 1.000 | \boldsymbol{G} \in \mathbb{C}^{n \times n}, \boldsymbol{c} \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO3522 | 272 | 1.000 | \lim _{k \rightarrow \infty} \boldsymbol{G}^{k}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO3523 | 272 | 0.991 | \boldsymbol{x}_{k+1}=\boldsymbol{G} \boldsymbol{x}_{k}+\boldsymbol{c} | | |
| lyche-numerical-linear-algebra_FO3524 | 272 | 0.991 | \boldsymbol{x}_{k+1}-\boldsymbol{x}= | | |
| lyche-numerical-linear-algebra_FO3525 | 272 | 1.000 | \boldsymbol{G}\left(\boldsymbol{x}_{k}-\boldsymbol{x}\right) | | |
| lyche-numerical-linear-algebra_FO3526 | 272 | 0.999 | \boldsymbol{x}_{k}-\boldsymbol{x} \rightarrow \mathbf{0} | | |
| lyche-numerical-linear-algebra_FO3527 | 272 | 0.999 | \boldsymbol{G}^{k} \rightarrow \mathbf{0} | | |
| lyche-numerical-linear-algebra_FO3528 | 272 | 1.000 | \boldsymbol{x}_{0}-\boldsymbol{x}=\boldsymbol{e}_{j} | | |
| lyche-numerical-linear-algebra_FO3529 | 272 | 1.000 | \boldsymbol{G}^{k} \boldsymbol{e}_{j} \rightarrow \mathbf{0} | | |
| lyche-numerical-linear-algebra_FO3530 | 272 | 1.000 | \|\boldsymbol{G}\|<1 | | |
| lyche-numerical-linear-algebra_FO3531 | 273 | 1.000 | \rho(\boldsymbol{G})<1 | | |
| lyche-numerical-linear-algebra_FO3532 | 273 | 1.000 | \boldsymbol{r}_{k}:=\boldsymbol{b}-\boldsymbol{A} \boldsymbol{x}_{k} | | |
| lyche-numerical-linear-algebra_FO3533 | 273 | 0.992 | \lambda_{1} \geq \lambda_{2} \geq \cdots \geq \lambda_{n}>0 | | |
| lyche-numerical-linear-algebra_FO3534 | 273 | 0.992 | \boldsymbol{x}_{k+1}=(\boldsymbol{I}-\alpha \boldsymbol{A}) \boldsymbol{x}_{k}+\boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO3535 | 273 | 1.000 | 0<\alpha<2 / \lambda_{1} | | |
| lyche-numerical-linear-algebra_FO3536 | 274 | 1.000 | \boldsymbol{I}-\alpha \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO3537 | 274 | 1.000 | 1-\alpha \lambda_{j}, \quad j=1, \ldots, n | | |
| lyche-numerical-linear-algebra_FO3538 | 274 | 1.000 | 1-\alpha \lambda_{n}=\alpha \lambda_{1}-1 | | |
| lyche-numerical-linear-algebra_FO3539 | 274 | 1.000 | \alpha=\alpha_{o} | | |
| lyche-numerical-linear-algebra_FO3540 | 274 | 1.000 | \rho_{\alpha}<1 | | |
| lyche-numerical-linear-algebra_FO3541 | 274 | 1.000 | \alpha \lambda_{1}-1<1 | | |
| lyche-numerical-linear-algebra_FO3542 | 274 | 1.000 | \rho_{\alpha}>\rho_{\alpha_{o}} | | |
| lyche-numerical-linear-algebra_FO3543 | 274 | 1.000 | \alpha \leq 0 | | |
| lyche-numerical-linear-algebra_FO3544 | 274 | 1.000 | \alpha \geq 2 / \lambda_{1} | | |
| lyche-numerical-linear-algebra_FO3545 | 274 | 1.000 | 0<\alpha<\alpha_{o} | | |
| lyche-numerical-linear-algebra_FO3546 | 274 | 1.000 | \rho_{\alpha}=1-\alpha \lambda_{n}>1-\alpha_{o} \lambda_{n}=\rho_{\alpha_{o}} | | |
| lyche-numerical-linear-algebra_FO3547 | 274 | 1.000 | \alpha_{o}<\alpha<2 / \lambda_{1} | | |
| lyche-numerical-linear-algebra_FO3548 | 274 | 0.998 | \rho_{\alpha}=\alpha \lambda_{1}-1>\alpha_{o} \lambda_{1}-1=\rho_{\alpha_{o}} | | |
| lyche-numerical-linear-algebra_FO3549 | 274 | 0.964 | \mathbf{R} | | |
| lyche-numerical-linear-algebra_FO3550 | 274 | 0.991 | \lambda_{\text {max }} | | |
| lyche-numerical-linear-algebra_FO3551 | 274 | 0.999 | 0<\alpha<2 / \lambda_{\text {max }} | | |
| lyche-numerical-linear-algebra_FO3552 | 274 | 1.000 | \alpha=\alpha_{o}:=\frac{2}{\lambda_{\text {max }}+\lambda_{\text {min }}} | | |
| lyche-numerical-linear-algebra_FO3553 | 274 | 0.960 | \kappa:=\lambda_{\text {max }} / \lambda_{\text {min }} | | |
| lyche-numerical-linear-algebra_FO3554 | 274 | 0.988 | \left\|\|_{2}\right. | | |
| lyche-numerical-linear-algebra_FO3555 | 274 | 0.988 | \| \boldsymbol{x}_{k}-\boldsymbol{x}\left\|_{2} \leq\right\| \boldsymbol{I}- | | |
| lyche-numerical-linear-algebra_FO3556 | 274 | 0.994 | \alpha_{o} \boldsymbol{A}\left\|_{2}^{k}\right\| \boldsymbol{x}_{0}-\boldsymbol{x} \|_{2} | | |
| lyche-numerical-linear-algebra_FO3557 | 275 | 1.000 | \omega \in(0,2) | | |
| lyche-numerical-linear-algebra_FO3558 | 275 | 1.000 | \boldsymbol{D} \boldsymbol{x}_{k+1}=\omega\left(\boldsymbol{A}_{L} \boldsymbol{x}_{k+1}+\boldsymbol{A}_{R} \boldsymbol{x}_{k}+\boldsymbol{b}\right)+(1-\omega) \boldsymbol{D} \boldsymbol{x}_{k} | | |
| lyche-numerical-linear-algebra_FO3559 | 275 | 1.000 | \boldsymbol{x}_{k+1}=\omega\left(\boldsymbol{L} \boldsymbol{x}_{k+1}+\boldsymbol{R} \boldsymbol{x}_{k}+\boldsymbol{D}^{-1} \boldsymbol{b}\right)+(1-\omega) \boldsymbol{x}_{k} | | |
| lyche-numerical-linear-algebra_FO3560 | 275 | 1.000 | \boldsymbol{L}:=\boldsymbol{D}^{-1} \boldsymbol{A}_{\boldsymbol{L}} | | |
| lyche-numerical-linear-algebra_FO3561 | 275 | 1.000 | \boldsymbol{R}:=\boldsymbol{D}^{-1} \boldsymbol{A}_{\boldsymbol{R}} | | |
| lyche-numerical-linear-algebra_FO3562 | 275 | 1.000 | (\boldsymbol{I}-\omega \boldsymbol{L}) \boldsymbol{x}_{k+1}=(\omega \boldsymbol{R}+(1-\omega) \boldsymbol{I}) \boldsymbol{x}_{k}+\omega \boldsymbol{D}^{-1} \boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO3563 | 275 | 1.000 | \boldsymbol{G}_{\omega} | | |
| lyche-numerical-linear-algebra_FO3564 | 275 | 1.000 | \boldsymbol{I}-\omega \boldsymbol{L} | | |
| lyche-numerical-linear-algebra_FO3565 | 275 | 1.000 | \omega \boldsymbol{R}+(1-\omega) \boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO3566 | 275 | 1.000 | 1-\omega | | |
| lyche-numerical-linear-algebra_FO3567 | 275 | 1.000 | (1-\omega)^{n} | | |
| lyche-numerical-linear-algebra_FO3568 | 275 | 0.998 | \operatorname{det}\left(\boldsymbol{G}_{\omega}\right)=(1-\omega)^{n} | | |
| lyche-numerical-linear-algebra_FO3569 | 275 | 1.000 | |\lambda| \geq|1-\omega| | | |
| lyche-numerical-linear-algebra_FO3570 | 275 | 1.000 | \rho\left(\boldsymbol{G}_{\omega}\right) \geq|\omega-1| | | |
| lyche-numerical-linear-algebra_FO3571 | 275 | 1.000 | \rho\left(\boldsymbol{G}_{\omega}\right) \geq 1 | | |
| lyche-numerical-linear-algebra_FO3572 | 275 | 1.000 | |\lambda|<1 | | |
| lyche-numerical-linear-algebra_FO3573 | 275 | 1.000 | \boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO3574 | 275 | 1.000 | \boldsymbol{x}^{*} \boldsymbol{D} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO3575 | 275 | 1.000 | x \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO3576 | 275 | 1.000 | a_{i i}= | | |
| lyche-numerical-linear-algebra_FO3577 | 275 | 1.000 | \boldsymbol{e}_{i}^{T} \boldsymbol{A} \boldsymbol{e}_{i}>0 | | |
| lyche-numerical-linear-algebra_FO3578 | 275 | 1.000 | |\lambda| \leq 1 | | |
| lyche-numerical-linear-algebra_FO3579 | 276 | 1.000 | \boldsymbol{G}_{\omega} \boldsymbol{x}=\lambda \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO3580 | 276 | 1.000 | \boldsymbol{x}-\left(\omega^{-1} \boldsymbol{D}-\boldsymbol{A}_{\boldsymbol{L}}\right)^{-1} \boldsymbol{A} \boldsymbol{x}=\lambda \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO3581 | 276 | 0.976 | \boldsymbol{A} \boldsymbol{x} \neq \mathbf{0} | | |
| lyche-numerical-linear-algebra_FO3582 | 276 | 0.976 | \lambda \neq 1 | | |
| lyche-numerical-linear-algebra_FO3583 | 276 | 1.000 | (\boldsymbol{A} \boldsymbol{x})^{*} \boldsymbol{y}=\boldsymbol{y}^{*}\left(\omega^{-1} \boldsymbol{D}-\boldsymbol{A}_{\boldsymbol{L}}\right)^{*} \boldsymbol{y}= | | |
| lyche-numerical-linear-algebra_FO3584 | 276 | 1.000 | \boldsymbol{y}^{*}\left(\omega^{-1} \boldsymbol{D}-\boldsymbol{A}_{\boldsymbol{R}}\right) \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO3585 | 276 | 1.000 | \boldsymbol{y}^{*}(\lambda \boldsymbol{A} \boldsymbol{x})=\boldsymbol{y}^{*} \boldsymbol{E} \boldsymbol{y}=\boldsymbol{y}^{*}\left(\omega^{-1} \boldsymbol{D}+\boldsymbol{A}_{\boldsymbol{R}}-\boldsymbol{D}\right) \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO3586 | 276 | 1.000 | (\boldsymbol{A} \boldsymbol{x})^{*}=\boldsymbol{x}^{*} \boldsymbol{A}^{*}=\boldsymbol{x}^{*} \boldsymbol{A}, \boldsymbol{y}:=(1-\lambda) \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO3587 | 276 | 1.000 | \boldsymbol{y}^{*}=(1-\bar{\lambda}) \boldsymbol{x}^{*} | | |
| lyche-numerical-linear-algebra_FO3588 | 276 | 0.997 | \boldsymbol{G}_{J}=\boldsymbol{I}-\boldsymbol{D}^{-1} \boldsymbol{A}=\boldsymbol{I}-\boldsymbol{A} / 4 | | |
| lyche-numerical-linear-algebra_FO3589 | 277 | 1.000 | \rho\left(\boldsymbol{G}_{J}\right)=\cos (\pi h)<1 | | |
| lyche-numerical-linear-algebra_FO3590 | 277 | 1.000 | \boldsymbol{G}_{J} | | |
| lyche-numerical-linear-algebra_FO3591 | 277 | 0.932 | \boldsymbol{x}_{k+1}=\boldsymbol{x}_{k}+\alpha \boldsymbol{r}_{k} | | |
| lyche-numerical-linear-algebra_FO3592 | 277 | 0.932 | \alpha=2 /\left(\lambda_{\text {max }}+\lambda_{\text {min }}\right)=1 / 4 | | |
| lyche-numerical-linear-algebra_FO3593 | 277 | 1.000 | \kappa | | |
| lyche-numerical-linear-algebra_FO3594 | 277 | 1.000 | \rho\left(\boldsymbol{G}_{\omega}\right) | | |
| lyche-numerical-linear-algebra_FO3595 | 277 | 1.000 | \omega \in | | |
| lyche-numerical-linear-algebra_FO3596 | 277 | 1.000 | \beta:=\rho\left(\boldsymbol{G}_{J}\right)=\cos (\pi h) | | |
| lyche-numerical-linear-algebra_FO3597 | 277 | 1.000 | \omega^{*} | | |
| lyche-numerical-linear-algebra_FO3598 | 277 | 1.000 | \omega=2 | | |
| lyche-numerical-linear-algebra_FO3599 | 277 | 1.000 | \beta=\cos (\pi h) | | |
| lyche-numerical-linear-algebra_FO3600 | 277 | 0.998 | \rho\left(\boldsymbol{G}_{1}\right)=\beta^{2}=\rho\left(\boldsymbol{G}_{J}\right)^{2}=\cos ^{2}(\pi h) | | |
| lyche-numerical-linear-algebra_FO3601 | 277 | 0.999 | \rho\left(\boldsymbol{G}_{J}\right), \rho\left(\boldsymbol{G}_{1}\right) | | |
| lyche-numerical-linear-algebra_FO3602 | 277 | 0.999 | \rho\left(\boldsymbol{G}_{\omega^{*}}\right)=\omega^{*}-1 | | |
| lyche-numerical-linear-algebra_FO3603 | 277 | 1.000 | \rho(\boldsymbol{G})^{k_{n}} \leq | | |
| lyche-numerical-linear-algebra_FO3606 | 278 | 1.000 | \boldsymbol{x}_{k}-\boldsymbol{x}=\boldsymbol{G}^{k}\left(\boldsymbol{x}_{0}-\boldsymbol{x}\right) | | |
| lyche-numerical-linear-algebra_FO3607 | 278 | 1.000 | \lim _{k \rightarrow \infty}\left\|\boldsymbol{G}^{k}\right\|^{1 / k}= | | |
| lyche-numerical-linear-algebra_FO3608 | 278 | 1.000 | \rho(\boldsymbol{G}) | | |
| lyche-numerical-linear-algebra_FO3609 | 278 | 1.000 | \left\|\boldsymbol{G}_{J}^{k}\right\|_{2}=\rho\left(\boldsymbol{G}_{J}\right)^{k} | | |
| lyche-numerical-linear-algebra_FO3610 | 278 | 1.000 | \rho(\boldsymbol{G})=1-\eta | | |
| lyche-numerical-linear-algebra_FO3611 | 278 | 1.000 | 0<\eta<1 | | |
| lyche-numerical-linear-algebra_FO3612 | 278 | 1.000 | s \in \mathbb{N} | | |
| lyche-numerical-linear-algebra_FO3613 | 278 | 1.000 | \rho(\boldsymbol{G})^{k} \leq 10^{-s} | | |
| lyche-numerical-linear-algebra_FO3614 | 279 | 1.000 | \tilde{k} | | |
| lyche-numerical-linear-algebra_FO3615 | 279 | 1.000 | \rho(\boldsymbol{G})^{k}=10^{-s} | | |
| lyche-numerical-linear-algebra_FO3616 | 279 | 1.000 | -\log (1-\eta) \approx \eta | | |
| lyche-numerical-linear-algebra_FO3617 | 279 | 1.000 | \eta | | |
| lyche-numerical-linear-algebra_FO3618 | 279 | 0.996 | \rho\left(\boldsymbol{G}_{J}\right)=\cos (\pi h)=1-\eta, \eta=1-\cos (\pi h)=\frac{1}{2} \pi^{2} h^{2}+O\left(h^{4}\right)= | | |
| lyche-numerical-linear-algebra_FO3619 | 279 | 1.000 | \frac{\pi^{2}}{2} / n+O\left(n^{-2}\right) | | |
| lyche-numerical-linear-algebra_FO3620 | 279 | 0.512 | \operatorname{GS}: \rho\left(\boldsymbol{G}_{1}\right)=\cos ^{2}(\pi h)=1-\eta, \eta=1-\cos ^{2}(\pi h)=\sin ^{2} \pi h=\pi^{2} h^{2}+ | | |
| lyche-numerical-linear-algebra_FO3621 | 279 | 1.000 | O\left(h^{4}\right)=\pi^{2} / n+O\left(n^{-2}\right) | | |
| lyche-numerical-linear-algebra_FO3622 | 279 | 0.906 | \rho\left(\boldsymbol{G}_{\omega^{*}}\right)=\frac{1-\sin (\pi h)}{1+\sin (\pi h)}=1-2 \pi h+O\left(h^{2}\right) | | |
| lyche-numerical-linear-algebra_FO3623 | 279 | 1.000 | \boldsymbol{G}^{k} | | |
| lyche-numerical-linear-algebra_FO3624 | 279 | 0.998 | \lim _{k \rightarrow \infty}\left\|\boldsymbol{G}^{k}\right\|^{1 / k}=\rho(\boldsymbol{G}) | | |
| lyche-numerical-linear-algebra_FO3625 | 279 | 1.000 | \left\|\boldsymbol{x}_{k+1}-\boldsymbol{x}_{k}\right\| | | |
| lyche-numerical-linear-algebra_FO3626 | 280 | 1.000 | \left\|\boldsymbol{x}_{k}-\boldsymbol{x}\right\| | | |
| lyche-numerical-linear-algebra_FO3627 | 280 | 1.000 | \|\boldsymbol{G}\| | | |
| lyche-numerical-linear-algebra_FO3628 | 280 | 1.000 | \boldsymbol{x}_{k}=\boldsymbol{G} \boldsymbol{x}_{k-1}+\boldsymbol{c}, \boldsymbol{x}=\boldsymbol{G} \boldsymbol{x}+\boldsymbol{c} | | |
| lyche-numerical-linear-algebra_FO3629 | 280 | 0.998 | (1-\|\boldsymbol{G}\|)\left\|\boldsymbol{x}_{k}-\boldsymbol{x}\right\| \leq\|\boldsymbol{G}\|\left\|\boldsymbol{x}_{k-1}-\boldsymbol{x}_{k}\right\| | | |
| lyche-numerical-linear-algebra_FO3630 | 280 | 0.997 | \boldsymbol{M} \boldsymbol{x}_{k+1}=\boldsymbol{M} \boldsymbol{x}_{k}-\boldsymbol{A} \boldsymbol{x}_{k}+\boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO3631 | 280 | 1.000 | \left\{\boldsymbol{A}^{k}\right\} | | |
| lyche-numerical-linear-algebra_FO3632 | 281 | 1.000 | \lim _{k \rightarrow \infty} \boldsymbol{A}^{k}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO3633 | 281 | 1.000 | \rho(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO3634 | 281 | 0.913 | \rho(\boldsymbol{A})<1 | | |
| lyche-numerical-linear-algebra_FO3635 | 281 | 1.000 | |\lambda| \geq 1 | | |
| lyche-numerical-linear-algebra_FO3636 | 281 | 1.000 | \boldsymbol{A}^{k} \boldsymbol{x}=\lambda^{k} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO3637 | 281 | 1.000 | \left\|\boldsymbol{A}^{k}\right\|_{2} \geq\left\|\boldsymbol{A}^{k} \boldsymbol{x}\right\|_{2}=\left\|\lambda^{k} \boldsymbol{x}\right\|_{2}=|\lambda|^{k} | | |
| lyche-numerical-linear-algebra_FO3638 | 281 | 1.000 | \|\boldsymbol{A}\|<1 | | |
| lyche-numerical-linear-algebra_FO3639 | 281 | 1.000 | \rho(\boldsymbol{A}) \leq | | |
| lyche-numerical-linear-algebra_FO3640 | 281 | 1.000 | \|\boldsymbol{A}\| | | |
| lyche-numerical-linear-algebra_FO3641 | 281 | 0.679 | \boldsymbol{A},\| \| | | |
| lyche-numerical-linear-algebra_FO3642 | 281 | 1.000 | \boldsymbol{X}:=[\boldsymbol{x}, \ldots, \boldsymbol{x}] \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO3643 | 281 | 1.000 | \lambda \boldsymbol{X}=\boldsymbol{A} \boldsymbol{X} | | |
| lyche-numerical-linear-algebra_FO3644 | 281 | 1.000 | |\lambda|\|\boldsymbol{X}\|= | | |
| lyche-numerical-linear-algebra_FO3645 | 281 | 0.999 | \|\lambda \boldsymbol{X}\|=\|\boldsymbol{A} \boldsymbol{X}\| \leq\|\boldsymbol{A}\|\|\boldsymbol{X}\| | | |
| lyche-numerical-linear-algebra_FO3646 | 281 | 0.999 | \|\boldsymbol{X}\| \neq 0 | | |
| lyche-numerical-linear-algebra_FO3647 | 281 | 0.999 | |\lambda| \leq\|\boldsymbol{A}\| | | |
| lyche-numerical-linear-algebra_FO3648 | 281 | 1.000 | \rho(\boldsymbol{A}) \leq\|\boldsymbol{A}\| \leq \rho(\boldsymbol{A})+\epsilon | | |
| lyche-numerical-linear-algebra_FO3649 | 281 | 1.000 | \boldsymbol{R}=\left[r_{i j}\right] | | |
| lyche-numerical-linear-algebra_FO3650 | 281 | 1.000 | \boldsymbol{U}^{*} \boldsymbol{A} \boldsymbol{U}=\boldsymbol{R} | | |
| lyche-numerical-linear-algebra_FO3651 | 281 | 1.000 | \boldsymbol{D}_{t}:=\operatorname{diag}\left(t, t^{2}, \ldots, t^{n}\right) \in \mathbb{R}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO3652 | 281 | 0.718 | (i, j) | | |
| lyche-numerical-linear-algebra_FO3653 | 281 | 0.718 | \boldsymbol{D}_{t} \boldsymbol{R} \boldsymbol{D}_{t}^{-1} | | |
| lyche-numerical-linear-algebra_FO3654 | 281 | 0.718 | t^{i-j} r_{i j} | | |
| lyche-numerical-linear-algebra_FO3655 | 281 | 0.877 | \|\boldsymbol{B}\|_{t}:=\left\|\boldsymbol{D}_{t} \boldsymbol{U}^{*} \boldsymbol{B} \boldsymbol{U} \boldsymbol{D}_{t}^{-1}\right\|_{1} | | |
| lyche-numerical-linear-algebra_FO3656 | 281 | 0.877 | \left\|\|_{t}\right. | | |
| lyche-numerical-linear-algebra_FO3657 | 281 | 1.000 | \|\boldsymbol{B}\|:=\|\boldsymbol{B}\|_{t} | | |
| lyche-numerical-linear-algebra_FO3658 | 282 | 1.000 | \lambda^{k} | | |
| lyche-numerical-linear-algebra_FO3659 | 282 | 1.000 | \rho(\boldsymbol{A})^{k}=\rho\left(\boldsymbol{A}^{k}\right) \leq\left\|\boldsymbol{A}^{k}\right\| | | |
| lyche-numerical-linear-algebra_FO3660 | 282 | 1.000 | \rho(\boldsymbol{A}) \leq\left\|\boldsymbol{A}^{k}\right\|^{1 / k} | | |
| lyche-numerical-linear-algebra_FO3661 | 282 | 1.000 | \boldsymbol{B}:=(\rho(\boldsymbol{A})+\epsilon)^{-1} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO3662 | 282 | 1.000 | \rho(\boldsymbol{B})=\rho(\boldsymbol{A}) /(\rho(\boldsymbol{A})+\epsilon)<1 | | |
| lyche-numerical-linear-algebra_FO3663 | 282 | 1.000 | \left\|\boldsymbol{B}^{k}\right\| \rightarrow 0 | | |
| lyche-numerical-linear-algebra_FO3664 | 282 | 1.000 | k \rightarrow \infty | | |
| lyche-numerical-linear-algebra_FO3665 | 282 | 1.000 | \left\|\boldsymbol{B}^{k}\right\|<1 | | |
| lyche-numerical-linear-algebra_FO3666 | 282 | 1.000 | k \geq N | | |
| lyche-numerical-linear-algebra_FO3667 | 282 | 1.000 | \rho(\boldsymbol{A}) \leq\left\|\boldsymbol{A}^{k}\right\|^{1 / k} \leq \rho(\boldsymbol{A})+\epsilon | | |
| lyche-numerical-linear-algebra_FO3668 | 282 | 1.000 | \sum_{k=0}^{\infty} \boldsymbol{A}_{k} | | |
| lyche-numerical-linear-algebra_FO3669 | 282 | 1.000 | \left\{\boldsymbol{S}_{m}\right\} | | |
| lyche-numerical-linear-algebra_FO3670 | 282 | 1.000 | \boldsymbol{S}_{m}=\sum_{k=0}^{m} \boldsymbol{A}_{k} | | |
| lyche-numerical-linear-algebra_FO3671 | 282 | 1.000 | \left\|\boldsymbol{S}_{l}-\boldsymbol{S}_{m}\right\|<\epsilon | | |
| lyche-numerical-linear-algebra_FO3672 | 282 | 1.000 | l>m \geq N | | |
| lyche-numerical-linear-algebra_FO3673 | 282 | 1.000 | \sum_{k=0}^{\infty} \boldsymbol{B}^{k} | | |
| lyche-numerical-linear-algebra_FO3674 | 282 | 1.000 | \rho(\boldsymbol{B})<1 | | |
| lyche-numerical-linear-algebra_FO3675 | 282 | 1.000 | (\boldsymbol{I}-\boldsymbol{B}) | | |
| lyche-numerical-linear-algebra_FO3676 | 282 | 1.000 | (\boldsymbol{I}-\boldsymbol{B})^{-1}=\sum_{k=0}^{\infty} \boldsymbol{B}^{k} | | |
| lyche-numerical-linear-algebra_FO3677 | 282 | 1.000 | \|\boldsymbol{B}\|<1 | | |
| lyche-numerical-linear-algebra_FO3678 | 283 | 1.000 | \boldsymbol{S}_{m}:=\sum_{k=0}^{m} \boldsymbol{B}^{k} | | |
| lyche-numerical-linear-algebra_FO3679 | 283 | 1.000 | l>m | | |
| lyche-numerical-linear-algebra_FO3680 | 283 | 1.000 | \frac{\|\boldsymbol{B}\|^{N+1}}{1-\|\boldsymbol{B}\|}<\epsilon | | |
| lyche-numerical-linear-algebra_FO3681 | 283 | 0.876 | \boldsymbol{S}_{m} \boldsymbol{x}= | | |
| lyche-numerical-linear-algebra_FO3682 | 283 | 1.000 | \sum_{k=0}^{m} \boldsymbol{B}^{k} \boldsymbol{x}=\left(\sum_{k=0}^{m} \lambda^{k}\right) \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO3683 | 283 | 1.000 | \sum_{k=0}^{\infty} \lambda^{k} | | |
| lyche-numerical-linear-algebra_FO3684 | 283 | 1.000 | \left\{\boldsymbol{S}_{m} \boldsymbol{x}\right\} | | |
| lyche-numerical-linear-algebra_FO3685 | 283 | 1.000 | \boldsymbol{B}^{m+1} \rightarrow 0 | | |
| lyche-numerical-linear-algebra_FO3686 | 283 | 0.947 | \left(\sum_{k=0}^{\infty} \boldsymbol{B}^{k}\right)(\boldsymbol{I}-\boldsymbol{B})=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO3687 | 283 | 0.999 | \left\|(\boldsymbol{I}-\boldsymbol{B})^{-1}\right\|=\left\|\sum_{k=0}^{\infty} \boldsymbol{B}^{k}\right\| \leq \sum_{k=0}^{\infty}\|\boldsymbol{B}\|^{k}=\frac{1}{1-\|\boldsymbol{B}\|} | | |
| lyche-numerical-linear-algebra_FO3689 | 283 | 0.546 | \boldsymbol{G}_{J} \boldsymbol{v}=\mu \boldsymbol{v} | | |
| lyche-numerical-linear-algebra_FO3690 | 283 | 1.000 | \boldsymbol{v}:=\operatorname{vec}(\boldsymbol{V}) \in \mathbb{R}^{m^{2}} | | |
| lyche-numerical-linear-algebra_FO3691 | 283 | 1.000 | v_{i, j}=0 | | |
| lyche-numerical-linear-algebra_FO3692 | 283 | 1.000 | i \in\{0, m+1\} | | |
| lyche-numerical-linear-algebra_FO3693 | 283 | 1.000 | j \in\{0, m+1\} | | |
| lyche-numerical-linear-algebra_FO3694 | 283 | 0.588 | (\lambda, \boldsymbol{w}) | | |
| lyche-numerical-linear-algebra_FO3695 | 283 | 0.588 | (12.21)(\boldsymbol{I}-\omega \boldsymbol{L})^{-1}(\omega \boldsymbol{R}+(1- | | |
| lyche-numerical-linear-algebra_FO3696 | 283 | 1.000 | \omega) \boldsymbol{I}) \boldsymbol{w}=\lambda \boldsymbol{w} | | |
| lyche-numerical-linear-algebra_FO3697 | 284 | 1.000 | \boldsymbol{w}=\operatorname{vec}(\boldsymbol{W}) | | |
| lyche-numerical-linear-algebra_FO3698 | 284 | 1.000 | \boldsymbol{W} \in \mathbb{C}^{m \times m} | | |
| lyche-numerical-linear-algebra_FO3699 | 284 | 1.000 | w_{i, j}=0 | | |
| lyche-numerical-linear-algebra_FO3700 | 284 | 0.430 | (\mu, \boldsymbol{v}) | | |
| lyche-numerical-linear-algebra_FO3701 | 284 | 1.000 | \boldsymbol{v}=(\boldsymbol{V}) | | |
| lyche-numerical-linear-algebra_FO3702 | 284 | 1.000 | v_{i, j}:=\lambda^{-(i+j) / 2} w_{i, j} | | |
| lyche-numerical-linear-algebra_FO3703 | 284 | 1.000 | w_{i, j} | | |
| lyche-numerical-linear-algebra_FO3704 | 284 | 1.000 | \lambda^{(i+j) / 2} v_{i, j} | | |
| lyche-numerical-linear-algebra_FO3705 | 284 | 1.000 | \lambda^{(i+j) / 2} | | |
| lyche-numerical-linear-algebra_FO3706 | 284 | 0.999 | \boldsymbol{v}=: \operatorname{vec}(\boldsymbol{V}), \boldsymbol{W}=\operatorname{vec}(\boldsymbol{W}) | | |
| lyche-numerical-linear-algebra_FO3707 | 284 | 0.999 | w_{i, j}:=\lambda^{(i+j) / 2} v_{i, j} | | |
| lyche-numerical-linear-algebra_FO3708 | 284 | 1.000 | \lambda^{-(i+j) / 2} | | |
| lyche-numerical-linear-algebra_FO3709 | 284 | 1.000 | \omega \lambda^{1 / 2} | | |
| lyche-numerical-linear-algebra_FO3710 | 284 | 0.952 | \omega \mu \lambda^{1 / 2}=\lambda+\omega-1 | | |
| lyche-numerical-linear-algebra_FO3711 | 284 | 0.952 | \lambda, \boldsymbol{w} | | |
| lyche-numerical-linear-algebra_FO3712 | 285 | 1.000 | \rho\left(\boldsymbol{G}_{\omega}\right)=|\lambda(\mu)| | | |
| lyche-numerical-linear-algebra_FO3713 | 285 | 1.000 | \lambda(\mu) | | |
| lyche-numerical-linear-algebra_FO3714 | 285 | 1.000 | \frac{1}{2} \cos (j \pi h)+\frac{1}{2} \cos (k \pi h) | | |
| lyche-numerical-linear-algebra_FO3715 | 285 | 1.000 | -\mu | | |
| lyche-numerical-linear-algebra_FO3716 | 285 | 1.000 | \lambda=\lambda(\mu) | | |
| lyche-numerical-linear-algebra_FO3717 | 285 | 0.998 | d(0)=4>0 | | |
| lyche-numerical-linear-algebra_FO3718 | 285 | 0.998 | d(2)=4 \mu^{2}-4<0 | | |
| lyche-numerical-linear-algebra_FO3719 | 285 | 0.998 | \omega, \lambda(\mu) | | |
| lyche-numerical-linear-algebra_FO3720 | 285 | 0.618 | d(\omega)=0 | | |
| lyche-numerical-linear-algebra_FO3721 | 285 | 1.000 | \tilde{\omega}(\mu) | | |
| lyche-numerical-linear-algebra_FO3722 | 285 | 1.000 | |\lambda(\mu)| | | |
| lyche-numerical-linear-algebra_FO3723 | 285 | 1.000 | \mu=\rho\left(\boldsymbol{G}_{J}\right)=: \beta | | |
| lyche-numerical-linear-algebra_FO3724 | 285 | 0.995 | 0<\omega \leq \tilde{\omega}(\beta)=\omega^{*} | | |
| lyche-numerical-linear-algebra_FO3725 | 285 | 1.000 | \rho\left(\boldsymbol{G}_{\omega}\right)=\omega-1 | | |
| lyche-numerical-linear-algebra_FO3726 | 285 | 1.000 | \omega^{*}<\omega<2 | | |
| lyche-numerical-linear-algebra_FO3727 | 285 | 1.000 | 0<\omega<\omega^{*} | | |
| lyche-numerical-linear-algebra_FO3728 | 285 | 1.000 | \beta^{2}\left(\omega^{2} \beta^{2}-4 \omega+4\right)<\left(2-\omega \beta^{2}\right)^{2} | | |
| lyche-numerical-linear-algebra_FO3729 | 286 | 1.000 | a_{i i}=d \neq 0 | | |
| lyche-numerical-linear-algebra_FO3730 | 286 | 1.000 | \alpha:=1 / d | | |
| lyche-numerical-linear-algebra_FO3731 | 286 | 1.000 | u_{j} | | |
| lyche-numerical-linear-algebra_FO3732 | 286 | 0.766 | 0<\alpha<\min _{j}\left(2 u_{j} /\left|\lambda_{j}\right|^{2}\right) | | |
| lyche-numerical-linear-algebra_FO3733 | 286 | 1.000 | \boldsymbol{A}=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO3734 | 286 | 0.945 | \boldsymbol{A}:=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right] \in \mathbb{R}^{2 \times 2} | | |
| lyche-numerical-linear-algebra_FO3735 | 286 | 1.000 | \left[\begin{array}{ccc}1 & a & a \\ a & 1 & a \\ a & a & 1\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO3736 | 286 | 1.000 | -1 / 2<a<1 | | |
| lyche-numerical-linear-algebra_FO3737 | 286 | 1.000 | 1 / 2<a<1 | | |
| lyche-numerical-linear-algebra_FO3738 | 286 | 0.995 | \boldsymbol{G}_{1} | | |
| lyche-numerical-linear-algebra_FO3739 | 286 | 0.924 | \boldsymbol{A}:=\left[\begin{array}{ccc}1 & 0 & 1 / 2 \\ 1 & 1 & 0 \\ -1 & 1 & 1\end{array}\right] .1 | | |
| lyche-numerical-linear-algebra_FO3740 | 286 | 0.996 | \boldsymbol{G}_{1}:=\left[\begin{array}{ccc}0 & 0 & -1 / 2 \\ 0 & 0 & 1 / 2 \\ 0 & 0 & -1\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO3741 | 286 | 1.000 | p(\lambda):=\operatorname{det}\left(\lambda \boldsymbol{I}-\boldsymbol{G}_{J}\right)=\lambda^{3}+\frac{1}{2} \lambda+\frac{1}{2} | | |
| lyche-numerical-linear-algebra_FO3742 | 286 | 0.997 | p(\lambda) \neq 0 | | |
| lyche-numerical-linear-algebra_FO3743 | 286 | 1.000 | \left|a_{i i}\right|>\sum_{j \neq i}\left|a_{i j}\right| | | |
| lyche-numerical-linear-algebra_FO3744 | 287 | 1.000 | r:=\max _{i} r_{i}<1 | | |
| lyche-numerical-linear-algebra_FO3745 | 287 | 1.000 | r_{i}=\sum_{j \neq i} \frac{\left|a_{i j}\right|}{\left|a_{i i}\right|} | | |
| lyche-numerical-linear-algebra_FO3746 | 287 | 1.000 | \left|\boldsymbol{\epsilon}_{k+1}(j)\right| \leq r\left\|\boldsymbol{\epsilon}_{k}\right\|_{\infty} | | |
| lyche-numerical-linear-algebra_FO3747 | 287 | 1.000 | j=1, \ldots, i | | |
| lyche-numerical-linear-algebra_FO3748 | 287 | 1.000 | a \in | | |
| lyche-numerical-linear-algebra_FO3749 | 287 | 1.000 | \boldsymbol{x}_{0}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO3750 | 287 | 1.000 | \boldsymbol{x}=[1,1]^{T} | | |
| lyche-numerical-linear-algebra_FO3751 | 287 | 1.000 | |a|<1 | | |
| lyche-numerical-linear-algebra_FO3752 | 287 | 1.000 | |a|>1 | | |
| lyche-numerical-linear-algebra_FO3753 | 287 | 1.000 | \eta=1-|a| | | |
| lyche-numerical-linear-algebra_FO3754 | 287 | 0.999 | a=0.9 | | |
| lyche-numerical-linear-algebra_FO3755 | 287 | 0.999 | s=16 | | |
| lyche-numerical-linear-algebra_FO3756 | 287 | 1.000 | |a|^{k} \leq 10^{-16} | | |
| lyche-numerical-linear-algebra_FO3757 | 287 | 1.000 | \rho\left(\boldsymbol{G}_{J}\right)=1 / 2 | | |
| lyche-numerical-linear-algebra_FO3758 | 287 | 1.000 | \boldsymbol{x}_{k}(1)=\boldsymbol{x}_{k}(2)=1-2^{-k} | | |
| lyche-numerical-linear-algebra_FO3759 | 287 | 1.000 | \boldsymbol{A}=\boldsymbol{I}-\boldsymbol{L}-\boldsymbol{L}^{T} | | |
| lyche-numerical-linear-algebra_FO3760 | 287 | 1.000 | l_{i, j}=0 | | |
| lyche-numerical-linear-algebra_FO3761 | 287 | 1.000 | j>=i | | |
| lyche-numerical-linear-algebra_FO3762 | 287 | 1.000 | \boldsymbol{N} | | |
| lyche-numerical-linear-algebra_FO3763 | 287 | 1.000 | \boldsymbol{M}^{-1} \boldsymbol{N} | | |
| lyche-numerical-linear-algebra_FO3764 | 288 | 1.000 | \boldsymbol{x}=[x(1), x(2), \ldots, x(n)]^{T} | | |
| lyche-numerical-linear-algebra_FO3765 | 288 | 0.996 | i=1,2, \ldots, n-1, n, n, n-1, n-2, \ldots, 1 | | |
| lyche-numerical-linear-algebra_FO3766 | 288 | 0.999 | \boldsymbol{b}:=[1,9,-2]^{T} | | |
| lyche-numerical-linear-algebra_FO3767 | 288 | 1.000 | \boldsymbol{x}_{0}=[1,1,1]^{t} | | |
| lyche-numerical-linear-algebra_FO3768 | 288 | 1.000 | \boldsymbol{x}_{1} \in \mathbb{R}^{3} | | |
| lyche-numerical-linear-algebra_FO3769 | 288 | 1.000 | \boldsymbol{E} \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO3770 | 288 | 1.000 | \rho\left(\boldsymbol{A}^{-1} \boldsymbol{E}\right)<1 | | |
| lyche-numerical-linear-algebra_FO3771 | 289 | 0.998 | \lim _{k \rightarrow \infty} \| | | |
| lyche-numerical-linear-algebra_FO3772 | 289 | 1.000 | \boldsymbol{A}^{k} \|^{1 / k}=\rho(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO3773 | 289 | 1.000 | |\lambda|=\rho(\boldsymbol{A})<1 | | |
| lyche-numerical-linear-algebra_FO3774 | 289 | 1.000 | (1, n) | | |
| lyche-numerical-linear-algebra_FO3775 | 289 | 1.000 | f(k):=\binom{k}{n-1} a^{n-1} \lambda^{k-n+1} | | |
| lyche-numerical-linear-algebra_FO3776 | 289 | 1.000 | k \geq n-1 | | |
| lyche-numerical-linear-algebra_FO3777 | 289 | 1.000 | n=5 | | |
| lyche-numerical-linear-algebra_FO3778 | 289 | 1.000 | f(k) | | |
| lyche-numerical-linear-algebra_FO3779 | 289 | 1.000 | \lambda=0.9, a=10 | | |
| lyche-numerical-linear-algebra_FO3780 | 289 | 1.000 | n-1 \leq k \leq 200 | | |
| lyche-numerical-linear-algebra_FO3781 | 289 | 1.000 | \max _{k} f(k) | | |
| lyche-numerical-linear-algebra_FO3782 | 289 | 1.000 | f(k)<10^{-8} | | |
| lyche-numerical-linear-algebra_FO3783 | 289 | 1.000 | \boldsymbol{E}:=(\boldsymbol{A}-\lambda \boldsymbol{I}) / a | | |
| lyche-numerical-linear-algebra_FO3784 | 289 | 1.000 | \boldsymbol{E}^{k}=\left[\begin{array}{cc}\mathbf{0} & \boldsymbol{I}_{n-k} \\ \mathbf{0} & \mathbf{0}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO3785 | 289 | 1.000 | \boldsymbol{E}^{n}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO3786 | 289 | 0.997 | \boldsymbol{A}^{k}=(a \boldsymbol{E}+\lambda \boldsymbol{I})^{k}=\sum_{j=0}^{\min \{k, n-1\}}\binom{k}{j} a^{j} \lambda^{k-j} \boldsymbol{E}^{j} | | |
| lyche-numerical-linear-algebra_FO3787 | 289 | 1.000 | \boldsymbol{A}^{n} \rightarrow \mathbf{0} | | |
| lyche-numerical-linear-algebra_FO3788 | 289 | 1.000 | \| \boldsymbol{A} \|<\rho(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO3789 | 290 | 0.969 | \kappa:=\frac{\lambda_{\text {max }}}{\lambda_{\text {min }}} | | |
| lyche-numerical-linear-algebra_FO3790 | 290 | 1.000 | \langle\boldsymbol{x}, \boldsymbol{y}\rangle=\boldsymbol{x}^{T} \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO3791 | 290 | 1.000 | \|\boldsymbol{x}\|_{2}=\sqrt{\boldsymbol{x}^{T} \boldsymbol{x}} | | |
| lyche-numerical-linear-algebra_FO3792 | 291 | 1.000 | \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z} \in | | |
| lyche-numerical-linear-algebra_FO3793 | 291 | 1.000 | \langle\boldsymbol{x}, \boldsymbol{x}\rangle_{\boldsymbol{A}}=\boldsymbol{x}^{T} \boldsymbol{A} \boldsymbol{x} \geq 0 | | |
| lyche-numerical-linear-algebra_FO3794 | 291 | 1.000 | \langle\boldsymbol{x}, \boldsymbol{x}\rangle_{\boldsymbol{A}}=0 | | |
| lyche-numerical-linear-algebra_FO3795 | 291 | 0.983 | \langle\boldsymbol{x}, \boldsymbol{y}\rangle_{\boldsymbol{A}}:=\boldsymbol{x}^{T} \boldsymbol{A} \boldsymbol{y}=\left(\boldsymbol{x}^{T} \boldsymbol{A} \boldsymbol{y}\right)^{T}=\boldsymbol{y}^{T} \boldsymbol{A}^{T} \boldsymbol{x}=\boldsymbol{y}^{T} \boldsymbol{A} \boldsymbol{x}=\langle\boldsymbol{y}, \boldsymbol{x}\rangle_{\boldsymbol{A}} | | |
| lyche-numerical-linear-algebra_FO3796 | 291 | 1.000 | \langle\boldsymbol{x}+\boldsymbol{y}, \boldsymbol{z}\rangle_{\boldsymbol{A}}:=\boldsymbol{x}^{T} \boldsymbol{A} \boldsymbol{z}+\boldsymbol{y}^{T} \boldsymbol{A} \boldsymbol{z}=\langle\boldsymbol{x}, \boldsymbol{z}\rangle_{\boldsymbol{A}}+\langle\boldsymbol{y}, \boldsymbol{z}\rangle_{\boldsymbol{A}} | | |
| lyche-numerical-linear-algebra_FO3797 | 291 | 1.000 | \boldsymbol{A} \in \mathbb{R}^{n \times n}, \boldsymbol{b} \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO3798 | 291 | 1.000 | c \in \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO3799 | 291 | 1.000 | Q: \mathbb{R}^{n} \rightarrow \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO3800 | 291 | 1.000 | \boldsymbol{y}, \boldsymbol{h} \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO3801 | 291 | 1.000 | \varepsilon \in \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO3802 | 291 | 0.996 | \boldsymbol{r}(\boldsymbol{y}):=\boldsymbol{b}-\boldsymbol{A} \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO3803 | 291 | 1.000 | \boldsymbol{r}(\boldsymbol{y})=-\nabla Q(\boldsymbol{y}) | | |
| lyche-numerical-linear-algebra_FO3804 | 291 | 1.000 | \nabla:=\left[\frac{\partial}{\partial y_{1}}, \ldots, \frac{\partial}{\partial y_{n}}\right]^{T} | | |
| lyche-numerical-linear-algebra_FO3805 | 291 | 0.998 | \boldsymbol{y}=\boldsymbol{x}, \varepsilon=1 | | |
| lyche-numerical-linear-algebra_FO3806 | 291 | 0.998 | Q(\boldsymbol{x}+\boldsymbol{h})= | | |
| lyche-numerical-linear-algebra_FO3807 | 291 | 1.000 | Q(\boldsymbol{x})+\frac{1}{2} \boldsymbol{h}^{T} \boldsymbol{A} \boldsymbol{h} | | |
| lyche-numerical-linear-algebra_FO3808 | 291 | 1.000 | Q(\boldsymbol{x}+\boldsymbol{h})>Q(\boldsymbol{x}) | | |
| lyche-numerical-linear-algebra_FO3809 | 291 | 0.982 | \boldsymbol{h} \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO3810 | 291 | 0.982 | \boldsymbol{A} \boldsymbol{x} \neq \boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO3811 | 292 | 1.000 | \boldsymbol{h}:=\boldsymbol{r}(\boldsymbol{x}) | | |
| lyche-numerical-linear-algebra_FO3812 | 292 | 1.000 | Q(\boldsymbol{x}+\varepsilon \boldsymbol{h})-Q(\boldsymbol{x})=-\varepsilon\left(\boldsymbol{h}^{T} \boldsymbol{r}(x)-\frac{1}{2} \varepsilon \boldsymbol{h}^{T} \boldsymbol{A} \boldsymbol{h}\right)<0 | | |
| lyche-numerical-linear-algebra_FO3813 | 292 | 1.000 | \boldsymbol{x}_{0} \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO3814 | 292 | 1.000 | \alpha_{k} | | |
| lyche-numerical-linear-algebra_FO3815 | 292 | 1.000 | \boldsymbol{x}_{k+1}=\boldsymbol{x}_{k}+\alpha_{k} \boldsymbol{p}_{k} | | |
| lyche-numerical-linear-algebra_FO3816 | 293 | 0.999 | Q\left(\boldsymbol{x}_{k+1}\right) | | |
| lyche-numerical-linear-algebra_FO3817 | 293 | 1.000 | Q\left(\boldsymbol{x}_{k}+\alpha \boldsymbol{p}_{k}\right)=Q\left(\boldsymbol{x}_{k}\right)-\alpha \boldsymbol{p}_{k}^{T} \boldsymbol{r}_{k}+\frac{1}{2} \alpha^{2} \boldsymbol{p}_{k}^{T} \boldsymbol{A} \boldsymbol{p}_{k} | | |
| lyche-numerical-linear-algebra_FO3818 | 293 | 1.000 | \boldsymbol{r}_{k}:= | | |
| lyche-numerical-linear-algebra_FO3819 | 293 | 1.000 | \boldsymbol{b}-\boldsymbol{A} \boldsymbol{x}_{k} | | |
| lyche-numerical-linear-algebra_FO3820 | 293 | 1.000 | \boldsymbol{p}_{k}^{T} \boldsymbol{A} \boldsymbol{p}_{k} \geq 0 | | |
| lyche-numerical-linear-algebra_FO3821 | 293 | 1.000 | \frac{\partial}{\partial \alpha} Q\left(\boldsymbol{x}_{k}+\alpha \boldsymbol{p}_{k}\right)=0 | | |
| lyche-numerical-linear-algebra_FO3822 | 293 | 1.000 | \boldsymbol{p}_{k}=\boldsymbol{r}_{k} | | |
| lyche-numerical-linear-algebra_FO3823 | 293 | 1.000 | k=0,1,2 \ldots | | |
| lyche-numerical-linear-algebra_FO3824 | 293 | 1.000 | \boldsymbol{r}_{k+1}^{T} \boldsymbol{r}_{k}=\left(\boldsymbol{r}_{k}-\alpha_{k} \boldsymbol{A} \boldsymbol{r}_{k}\right)^{T} \boldsymbol{r}_{k}=0 | | |
| lyche-numerical-linear-algebra_FO3825 | 293 | 1.000 | \alpha_{k}= | | |
| lyche-numerical-linear-algebra_FO3826 | 293 | 0.887 | \frac{\boldsymbol{r}_{k}^{T} \boldsymbol{r}_{k}}{\boldsymbol{r}_{k}^{T} \boldsymbol{A} \boldsymbol{r}_{k}} | | |
| lyche-numerical-linear-algebra_FO3827 | 293 | 1.000 | \boldsymbol{x}_{0}=[-1,-1 / 2]^{T} | | |
| lyche-numerical-linear-algebra_FO3828 | 293 | 1.000 | \boldsymbol{r}_{0}=-\boldsymbol{A} \boldsymbol{x}_{0}=[3 / 2,0]^{T} | | |
| lyche-numerical-linear-algebra_FO3829 | 294 | 1.000 | k \geq 1 | | |
| lyche-numerical-linear-algebra_FO3830 | 294 | 1.000 | \alpha_{k}=1 / 2 | | |
| lyche-numerical-linear-algebra_FO3831 | 294 | 1.000 | \left\|\boldsymbol{x}_{j+1}\right\|_{2} /\left\|\boldsymbol{x}_{j}\right\|=\left\|\boldsymbol{r}_{j+1}\right\|_{2} /\left\|\boldsymbol{r}_{j}\right\|_{2}=1 / 2 | | |
| lyche-numerical-linear-algebra_FO3832 | 294 | 1.000 | \boldsymbol{p}_{i}^{T} \boldsymbol{A} \boldsymbol{p}_{j}=0 | | |
| lyche-numerical-linear-algebra_FO3833 | 294 | 1.000 | \boldsymbol{r}_{0}= | | |
| lyche-numerical-linear-algebra_FO3834 | 294 | 1.000 | \boldsymbol{b}-\boldsymbol{A} \boldsymbol{x}_{0}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO3835 | 294 | 1.000 | \boldsymbol{r}_{1}^{T} \boldsymbol{r}_{0}=0 | | |
| lyche-numerical-linear-algebra_FO3836 | 294 | 1.000 | \boldsymbol{p}_{0}:=\boldsymbol{r}_{0} | | |
| lyche-numerical-linear-algebra_FO3837 | 294 | 1.000 | j \geq 0 | | |
| lyche-numerical-linear-algebra_FO3838 | 294 | 1.000 | \boldsymbol{p}_{j} | | |
| lyche-numerical-linear-algebra_FO3839 | 294 | 0.999 | \boldsymbol{r}_{0}, \ldots, \boldsymbol{r}_{j} | | |
| lyche-numerical-linear-algebra_FO3840 | 294 | 1.000 | \alpha_{j} | | |
| lyche-numerical-linear-algebra_FO3841 | 295 | 1.000 | \boldsymbol{x}_{j} | | |
| lyche-numerical-linear-algebra_FO3842 | 295 | 1.000 | \boldsymbol{r}_{j} \neq 0 | | |
| lyche-numerical-linear-algebra_FO3843 | 295 | 1.000 | \boldsymbol{r}_{i}^{T} \boldsymbol{r}_{j}=0 | | |
| lyche-numerical-linear-algebra_FO3844 | 295 | 1.000 | i, j=0,1, \ldots, k, i \neq j | | |
| lyche-numerical-linear-algebra_FO3845 | 295 | 1.000 | \boldsymbol{r}_{k+1}^{T} \boldsymbol{r}_{j}=0 | | |
| lyche-numerical-linear-algebra_FO3846 | 295 | 1.000 | j=0,1, \ldots, k | | |
| lyche-numerical-linear-algebra_FO3847 | 295 | 1.000 | \boldsymbol{r}_{j} | | |
| lyche-numerical-linear-algebra_FO3848 | 295 | 1.000 | j \leq k | | |
| lyche-numerical-linear-algebra_FO3849 | 295 | 1.000 | \boldsymbol{p}_{k}^{T} \boldsymbol{A} \boldsymbol{p}_{i}=0 | | |
| lyche-numerical-linear-algebra_FO3850 | 295 | 1.000 | i<k | | |
| lyche-numerical-linear-algebra_FO3851 | 295 | 1.000 | \boldsymbol{r}_{k+1} | | |
| lyche-numerical-linear-algebra_FO3852 | 295 | 1.000 | j<k | | |
| lyche-numerical-linear-algebra_FO3853 | 295 | 1.000 | \operatorname{dim} \mathbb{R}^{n}=n | | |
| lyche-numerical-linear-algebra_FO3854 | 295 | 1.000 | \boldsymbol{r}_{0}, \ldots, \boldsymbol{r}_{n} | | |
| lyche-numerical-linear-algebra_FO3855 | 295 | 1.000 | \boldsymbol{r}_{k}=0 | | |
| lyche-numerical-linear-algebra_FO3856 | 295 | 1.000 | j=k+1 | | |
| lyche-numerical-linear-algebra_FO3857 | 295 | 1.000 | \boldsymbol{p}_{k+1}=\boldsymbol{r}_{k+1}+\beta_{k} \boldsymbol{p}_{k} | | |
| lyche-numerical-linear-algebra_FO3858 | 295 | 1.000 | \boldsymbol{x}_{0}, \boldsymbol{p}_{0}=\boldsymbol{r}_{0}= | | |
| lyche-numerical-linear-algebra_FO3859 | 295 | 1.000 | \boldsymbol{b}-\boldsymbol{A} \boldsymbol{x}_{0} | | |
| lyche-numerical-linear-algebra_FO3860 | 296 | 1.000 | \left[\begin{array}{cc}2 & -1 \\ -1 & 2\end{array}\right]\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]=\left[\begin{array}{l}0 \\ 0\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO3861 | 296 | 1.000 | \boldsymbol{x}_{0}=\left[\begin{array}{c}-1 \\ -1 / 2\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO3862 | 296 | 1.000 | \boldsymbol{p}_{0}=\boldsymbol{r}_{0}=\left[\begin{array}{c}3 / 2 \\ 0\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO3863 | 296 | 1.000 | \boldsymbol{x}_{2} | | |
| lyche-numerical-linear-algebra_FO3864 | 296 | 0.969 | \left\|\boldsymbol{r}_{k}\right\|_{2} /\|\boldsymbol{b}\|_{2} \leq | | |
| lyche-numerical-linear-algebra_FO3865 | 297 | 0.803 | k> | | |
| lyche-numerical-linear-algebra_FO3866 | 297 | 0.996 | (\boldsymbol{t}=\boldsymbol{A} \boldsymbol{p}) | | |
| lyche-numerical-linear-algebra_FO3867 | 297 | 1.000 | \left(\boldsymbol{p}^{T} \boldsymbol{t}\right. | | |
| lyche-numerical-linear-algebra_FO3868 | 297 | 1.000 | \left.\boldsymbol{r}^{T} \boldsymbol{r}\right) | | |
| lyche-numerical-linear-algebra_FO3869 | 297 | 0.758 | \boldsymbol{x}=\boldsymbol{x}+a \boldsymbol{p}, \boldsymbol{r}=\boldsymbol{r}-a \boldsymbol{t} | | |
| lyche-numerical-linear-algebra_FO3870 | 297 | 0.758 | \boldsymbol{p}=\boldsymbol{r}+ | | |
| lyche-numerical-linear-algebra_FO3871 | 297 | 1.000 | \boldsymbol{t}=\boldsymbol{A} \boldsymbol{p} | | |
| lyche-numerical-linear-algebra_FO3872 | 297 | 1.000 | \boldsymbol{T}_{2}:=\boldsymbol{T}_{1} \otimes \boldsymbol{I}+\boldsymbol{I} \otimes \boldsymbol{T}_{1} | | |
| lyche-numerical-linear-algebra_FO3873 | 297 | 1.000 | \boldsymbol{T}_{1}:=\operatorname{tridiag}_{m}(a, d, a) \in \mathbb{R}^{m \times m} | | |
| lyche-numerical-linear-algebra_FO3874 | 297 | 1.000 | \boldsymbol{f}=[1, \ldots, 1]^{T} \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO3875 | 297 | 1.000 | a=1 / 9, d=5 / 18 | | |
| lyche-numerical-linear-algebra_FO3876 | 297 | 1.000 | a=-1, d=2 | | |
| lyche-numerical-linear-algebra_FO3877 | 297 | 1.000 | O(5 n) | | |
| lyche-numerical-linear-algebra_FO3878 | 297 | 1.000 | \boldsymbol{t} | | |
| lyche-numerical-linear-algebra_FO3879 | 298 | 1.000 | \boldsymbol{V}, \boldsymbol{R}, \boldsymbol{P}, \boldsymbol{B}, \boldsymbol{T} \in | | |
| lyche-numerical-linear-algebra_FO3880 | 298 | 1.000 | \boldsymbol{x}=\operatorname{vec}(\boldsymbol{V}), \boldsymbol{r}=\operatorname{vec}(\boldsymbol{R}), \boldsymbol{p}=\operatorname{vec}(\boldsymbol{P}), \boldsymbol{t}=\operatorname{vec}(\boldsymbol{T}) | | |
| lyche-numerical-linear-algebra_FO3881 | 298 | 1.000 | h^{2} \boldsymbol{f}=\operatorname{vec}(\boldsymbol{B}) | | |
| lyche-numerical-linear-algebra_FO3882 | 298 | 1.000 | \boldsymbol{T}_{2} \boldsymbol{x}=h^{2} \boldsymbol{f} \Longleftrightarrow \boldsymbol{T}_{1} \boldsymbol{V}+\boldsymbol{V} \boldsymbol{T}_{1}=\boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO3883 | 298 | 1.000 | \boldsymbol{t}=\boldsymbol{T}_{2} \boldsymbol{p} \Longleftrightarrow \boldsymbol{T}=\boldsymbol{T}_{1} \boldsymbol{P}+\boldsymbol{P} \boldsymbol{T}_{1} | | |
| lyche-numerical-linear-algebra_FO3884 | 298 | 0.958 | t o l=10^{-8} | | |
| lyche-numerical-linear-algebra_FO3885 | 298 | 1.000 | K / \sqrt{n} | | |
| lyche-numerical-linear-algebra_FO3886 | 299 | 1.000 | \|\boldsymbol{x}\|_{\boldsymbol{A}}:=\sqrt{\boldsymbol{x}^{T} \boldsymbol{A} \boldsymbol{x}} | | |
| lyche-numerical-linear-algebra_FO3887 | 299 | 0.995 | \kappa=\operatorname{cond}_{2}(\boldsymbol{A}):=\lambda_{\text {max }} / \lambda_{\text {min }} | | |
| lyche-numerical-linear-algebra_FO3888 | 299 | 1.000 | \frac{\kappa-1}{\kappa+1}<1 | | |
| lyche-numerical-linear-algebra_FO3889 | 299 | 1.000 | \frac{\kappa-1}{\kappa+1} | | |
| lyche-numerical-linear-algebra_FO3890 | 300 | 1.000 | d=5 / 18, a=1 / 9 | | |
| lyche-numerical-linear-algebra_FO3891 | 300 | 0.999 | \lambda_{\text {max }}=\frac{5}{9}+\frac{4}{9} \cos (\pi h) | | |
| lyche-numerical-linear-algebra_FO3892 | 300 | 0.999 | \lambda_{\text {min }}=\frac{5}{9}-\frac{4}{9} \cos (\pi h) | | |
| lyche-numerical-linear-algebra_FO3893 | 300 | 1.000 | \mathbb{W}_{0}=\{\mathbf{0}\} | | |
| lyche-numerical-linear-algebra_FO3894 | 301 | 1.000 | \operatorname{dim}\left(\mathbb{W}_{k}\right) \leq k | | |
| lyche-numerical-linear-algebra_FO3895 | 301 | 1.000 | \boldsymbol{v} \in \mathbb{W}_{k} | | |
| lyche-numerical-linear-algebra_FO3896 | 301 | 1.000 | \boldsymbol{A} \boldsymbol{v} \in \mathbb{W}_{k+1} | | |
| lyche-numerical-linear-algebra_FO3897 | 301 | 1.000 | \boldsymbol{p}_{0}=\boldsymbol{r}_{0} | | |
| lyche-numerical-linear-algebra_FO3898 | 301 | 1.000 | \boldsymbol{r}_{k+1}=\boldsymbol{r}_{k}-\alpha_{k} \boldsymbol{A} \boldsymbol{p}_{k} \in \mathbb{W}_{k+2}, \boldsymbol{p}_{k+1}=\boldsymbol{r}_{k+1}+\beta_{k} \boldsymbol{p}_{k} \in | | |
| lyche-numerical-linear-algebra_FO3899 | 301 | 0.995 | \mathbb{W}_{k+2} | | |
| lyche-numerical-linear-algebra_FO3900 | 301 | 0.995 | \boldsymbol{x}_{k+1}-\boldsymbol{x}_{0} \stackrel{\text { (13.12) }}{=} \boldsymbol{x}_{k}-\boldsymbol{x}_{0}+\alpha_{k} \boldsymbol{p}_{k} \in \mathbb{W}_{k+1} | | |
| lyche-numerical-linear-algebra_FO3901 | 301 | 0.984 | \boldsymbol{w} \in \mathbb{W}_{k} | | |
| lyche-numerical-linear-algebra_FO3902 | 301 | 1.000 | \left\{\boldsymbol{r}_{0}, \boldsymbol{r}_{1}, \ldots, \boldsymbol{r}_{k-1}\right\} | | |
| lyche-numerical-linear-algebra_FO3903 | 301 | 1.000 | \left\{\boldsymbol{p}_{0}, \boldsymbol{p}_{1}, \ldots, \boldsymbol{p}_{k-1}\right\} | | |
| lyche-numerical-linear-algebra_FO3904 | 301 | 0.995 | \boldsymbol{u}:=\boldsymbol{x}_{k}-\boldsymbol{x}_{0}-\boldsymbol{w} | | |
| lyche-numerical-linear-algebra_FO3905 | 301 | 0.995 | \boldsymbol{u} \in \mathbb{W}_{k} | | |
| lyche-numerical-linear-algebra_FO3906 | 301 | 1.000 | \left\langle\boldsymbol{x}-\boldsymbol{x}_{k}, \boldsymbol{u}\right\rangle=\boldsymbol{r}_{k}^{T} \boldsymbol{u}=0 | | |
| lyche-numerical-linear-algebra_FO3907 | 301 | 1.000 | \boldsymbol{u}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO3908 | 301 | 1.000 | \boldsymbol{x}_{0} \neq \mathbf{0} | | |
| lyche-numerical-linear-algebra_FO3909 | 301 | 1.000 | \boldsymbol{x}-\boldsymbol{x}_{k}=\left(\boldsymbol{x}-\boldsymbol{x}_{0}\right)- | | |
| lyche-numerical-linear-algebra_FO3910 | 301 | 1.000 | \left(\boldsymbol{x}_{k}-\boldsymbol{x}_{0}\right) | | |
| lyche-numerical-linear-algebra_FO3911 | 301 | 1.000 | \boldsymbol{x}_{k}-\boldsymbol{x}_{0} | | |
| lyche-numerical-linear-algebra_FO3912 | 301 | 1.000 | p(t):=\sum_{j=0}^{m} a_{j} t^{m} | | |
| lyche-numerical-linear-algebra_FO3913 | 301 | 0.808 | p(\boldsymbol{A}):=a_{0} \boldsymbol{I}+a_{1} \boldsymbol{A}+\cdots+a_{m} \boldsymbol{A}^{m} | | |
| lyche-numerical-linear-algebra_FO3914 | 301 | 0.979 | \left(p\left(\lambda_{j}\right), \boldsymbol{u}_{j}\right) | | |
| lyche-numerical-linear-algebra_FO3915 | 301 | 0.979 | p(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO3916 | 301 | 0.991 | \left(\lambda_{j}, \boldsymbol{u}_{j}\right), j=1,2, \ldots, n | | |
| lyche-numerical-linear-algebra_FO3917 | 301 | 1.000 | \boldsymbol{r}_{0}:=\boldsymbol{b}-\boldsymbol{A} \boldsymbol{x}_{0} | | |
| lyche-numerical-linear-algebra_FO3918 | 302 | 1.000 | P(t):=\sum_{j=0}^{k-1} a_{j} t^{k-1} | | |
| lyche-numerical-linear-algebra_FO3919 | 302 | 1.000 | \boldsymbol{w}=P(\boldsymbol{A}) \boldsymbol{r}_{0} | | |
| lyche-numerical-linear-algebra_FO3920 | 302 | 1.000 | \sum_{j=1}^{n} \sigma_{j} \boldsymbol{u}_{j} | | |
| lyche-numerical-linear-algebra_FO3921 | 302 | 1.000 | \boldsymbol{w}=a_{0} \boldsymbol{r}_{0}+a_{1} \boldsymbol{A} \boldsymbol{r}_{0}+\cdots+a_{k-1} \boldsymbol{A}^{k-1} \boldsymbol{r}_{0} | | |
| lyche-numerical-linear-algebra_FO3922 | 302 | 1.000 | a_{0}, \ldots, a_{k-1} | | |
| lyche-numerical-linear-algebra_FO3923 | 302 | 1.000 | \boldsymbol{x}-\boldsymbol{x}_{0}-P(\boldsymbol{A}) \boldsymbol{r}_{0}=\boldsymbol{A}^{-1}\left(\boldsymbol{r}_{0}-\right. | | |
| lyche-numerical-linear-algebra_FO3924 | 302 | 1.000 | \boldsymbol{A} P(\boldsymbol{A})) \boldsymbol{r}_{0}=\boldsymbol{A}^{-1} Q(\boldsymbol{A}) \boldsymbol{r}_{0} | | |
| lyche-numerical-linear-algebra_FO3925 | 302 | 1.000 | \boldsymbol{A}\left(\boldsymbol{x}-\boldsymbol{x}_{0}-P(\boldsymbol{A}) \boldsymbol{r}_{0}\right)=Q(\boldsymbol{A}) \boldsymbol{r}_{0} | | |
| lyche-numerical-linear-algebra_FO3926 | 302 | 1.000 | \boldsymbol{r}_{0} | | |
| lyche-numerical-linear-algebra_FO3927 | 303 | 0.978 | 0<a<b | | |
| lyche-numerical-linear-algebra_FO3928 | 303 | 0.978 | A | | |
| lyche-numerical-linear-algebra_FO3929 | 303 | 1.000 | \Pi_{k} | | |
| lyche-numerical-linear-algebra_FO3930 | 303 | 1.000 | \leq k | | |
| lyche-numerical-linear-algebra_FO3931 | 303 | 0.998 | Q(t)=1 | | |
| lyche-numerical-linear-algebra_FO3932 | 303 | 0.998 | P(\boldsymbol{A})=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO3933 | 303 | 0.998 | \| \boldsymbol{x}- | | |
| lyche-numerical-linear-algebra_FO3934 | 303 | 0.969 | \boldsymbol{x}_{0} \|_{\boldsymbol{A}}^{2}=\sum_{j=1}^{n} \frac{\sigma_{j}^{2}}{\lambda_{j}} | | |
| lyche-numerical-linear-algebra_FO3935 | 303 | 1.000 | Q \in \Pi_{k} | | |
| lyche-numerical-linear-algebra_FO3936 | 303 | 1.000 | Q(0)=1 | | |
| lyche-numerical-linear-algebra_FO3937 | 303 | 0.999 | \kappa=\lambda_{\text {max }} / \lambda_{\text {min }} | | |
| lyche-numerical-linear-algebra_FO3938 | 303 | 1.000 | y=1 / x | | |
| lyche-numerical-linear-algebra_FO3939 | 303 | 1.000 | 2^{k} / k!=2, k=1,2 | | |
| lyche-numerical-linear-algebra_FO3940 | 303 | 1.000 | 2^{k} / k!<2 | | |
| lyche-numerical-linear-algebra_FO3941 | 303 | 1.000 | k>2 | | |
| lyche-numerical-linear-algebra_FO3942 | 304 | 1.000 | c \notin[a, b] | | |
| lyche-numerical-linear-algebra_FO3943 | 304 | 1.000 | \mathcal{S}_{k} | | |
| lyche-numerical-linear-algebra_FO3944 | 304 | 1.000 | Q(c)=1 | | |
| lyche-numerical-linear-algebra_FO3945 | 304 | 1.000 | Q^{*} \in \mathcal{S}_{k} | | |
| lyche-numerical-linear-algebra_FO3946 | 304 | 1.000 | Q^{*} | | |
| lyche-numerical-linear-algebra_FO3947 | 304 | 1.000 | T_{n} | | |
| lyche-numerical-linear-algebra_FO3948 | 304 | 0.999 | T_{0}(t)=1 | | |
| lyche-numerical-linear-algebra_FO3949 | 304 | 0.999 | T_{1}(t)=t | | |
| lyche-numerical-linear-algebra_FO3950 | 304 | 0.999 | T_{2}(t)=2 t^{2}-1, T_{3}(t)=4 t^{3}-3 t | | |
| lyche-numerical-linear-algebra_FO3951 | 304 | 1.000 | T_{n}(t)=\cos (n \arccos t) | | |
| lyche-numerical-linear-algebra_FO3952 | 304 | 1.000 | t \in[-1,1] | | |
| lyche-numerical-linear-algebra_FO3953 | 304 | 1.000 | T_{n}(t)=\frac{1}{2}\left[\left(t+\sqrt{t^{2}-1}\right)^{n}+\left(t+\sqrt{t^{2}-1}\right)^{-n}\right] | | |
| lyche-numerical-linear-algebra_FO3954 | 304 | 1.000 | |t| \geq 1 | | |
| lyche-numerical-linear-algebra_FO3955 | 304 | 1.000 | P_{n}(t)=\cos (n \arccos t) | | |
| lyche-numerical-linear-algebra_FO3956 | 304 | 1.000 | P_{n}(t)=\cos n \phi | | |
| lyche-numerical-linear-algebra_FO3957 | 304 | 1.000 | t=\cos \phi | | |
| lyche-numerical-linear-algebra_FO3958 | 304 | 1.000 | P_{n} | | |
| lyche-numerical-linear-algebra_FO3959 | 304 | 1.000 | P_{0}=T_{0} | | |
| lyche-numerical-linear-algebra_FO3960 | 304 | 1.000 | P_{1}=T_{1} | | |
| lyche-numerical-linear-algebra_FO3961 | 304 | 1.000 | P_{n}=T_{n} | | |
| lyche-numerical-linear-algebra_FO3962 | 304 | 1.000 | x_{n}:=T_{n}(t) | | |
| lyche-numerical-linear-algebra_FO3963 | 305 | 0.711 | x_{n}=z^{n} | | |
| lyche-numerical-linear-algebra_FO3964 | 305 | 0.711 | z^{n+1}- | | |
| lyche-numerical-linear-algebra_FO3965 | 305 | 1.000 | 2 t z^{n}+z^{n-1}=0 | | |
| lyche-numerical-linear-algebra_FO3966 | 305 | 1.000 | z^{2}-2 t z+1=0 | | |
| lyche-numerical-linear-algebra_FO3967 | 305 | 1.000 | z_{1}^{n}, z_{2}^{n} | | |
| lyche-numerical-linear-algebra_FO3968 | 305 | 1.000 | c_{1} z_{1}^{n}+c_{2} z_{2}^{n} | | |
| lyche-numerical-linear-algebra_FO3969 | 305 | 1.000 | c_{1} | | |
| lyche-numerical-linear-algebra_FO3970 | 305 | 1.000 | c_{2} | | |
| lyche-numerical-linear-algebra_FO3971 | 305 | 1.000 | x_{0}= | | |
| lyche-numerical-linear-algebra_FO3972 | 305 | 1.000 | c_{1}+c_{2}=1 | | |
| lyche-numerical-linear-algebra_FO3973 | 305 | 1.000 | x_{1}=c_{1} z_{1}+c_{2} z_{2}=t | | |
| lyche-numerical-linear-algebra_FO3974 | 305 | 1.000 | z_{1}+z_{2}=2 t | | |
| lyche-numerical-linear-algebra_FO3975 | 305 | 1.000 | c_{1}=c_{2}=\frac{1}{2} | | |
| lyche-numerical-linear-algebra_FO3976 | 305 | 1.000 | a<b, c \notin[a, b] | | |
| lyche-numerical-linear-algebra_FO3977 | 305 | 1.000 | k \in | | |
| lyche-numerical-linear-algebra_FO3978 | 305 | 1.000 | \mathbb{N} | | |
| lyche-numerical-linear-algebra_FO3979 | 305 | 1.000 | Q \in \mathcal{S}_{k} | | |
| lyche-numerical-linear-algebra_FO3980 | 305 | 1.000 | Q \neq Q^{*} | | |
| lyche-numerical-linear-algebra_FO3981 | 305 | 1.000 | \|Q\|_{\infty}>\left\|Q^{*}\right\|_{\infty} | | |
| lyche-numerical-linear-algebra_FO3982 | 305 | 1.000 | p(z)=p^{\prime}(z)=0 | | |
| lyche-numerical-linear-algebra_FO3983 | 305 | 1.000 | \left|Q^{*}\right| | | |
| lyche-numerical-linear-algebra_FO3984 | 305 | 1.000 | 1 /\left|T_{k}(u(c))\right| | | |
| lyche-numerical-linear-algebra_FO3985 | 305 | 1.000 | \mu_{0}, \ldots, \mu_{k} | | |
| lyche-numerical-linear-algebra_FO3986 | 305 | 1.000 | u\left(\mu_{i}\right)=\cos (i \pi / k) | | |
| lyche-numerical-linear-algebra_FO3987 | 305 | 1.000 | i=0,1, \ldots, k | | |
| lyche-numerical-linear-algebra_FO3988 | 305 | 1.000 | Q \in S_{k} | | |
| lyche-numerical-linear-algebra_FO3989 | 305 | 1.000 | \|Q\|_{\infty} \leq\left\|Q^{*}\right\|_{\infty} | | |
| lyche-numerical-linear-algebra_FO3990 | 305 | 1.000 | Q \equiv Q^{*} | | |
| lyche-numerical-linear-algebra_FO3991 | 305 | 1.000 | f \equiv Q-Q^{*} | | |
| lyche-numerical-linear-algebra_FO3992 | 305 | 1.000 | f(c)=0 | | |
| lyche-numerical-linear-algebra_FO3993 | 305 | 1.000 | f \equiv 0 | | |
| lyche-numerical-linear-algebra_FO3994 | 305 | 1.000 | I_{j}=\left[\mu_{j-1}, \mu_{j}\right] | | |
| lyche-numerical-linear-algebra_FO3995 | 305 | 1.000 | \sigma_{j} \leq 0 | | |
| lyche-numerical-linear-algebra_FO3996 | 305 | 1.000 | Q^{*}\left(\mu_{j}\right)>0 | | |
| lyche-numerical-linear-algebra_FO3997 | 305 | 1.000 | f\left(\mu_{j}\right) \leq 0 | | |
| lyche-numerical-linear-algebra_FO3998 | 305 | 1.000 | f\left(\mu_{j-1}\right) \geq 0 | | |
| lyche-numerical-linear-algebra_FO3999 | 305 | 1.000 | Q^{*}\left(\mu_{j}\right)<0 | | |
| lyche-numerical-linear-algebra_FO4000 | 305 | 1.000 | \sigma_{j}<0, f | | |
| lyche-numerical-linear-algebra_FO4001 | 305 | 1.000 | I_{j} | | |
| lyche-numerical-linear-algebra_FO4002 | 305 | 1.000 | \sigma_{j}=0 | | |
| lyche-numerical-linear-algebra_FO4003 | 305 | 1.000 | f\left(\mu_{j-1}\right)=0 | | |
| lyche-numerical-linear-algebra_FO4004 | 305 | 1.000 | f\left(\mu_{j}\right)=0 | | |
| lyche-numerical-linear-algebra_FO4005 | 305 | 1.000 | Q\left(\mu_{j}\right)=Q^{*}\left(\mu_{j}\right) | | |
| lyche-numerical-linear-algebra_FO4006 | 305 | 0.998 | \mu_{j} \in(a, b) | | |
| lyche-numerical-linear-algebra_FO4007 | 305 | 0.998 | Q^{\prime}\left(\mu_{j}\right)= | | |
| lyche-numerical-linear-algebra_FO4008 | 306 | 1.000 | Q^{* \prime}\left(\mu_{j}\right)=0 | | |
| lyche-numerical-linear-algebra_FO4009 | 306 | 1.000 | f\left(\mu_{j}\right)=f^{\prime}\left(\mu_{j}\right)=0 | | |
| lyche-numerical-linear-algebra_FO4010 | 306 | 1.000 | I_{j+1} | | |
| lyche-numerical-linear-algebra_FO4011 | 306 | 1.000 | \mu_{j}=b | | |
| lyche-numerical-linear-algebra_FO4012 | 306 | 1.000 | \mu_{j-1} | | |
| lyche-numerical-linear-algebra_FO4013 | 306 | 1.000 | \mu_{j-1} \in(a, b) | | |
| lyche-numerical-linear-algebra_FO4014 | 306 | 1.000 | I_{j-1} | | |
| lyche-numerical-linear-algebra_FO4015 | 306 | 1.000 | I_{j}, j= | | |
| lyche-numerical-linear-algebra_FO4016 | 306 | 0.999 | 1,2, \ldots, k | | |
| lyche-numerical-linear-algebra_FO4017 | 306 | 1.000 | c=0 | | |
| lyche-numerical-linear-algebra_FO4018 | 306 | 1.000 | t=(b+a) /(b-a) | | |
| lyche-numerical-linear-algebra_FO4019 | 307 | 1.000 | M:=\lambda_{\text {max }} | | |
| lyche-numerical-linear-algebra_FO4020 | 307 | 1.000 | m:=\lambda_{\text {min }} | | |
| lyche-numerical-linear-algebra_FO4021 | 307 | 0.997 | \left(\lambda_{j}^{-1}, \boldsymbol{u}_{j}\right) | | |
| lyche-numerical-linear-algebra_FO4022 | 307 | 1.000 | \boldsymbol{A}^{-1}, j=1, \ldots, n | | |
| lyche-numerical-linear-algebra_FO4023 | 307 | 1.000 | \boldsymbol{y}=\sum_{j=1}^{n} c_{j} \boldsymbol{u}_{j} | | |
| lyche-numerical-linear-algebra_FO4024 | 307 | 1.000 | f:[m c, M c] \rightarrow \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO4025 | 307 | 1.000 | f(x):=x+1 / x | | |
| lyche-numerical-linear-algebra_FO4026 | 307 | 1.000 | f \in C^{2} | | |
| lyche-numerical-linear-algebra_FO4027 | 307 | 1.000 | f^{\prime \prime} | | |
| lyche-numerical-linear-algebra_FO4028 | 308 | 1.000 | c:=1 / \sqrt{m M} | | |
| lyche-numerical-linear-algebra_FO4029 | 308 | 1.000 | f(m c)=f(M c)=\sqrt{\frac{M}{m}}+\sqrt{\frac{m}{M}}=\frac{M+m}{\sqrt{m M}} | | |
| lyche-numerical-linear-algebra_FO4030 | 308 | 1.000 | \boldsymbol{\epsilon}_{j}:=\boldsymbol{x}-\boldsymbol{x}_{j}, j=0,1, \ldots | | |
| lyche-numerical-linear-algebra_FO4031 | 308 | 1.000 | \left\|\boldsymbol{\epsilon}_{k}\right\|_{\boldsymbol{A}} \leq\left(\frac{\kappa-1}{\kappa+1}\right)\left\|\boldsymbol{\epsilon}_{k-1}\right\| \leq \cdots \leq\left(\frac{\kappa-1}{\kappa+1}\right)^{k}\left\|\boldsymbol{\epsilon}_{0}\right\| | | |
| lyche-numerical-linear-algebra_FO4032 | 309 | 1.000 | \boldsymbol{r}_{m+1}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO4033 | 309 | 1.000 | k \leq m | | |
| lyche-numerical-linear-algebra_FO4034 | 309 | 1.000 | \left\|\boldsymbol{\epsilon}_{k+1}\right\|_{2}<\left\|\boldsymbol{\epsilon}_{k}\right\|_{2} | | |
| lyche-numerical-linear-algebra_FO4035 | 309 | 0.983 | \boldsymbol{\epsilon}_{j}=\boldsymbol{x}-\boldsymbol{x}_{j} | | |
| lyche-numerical-linear-algebra_FO4036 | 309 | 1.000 | j \leq m | | |
| lyche-numerical-linear-algebra_FO4037 | 309 | 1.000 | \beta_{i-1} \cdots \beta_{k}=\left(\boldsymbol{r}_{i}^{T} \boldsymbol{r}_{i}\right) /\left(\boldsymbol{r}_{k}^{T} \boldsymbol{r}_{k}\right) | | |
| lyche-numerical-linear-algebra_FO4038 | 310 | 1.000 | \operatorname{cond}_{2}(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO4039 | 310 | 1.000 | \boldsymbol{B} \boldsymbol{A} \boldsymbol{x}=\boldsymbol{B} \boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO4040 | 310 | 1.000 | \operatorname{cond}_{2}(\boldsymbol{B} \boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO4041 | 310 | 0.578 | \boldsymbol{B}=\boldsymbol{M}^{-1} | | |
| lyche-numerical-linear-algebra_FO4042 | 310 | 1.000 | \boldsymbol{B}=\boldsymbol{C}^{T} \boldsymbol{C} | | |
| lyche-numerical-linear-algebra_FO4043 | 310 | 1.000 | \boldsymbol{y}:=\boldsymbol{C}^{-T} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO4044 | 311 | 1.000 | \boldsymbol{C} \boldsymbol{A} \boldsymbol{C}^{T} | | |
| lyche-numerical-linear-algebra_FO4045 | 311 | 1.000 | \left(\boldsymbol{C} \boldsymbol{A} \boldsymbol{C}^{T}\right) \boldsymbol{y}=\boldsymbol{C} \boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO4046 | 311 | 0.997 | \boldsymbol{q}_{k} | | |
| lyche-numerical-linear-algebra_FO4047 | 311 | 0.997 | \boldsymbol{z}_{k}:=\boldsymbol{C} \boldsymbol{b}-\boldsymbol{C} \boldsymbol{A} \boldsymbol{C}^{T} \boldsymbol{y}_{k} | | |
| lyche-numerical-linear-algebra_FO4048 | 311 | 0.998 | \boldsymbol{A} \boldsymbol{x}=\boldsymbol{b}, \boldsymbol{r}_{k}=\boldsymbol{b}-\boldsymbol{A} \boldsymbol{x}_{k} | | |
| lyche-numerical-linear-algebra_FO4049 | 311 | 0.573 | \boldsymbol{s}_{k}=\boldsymbol{C}^{T} \boldsymbol{z}_{k}=\boldsymbol{C}^{T}\left(\boldsymbol{C}-\boldsymbol{C} \boldsymbol{A} \boldsymbol{C}^{T}\right) \boldsymbol{y}_{k}=\boldsymbol{B} \boldsymbol{b}- | | |
| lyche-numerical-linear-algebra_FO4050 | 311 | 1.000 | \boldsymbol{B} \boldsymbol{A} \boldsymbol{x}_{k} | | |
| lyche-numerical-linear-algebra_FO4051 | 311 | 1.000 | \boldsymbol{r}_{0}=\boldsymbol{b}-\boldsymbol{A} \boldsymbol{x}_{0}, \boldsymbol{p}_{0}= | | |
| lyche-numerical-linear-algebra_FO4052 | 311 | 0.835 | \boldsymbol{s}_{0}=\boldsymbol{B r}_{0} | | |
| lyche-numerical-linear-algebra_FO4053 | 311 | 0.995 | \mathrm{x}\left(=\boldsymbol{x}_{0}\right) | | |
| lyche-numerical-linear-algebra_FO4054 | 312 | 1.000 | \rho | | |
| lyche-numerical-linear-algebra_FO4055 | 312 | 1.000 | w=B * t | | |
| lyche-numerical-linear-algebra_FO4056 | 312 | 1.000 | \boldsymbol{y}_{k} | | |
| lyche-numerical-linear-algebra_FO4057 | 313 | 0.998 | \boldsymbol{B} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO4058 | 313 | 1.000 | u=u(x, y) | | |
| lyche-numerical-linear-algebra_FO4059 | 313 | 1.000 | c(x, y)>0 | | |
| lyche-numerical-linear-algebra_FO4060 | 313 | 0.991 | (x, y) \in \Omega | | |
| lyche-numerical-linear-algebra_FO4061 | 313 | 1.000 | c(x, y)=1 | | |
| lyche-numerical-linear-algebra_FO4062 | 313 | 1.000 | m \in \mathbb{N} | | |
| lyche-numerical-linear-algebra_FO4063 | 313 | 1.000 | v_{j, k} \approx u\left(x_{j}, y_{k}\right) | | |
| lyche-numerical-linear-algebra_FO4064 | 314 | 1.000 | f, g | | |
| lyche-numerical-linear-algebra_FO4065 | 314 | 1.000 | f_{j, k}:=f\left(x_{j}, y_{k}\right),(d v)_{j, k}:=\left(d_{1} v\right)_{j, k}+\left(d_{2} v\right)_{j, k} | | |
| lyche-numerical-linear-algebra_FO4066 | 314 | 1.000 | c_{p, q}=c(p h, q h) | | |
| lyche-numerical-linear-algebra_FO4067 | 314 | 1.000 | p, q \in \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO4068 | 314 | 0.997 | h^{2} L_{h} v=h^{2} \boldsymbol{F} | | |
| lyche-numerical-linear-algebra_FO4069 | 315 | 1.000 | \boldsymbol{V}, \boldsymbol{W} \in \mathbb{R}^{m \times m} | | |
| lyche-numerical-linear-algebra_FO4070 | 315 | 1.000 | v_{j, k}=w_{j, k}=0 | | |
| lyche-numerical-linear-algebra_FO4071 | 315 | 1.000 | (j, k) \in \partial I_{m} | | |
| lyche-numerical-linear-algebra_FO4072 | 315 | 0.994 | m \in \mathbb{N}, a_{i}, b_{i}, c_{i} \in \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO4073 | 315 | 0.994 | i=0, \ldots, m | | |
| lyche-numerical-linear-algebra_FO4074 | 315 | 0.994 | b_{0}=c_{0}=b_{m+1}=c_{m+1}=0 | | |
| lyche-numerical-linear-algebra_FO4075 | 315 | 1.000 | \left(d_{1} v\right)_{j, k} w_{j, k} | | |
| lyche-numerical-linear-algebra_FO4076 | 315 | 1.000 | \left(d_{2} v\right)_{j, k} w_{j, k} | | |
| lyche-numerical-linear-algebra_FO4077 | 315 | 1.000 | \left\langle L_{h} v, \boldsymbol{W}\right\rangle=\left\langle\boldsymbol{W}, L_{h} v\right\rangle | | |
| lyche-numerical-linear-algebra_FO4078 | 315 | 1.000 | \boldsymbol{W}= | | |
| lyche-numerical-linear-algebra_FO4079 | 315 | 1.000 | c_{j+\frac{1}{2}, k} | | |
| lyche-numerical-linear-algebra_FO4080 | 315 | 1.000 | c_{j, k+\frac{1}{2}} | | |
| lyche-numerical-linear-algebra_FO4081 | 315 | 1.000 | \left\langle L_{h} v, \boldsymbol{V}\right\rangle \geq 0 | | |
| lyche-numerical-linear-algebra_FO4082 | 315 | 1.000 | \boldsymbol{V} \in \mathbb{R}^{m \times m} | | |
| lyche-numerical-linear-algebra_FO4083 | 315 | 1.000 | \left\langle L_{h} v, \boldsymbol{V}\right\rangle=0 | | |
| lyche-numerical-linear-algebra_FO4084 | 315 | 1.000 | v_{j, k+1}=v_{j, k} | | |
| lyche-numerical-linear-algebra_FO4085 | 315 | 1.000 | k=0,1, \ldots, m | | |
| lyche-numerical-linear-algebra_FO4086 | 315 | 1.000 | v_{j, 0}=v_{j, m+1}=0 | | |
| lyche-numerical-linear-algebra_FO4087 | 315 | 1.000 | \boldsymbol{V}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO4088 | 316 | 1.000 | b \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO4089 | 316 | 1.000 | x \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO4090 | 316 | 1.000 | m=\sqrt{n} | | |
| lyche-numerical-linear-algebra_FO4091 | 316 | 1.000 | c(x, y) \equiv 1 | | |
| lyche-numerical-linear-algebra_FO4092 | 316 | 1.000 | \boldsymbol{A}_{p} | | |
| lyche-numerical-linear-algebra_FO4093 | 316 | 1.000 | \boldsymbol{B}=\boldsymbol{A}_{p}^{-1} | | |
| lyche-numerical-linear-algebra_FO4094 | 316 | 0.928 | \boldsymbol{w}=\boldsymbol{B} \boldsymbol{t} | | |
| lyche-numerical-linear-algebra_FO4095 | 316 | 1.000 | \boldsymbol{A}_{p} \boldsymbol{w}_{k}=\boldsymbol{t}_{k} | | |
| lyche-numerical-linear-algebra_FO4096 | 316 | 1.000 | \boldsymbol{A}_{p} \boldsymbol{w}=\boldsymbol{t} | | |
| lyche-numerical-linear-algebra_FO4097 | 316 | 1.000 | \boldsymbol{x}_{0}=0 | | |
| lyche-numerical-linear-algebra_FO4098 | 316 | 1.000 | \epsilon=10^{-8} | | |
| lyche-numerical-linear-algebra_FO4099 | 316 | 1.000 | O\left(n \log _{2} n\right) | | |
| lyche-numerical-linear-algebra_FO4101 | 317 | 1.000 | 0<c_{0} \leq | | |
| lyche-numerical-linear-algebra_FO4102 | 317 | 0.973 | c(x, y) \leq c_{1} | | |
| lyche-numerical-linear-algebra_FO4103 | 317 | 0.973 | (x, y) \in[0,1]^{2} | | |
| lyche-numerical-linear-algebra_FO4104 | 317 | 0.973 | \boldsymbol{B} \boldsymbol{A}=\boldsymbol{A}_{p}^{-1} \boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO4105 | 317 | 1.000 | \boldsymbol{A}_{p}^{-1} \boldsymbol{A} \boldsymbol{x}=\lambda x | | |
| lyche-numerical-linear-algebra_FO4106 | 317 | 1.000 | \boldsymbol{x} \in \mathbb{R}^{n} \backslash\{0\} | | |
| lyche-numerical-linear-algebra_FO4107 | 317 | 1.000 | \boldsymbol{A} \boldsymbol{x}=\lambda \boldsymbol{A}_{p} x | | |
| lyche-numerical-linear-algebra_FO4108 | 317 | 1.000 | \boldsymbol{x}^{T} \boldsymbol{A} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO4109 | 317 | 1.000 | \boldsymbol{x}^{T} \boldsymbol{A}_{p} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO4110 | 317 | 0.997 | \boldsymbol{x}^{T} \boldsymbol{A}_{p} x>0 | | |
| lyche-numerical-linear-algebra_FO4111 | 317 | 0.997 | c_{0} | | |
| lyche-numerical-linear-algebra_FO4112 | 317 | 1.000 | c_{0} \leq \lambda \leq c_{1} | | |
| lyche-numerical-linear-algebra_FO4113 | 317 | 0.998 | c(x, y)=e^{-x+y} | | |
| lyche-numerical-linear-algebra_FO4114 | 317 | 0.998 | c_{0}=e^{-2} | | |
| lyche-numerical-linear-algebra_FO4115 | 317 | 0.998 | c_{1}=1 | | |
| lyche-numerical-linear-algebra_FO4116 | 317 | 0.998 | \kappa \leq e^{2} \approx | | |
| lyche-numerical-linear-algebra_FO4117 | 317 | 1.000 | \boldsymbol{A}=\boldsymbol{U} \boldsymbol{D} \boldsymbol{U}^{T} | | |
| lyche-numerical-linear-algebra_FO4118 | 318 | 0.966 | \boldsymbol{v}=\left[v_{1}, \ldots, v_{n}\right]^{T}:=\boldsymbol{U}^{T} \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO4119 | 318 | 0.966 | \boldsymbol{c}:=\boldsymbol{U}^{T} \boldsymbol{b}=\left[c_{1}, \ldots, c_{n}\right]^{T} | | |
| lyche-numerical-linear-algebra_FO4120 | 318 | 0.999 | \boldsymbol{x}_{k+1}=\boldsymbol{x}_{k}+\alpha_{k} \boldsymbol{r}_{k} | | |
| lyche-numerical-linear-algebra_FO4121 | 318 | 0.999 | \boldsymbol{r}_{k}=\boldsymbol{b}-\boldsymbol{A} \boldsymbol{x}_{k} | | |
| lyche-numerical-linear-algebra_FO4122 | 318 | 0.973 | \alpha_{k}=\frac{\boldsymbol{r}_{k}^{T} \boldsymbol{r}_{k}}{\boldsymbol{r}_{k}^{T} \boldsymbol{A r}_{k}} | | |
| lyche-numerical-linear-algebra_FO4123 | 318 | 0.927 | \boldsymbol{A}=\left[\begin{array}{cc}2 & -1 \\ -1 & 2\end{array}\right], \boldsymbol{b}=\left[\begin{array}{ll}1 & 1\end{array}\right]^{T} | | |
| lyche-numerical-linear-algebra_FO4124 | 318 | 1.000 | \boldsymbol{e}_{k}=\boldsymbol{x}_{k}-\boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO4125 | 318 | 0.907 | \boldsymbol{x}_{1}=\left(\frac{\boldsymbol{b}^{T} \boldsymbol{b}}{\boldsymbol{b}^{T} \boldsymbol{A} \boldsymbol{b}}\right) \boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO4126 | 318 | 0.990 | \boldsymbol{A} \boldsymbol{y}=\boldsymbol{r}_{0}:=\boldsymbol{b}-\boldsymbol{A} \boldsymbol{x}_{0} | | |
| lyche-numerical-linear-algebra_FO4127 | 318 | 0.990 | \boldsymbol{y}_{0}=\mathbf{0} .^{3} | | |
| lyche-numerical-linear-algebra_FO4128 | 318 | 1.000 | \boldsymbol{p}_{k}=\boldsymbol{r}_{k}+\sum_{j=0}^{k-1} a_{k, j} \boldsymbol{r}_{j} | | |
| lyche-numerical-linear-algebra_FO4129 | 318 | 0.881 | \boldsymbol{A} \boldsymbol{y}=\boldsymbol{r}_{0} | | |
| lyche-numerical-linear-algebra_FO4130 | 318 | 0.881 | \boldsymbol{y}_{k+1}:=\boldsymbol{y}_{k}+\gamma_{k} \boldsymbol{q}_{k}, \gamma_{k}:= | | |
| lyche-numerical-linear-algebra_FO4131 | 318 | 0.888 | \frac{\boldsymbol{s}_{k}^{T} \boldsymbol{s}_{k}}{\boldsymbol{q}_{k}^{T} \boldsymbol{A} \boldsymbol{q}_{k}}, \boldsymbol{s}_{k+1}:=\boldsymbol{s}_{k}-\gamma_{k} \boldsymbol{A} \boldsymbol{q}_{k}, \boldsymbol{q}_{k+1}:=\boldsymbol{s}_{k+1}+\delta_{k} \boldsymbol{q}_{k}, \delta_{k}:=\frac{\boldsymbol{s}_{k+1}^{T} \boldsymbol{s}_{k+1}}{\boldsymbol{s}_{k}^{T} \boldsymbol{s}_{k}} | | |
| lyche-numerical-linear-algebra_FO4132 | 318 | 0.888 | \boldsymbol{y}_{k}=\boldsymbol{x}_{k}-\boldsymbol{x}_{0} | | |
| lyche-numerical-linear-algebra_FO4133 | 318 | 0.977 | \boldsymbol{s}_{k}=\boldsymbol{r}_{k} | | |
| lyche-numerical-linear-algebra_FO4134 | 318 | 0.977 | \boldsymbol{q}_{k}=\boldsymbol{p}_{k} | | |
| lyche-numerical-linear-algebra_FO4135 | 319 | 1.000 | n=100,400,1600,2500 | | |
| lyche-numerical-linear-algebra_FO4136 | 319 | 1.000 | K / n | | |
| lyche-numerical-linear-algebra_FO4137 | 319 | 1.000 | \boldsymbol{A} \in \mathbb{R}^{m, n}, \boldsymbol{b} \in \mathbb{R}^{m} | | |
| lyche-numerical-linear-algebra_FO4138 | 319 | 0.996 | \mathrm{r}=\mathrm{A}^{\prime}\left(\mathrm{b}-\mathrm{A}^{*} \mathrm{x}\right) ; \mathrm{p}=\mathrm{r} | | |
| lyche-numerical-linear-algebra_FO4139 | 319 | 0.926 | \mathrm{r}=\mathrm{r}-\mathrm{a}^{*} \mathrm{~A}^{*} \mathrm{t} | | |
| lyche-numerical-linear-algebra_FO4140 | 319 | 1.000 | \operatorname{cond}_{2}(\boldsymbol{A})^{2} | | |
| lyche-numerical-linear-algebra_FO4141 | 319 | 1.000 | \left\{\boldsymbol{v}_{i}\right\}_{i=1}^{k} | | |
| lyche-numerical-linear-algebra_FO4142 | 319 | 1.000 | n_{i j}=\left\langle\boldsymbol{v}_{i}, \boldsymbol{v}_{j}\right\rangle | | |
| lyche-numerical-linear-algebra_FO4143 | 319 | 0.997 | \mathbb{W} \subset \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO4144 | 319 | 1.000 | \hat{\boldsymbol{x}} \in \mathbb{W} | | |
| lyche-numerical-linear-algebra_FO4145 | 319 | 1.000 | \hat{\boldsymbol{x}} | | |
| lyche-numerical-linear-algebra_FO4146 | 319 | 1.000 | \mathbb{W} | | |
| lyche-numerical-linear-algebra_FO4147 | 320 | 1.000 | \boldsymbol{x}_{k} \in \mathbb{W}_{k} | | |
| lyche-numerical-linear-algebra_FO4148 | 320 | 1.000 | \boldsymbol{p}_{k} \in \mathbb{W}_{k+1} | | |
| lyche-numerical-linear-algebra_FO4149 | 320 | 1.000 | \left\|\boldsymbol{A} \boldsymbol{p}_{k}\right\|_{2}= | | |
| lyche-numerical-linear-algebra_FO4150 | 320 | 1.000 | \left\|\boldsymbol{p}_{k}\right\|_{\boldsymbol{A}}=1 | | |
| lyche-numerical-linear-algebra_FO4151 | 320 | 1.000 | \alpha_{k} \in \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO4152 | 320 | 1.000 | \boldsymbol{p}_{k-2}, \boldsymbol{p}_{k-1} | | |
| lyche-numerical-linear-algebra_FO4153 | 320 | 1.000 | k \leq 3 | | |
| lyche-numerical-linear-algebra_FO4154 | 320 | 1.000 | \left[\boldsymbol{b}, \boldsymbol{A} \boldsymbol{b}, \boldsymbol{A}^{2} \boldsymbol{b}\right]=\left[\begin{array}{rrr}4 & 8 & 20 \\ 0 & -4 & -16 \\ 0 & 0 & 4\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO4155 | 321 | 0.983 | \operatorname{dim}\left(\mathbb{W}_{k}\right)=k | | |
| lyche-numerical-linear-algebra_FO4156 | 321 | 0.983 | k=0,1,2,3 | | |
| lyche-numerical-linear-algebra_FO4157 | 321 | 0.987 | \boldsymbol{x}_{3} | | |
| lyche-numerical-linear-algebra_FO4158 | 321 | 1.000 | \boldsymbol{r}_{0}, \ldots, \boldsymbol{r}_{k-1} | | |
| lyche-numerical-linear-algebra_FO4159 | 321 | 1.000 | k=1,2,3 | | |
| lyche-numerical-linear-algebra_FO4160 | 321 | 1.000 | \boldsymbol{p}_{0}, \ldots, \boldsymbol{p}_{k-1} | | |
| lyche-numerical-linear-algebra_FO4161 | 321 | 1.000 | \left\{\left\|\boldsymbol{r}_{k}\right\|\right. | | |
| lyche-numerical-linear-algebra_FO4162 | 321 | 0.991 | \left\{\left\|\boldsymbol{x}_{k}-\boldsymbol{x}\right\|\right. | | |
| lyche-numerical-linear-algebra_FO4163 | 321 | 1.000 | \boldsymbol{B}^{T}=-\boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO4164 | 321 | 1.000 | \boldsymbol{A}:=\boldsymbol{I}-\boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO4165 | 321 | 0.998 | \|\boldsymbol{A} \boldsymbol{x}\|_{2}^{2}=\|\boldsymbol{x}\|_{2}^{2}+\|\boldsymbol{B} \boldsymbol{x}\|_{2}^{2} | | |
| lyche-numerical-linear-algebra_FO4166 | 321 | 0.998 | \|\boldsymbol{A}\|_{2}=\sqrt{1+\|\boldsymbol{B}\|_{2}^{2}} | | |
| lyche-numerical-linear-algebra_FO4167 | 321 | 1.000 | \|\boldsymbol{A}\|_{2} \leq 1 | | |
| lyche-numerical-linear-algebra_FO4168 | 321 | 1.000 | 1 \leq k \leq n, \mathcal{W}=\operatorname{span}\left(\boldsymbol{w}_{1}, \ldots, \boldsymbol{w}_{k}\right) | | |
| lyche-numerical-linear-algebra_FO4169 | 321 | 1.000 | \boldsymbol{x} \in \mathcal{W} | | |
| lyche-numerical-linear-algebra_FO4170 | 321 | 0.835 | \|\boldsymbol{x}\|_{2} \leq\|\boldsymbol{b}\|_{2} | | |
| lyche-numerical-linear-algebra_FO4171 | 322 | 1.000 | \boldsymbol{x}:=\sum_{j=1}^{k} x_{j} \boldsymbol{w}_{j} | | |
| lyche-numerical-linear-algebra_FO4172 | 322 | 1.000 | x_{1}, \ldots, x_{k} | | |
| lyche-numerical-linear-algebra_FO4173 | 322 | 0.996 | \boldsymbol{x}^{*}:=\boldsymbol{A}^{-1} \boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO4174 | 322 | 1.000 | \boldsymbol{x}_{k} \in \mathcal{W}_{k} | | |
| lyche-numerical-linear-algebra_FO4175 | 322 | 0.986 | \boldsymbol{x} * | | |
| lyche-numerical-linear-algebra_FO4176 | 322 | 1.000 | k=0, \ldots, n | | |
| lyche-numerical-linear-algebra_FO4177 | 322 | 1.000 | \omega_{0}, \omega_{1}, \ldots, \omega_{m} | | |
| lyche-numerical-linear-algebra_FO4178 | 322 | 1.000 | k=1,2, \ldots, m | | |
| lyche-numerical-linear-algebra_FO4179 | 323 | 1.000 | k=0,1, \ldots, m-1 | | |
| lyche-numerical-linear-algebra_FO4180 | 323 | 1.000 | \boldsymbol{x}_{0}=\boldsymbol{x}_{-1}=0 | | |
| lyche-numerical-linear-algebra_FO4181 | 323 | 1.000 | \boldsymbol{r}_{0}=\boldsymbol{r}_{-1}=\boldsymbol{b} | | |
| lyche-numerical-linear-algebra_FO4182 | 323 | 1.000 | 0<\omega_{k}<1 | | |
| lyche-numerical-linear-algebra_FO4183 | 323 | 1.000 | \left\langle\boldsymbol{r}_{k}, \boldsymbol{r}_{j}\right\rangle=0 | | |
| lyche-numerical-linear-algebra_FO4184 | 323 | 1.000 | j=0,1, \ldots, k-1 | | |
| lyche-numerical-linear-algebra_FO4185 | 323 | 1.000 | k \leq m+1 | | |
| lyche-numerical-linear-algebra_FO4186 | 323 | 1.000 | \mathcal{W}_{k}=\operatorname{span}\left(\boldsymbol{r}_{0}, \boldsymbol{r}_{1}, \ldots, \boldsymbol{r}_{k-1}\right) | | |
| lyche-numerical-linear-algebra_FO4187 | 323 | 1.000 | \operatorname{dim} \mathcal{W}_{k}=k | | |
| lyche-numerical-linear-algebra_FO4188 | 323 | 1.000 | 1 \leq k \leq m-1 | | |
| lyche-numerical-linear-algebra_FO4189 | 323 | 0.605 | \alpha_{k}:=\left\langle\boldsymbol{B r}_{k}, \boldsymbol{r}_{k+1}\right\rangle / \rho_{k+1} | | |
| lyche-numerical-linear-algebra_FO4190 | 323 | 0.605 | \beta_{k}:=\left\langle\boldsymbol{B r}_{k}, \boldsymbol{r}_{k-1}\right\rangle / \rho_{k-1} | | |
| lyche-numerical-linear-algebra_FO4191 | 323 | 0.865 | \alpha_{0}:=\left\langle\boldsymbol{B r}_{0}, \boldsymbol{r}_{1}\right\rangle / \rho_{1} | | |
| lyche-numerical-linear-algebra_FO4192 | 323 | 0.865 | \alpha_{0}=1 | | |
| lyche-numerical-linear-algebra_FO4193 | 323 | 1.000 | \beta_{k}=-\alpha_{k-1} \rho_{k} / \rho_{k-1} | | |
| lyche-numerical-linear-algebra_FO4194 | 323 | 1.000 | \alpha_{k}+\beta_{k}=1 | | |
| lyche-numerical-linear-algebra_FO4195 | 323 | 1.000 | k=1,2, \ldots, m-1 | | |
| lyche-numerical-linear-algebra_FO4196 | 323 | 1.000 | \alpha_{k} \geq 1 | | |
| lyche-numerical-linear-algebra_FO4197 | 323 | 1.000 | \boldsymbol{x}_{k}, \boldsymbol{r}_{k} | | |
| lyche-numerical-linear-algebra_FO4198 | 323 | 1.000 | \boldsymbol{\omega}_{k} | | |
| lyche-numerical-linear-algebra_FO4199 | 323 | 0.887 | \operatorname{arccosh} | | |
| lyche-numerical-linear-algebra_FO4200 | 323 | 0.887 | \cosh x:=\left(e^{x}+e^{-x}\right) / 2 | | |
| lyche-numerical-linear-algebra_FO4201 | 323 | 1.000 | \boldsymbol{A}^{-1}\left(\boldsymbol{r}_{k+1}-\boldsymbol{r}_{j}\right) \in \mathcal{W}_{k+1} | | |
| lyche-numerical-linear-algebra_FO4202 | 324 | 1.000 | f:[a, b] \rightarrow \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO4203 | 324 | 1.000 | \max _{a \leq x \leq b} f(x) \leq \max \{f(a), f(b)\} | | |
| lyche-numerical-linear-algebra_FO4204 | 326 | 1.000 | \boldsymbol{A} \boldsymbol{e}_{i}=a_{i i} \boldsymbol{e}_{i} | | |
| lyche-numerical-linear-algebra_FO4205 | 326 | 0.965 | \left(\lambda_{i}, \boldsymbol{e}_{i}\right) | | |
| lyche-numerical-linear-algebra_FO4206 | 326 | 0.965 | \lambda_{i}=a_{i i} | | |
| lyche-numerical-linear-algebra_FO4207 | 326 | 1.000 | \operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})=\prod_{i=1}^{n}\left(a_{i i}-\lambda\right) | | |
| lyche-numerical-linear-algebra_FO4208 | 326 | 1.000 | \boldsymbol{A}_{i} \boldsymbol{X}_{i}=\boldsymbol{X}_{i} \boldsymbol{D}_{i} | | |
| lyche-numerical-linear-algebra_FO4209 | 326 | 1.000 | \boldsymbol{A}_{i} | | |
| lyche-numerical-linear-algebra_FO4210 | 326 | 1.000 | \boldsymbol{X}:=\operatorname{diag}\left(\boldsymbol{X}_{1}, \ldots, \boldsymbol{X}_{r}\right) | | |
| lyche-numerical-linear-algebra_FO4211 | 326 | 1.000 | \boldsymbol{D}:=\operatorname{diag}\left(\boldsymbol{D}_{1}, \ldots, \boldsymbol{D}_{r}\right) | | |
| lyche-numerical-linear-algebra_FO4212 | 327 | 0.988 | \boldsymbol{A}_{11}, \boldsymbol{A}_{22}, \ldots, \boldsymbol{A}_{r r} | | |
| lyche-numerical-linear-algebra_FO4213 | 327 | 1.000 | \pi_{\boldsymbol{A}}(\lambda) | | |
| lyche-numerical-linear-algebra_FO4214 | 327 | 0.405 | \left.\pi_{\boldsymbol{A}}(\lambda):\right)=\lambda^{16} | | |
| lyche-numerical-linear-algebra_FO4215 | 327 | 0.405 | q(\lambda)=\lambda^{16}-10^{-16} | | |
| lyche-numerical-linear-algebra_FO4216 | 327 | 0.999 | \lambda_{j}=10^{-1} e^{2 \pi i j / 16} | | |
| lyche-numerical-linear-algebra_FO4217 | 327 | 1.000 | j=1, \ldots, 16 | | |
| lyche-numerical-linear-algebra_FO4218 | 327 | 1.000 | 10^{-16} | | |
| lyche-numerical-linear-algebra_FO4219 | 328 | 1.000 | R \cap C | | |
| lyche-numerical-linear-algebra_FO4220 | 328 | 1.000 | R=R_{1} \cup R_{2} \cup \cdots \cup R_{n} | | |
| lyche-numerical-linear-algebra_FO4221 | 328 | 1.000 | C=C_{1} \cup C_{2} \cup \cdots \cup C_{n} | | |
| lyche-numerical-linear-algebra_FO4222 | 328 | 0.621 | \lambda \in R_{i} | | |
| lyche-numerical-linear-algebra_FO4223 | 328 | 0.997 | \left|x_{i}\right|=\|\boldsymbol{x}\|_{\infty} | | |
| lyche-numerical-linear-algebra_FO4224 | 328 | 0.997 | \sum_{j} a_{i j} x_{j}=\lambda x_{i} | | |
| lyche-numerical-linear-algebra_FO4225 | 328 | 0.997 | \left(\lambda-a_{i i}\right) x_{i}= | | |
| lyche-numerical-linear-algebra_FO4226 | 328 | 1.000 | \sum_{j \neq i} a_{i j} x_{j} | | |
| lyche-numerical-linear-algebra_FO4227 | 328 | 1.000 | \left|x_{j} / x_{i}\right| \leq 1 | | |
| lyche-numerical-linear-algebra_FO4228 | 328 | 1.000 | C_{j} | | |
| lyche-numerical-linear-algebra_FO4229 | 328 | 1.000 | \lambda \in C_{j} | | |
| lyche-numerical-linear-algebra_FO4230 | 328 | 1.000 | r_{i} | | |
| lyche-numerical-linear-algebra_FO4231 | 328 | 1.000 | R_{1}, \ldots, R_{n} | | |
| lyche-numerical-linear-algebra_FO4232 | 328 | 1.000 | C_{1}, \ldots, C_{n} | | |
| lyche-numerical-linear-algebra_FO4233 | 328 | 1.000 | R_{i}=C_{i} | | |
| lyche-numerical-linear-algebra_FO4234 | 328 | 1.000 | R_{1}=R_{m}=\{z \in \mathbb{R}:|z-2| \leq 1\} | | |
| lyche-numerical-linear-algebra_FO4235 | 328 | 1.000 | \lambda \in[0,4] | | |
| lyche-numerical-linear-algebra_FO4236 | 329 | 1.000 | 4\left[\sin \frac{\pi}{2(m+1)}\right]^{2} | | |
| lyche-numerical-linear-algebra_FO4237 | 329 | 0.969 | 4\left[\sin \frac{m \pi}{2(m+1)}\right]^{2} | | |
| lyche-numerical-linear-algebra_FO4238 | 329 | 1.000 | \boldsymbol{A}(0)=\boldsymbol{D} | | |
| lyche-numerical-linear-algebra_FO4239 | 329 | 1.000 | \boldsymbol{A}(1)=\boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO4240 | 329 | 1.000 | \boldsymbol{A}(t) | | |
| lyche-numerical-linear-algebra_FO4241 | 329 | 1.000 | R_{i}(t) | | |
| lyche-numerical-linear-algebra_FO4242 | 329 | 1.000 | 0 \leq t_{1}<t_{2} \leq 1 | | |
| lyche-numerical-linear-algebra_FO4243 | 329 | 1.000 | R_{i}\left(t_{1}\right) \subset R_{i}\left(t_{2}\right) | | |
| lyche-numerical-linear-algebra_FO4244 | 329 | 1.000 | R_{i}(1) | | |
| lyche-numerical-linear-algebra_FO4245 | 329 | 0.992 | \bigcup_{k=1}^{p} R_{i_{k}}(1) | | |
| lyche-numerical-linear-algebra_FO4246 | 329 | 0.992 | R^{p}(t):=\bigcup_{k=1}^{p} R_{i_{k}}(t) | | |
| lyche-numerical-linear-algebra_FO4247 | 329 | 0.992 | t \in[0,1] | | |
| lyche-numerical-linear-algebra_FO4248 | 329 | 0.992 | R^{p}(0) | | |
| lyche-numerical-linear-algebra_FO4249 | 329 | 1.000 | a_{i_{1}, i_{1}}, \ldots, a_{i_{p}, i_{p}} | | |
| lyche-numerical-linear-algebra_FO4250 | 329 | 1.000 | R^{p}(t) | | |
| lyche-numerical-linear-algebra_FO4251 | 329 | 1.000 | R^{p}(1) | | |
| lyche-numerical-linear-algebra_FO4252 | 329 | 1.000 | \boldsymbol{A}=\left[\begin{array}{ccc}1 & \epsilon_{1} & \epsilon_{2} \\ \epsilon_{3} & 2 & \epsilon_{4} \\ \epsilon_{5} & \epsilon_{6} & 3\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO4253 | 329 | 1.000 | \left|\epsilon_{i}\right| \leq 10^{-15} | | |
| lyche-numerical-linear-algebra_FO4254 | 329 | 1.000 | \lambda_{1}, \lambda_{2}, \lambda_{3} | | |
| lyche-numerical-linear-algebra_FO4255 | 329 | 1.000 | \left|\lambda_{j}-j\right| \leq | | |
| lyche-numerical-linear-algebra_FO4256 | 329 | 1.000 | 2 \times 10^{-15} | | |
| lyche-numerical-linear-algebra_FO4257 | 329 | 1.000 | j=1,2,3 | | |
| lyche-numerical-linear-algebra_FO4258 | 329 | 0.979 | \boldsymbol{A}+ | | |
| lyche-numerical-linear-algebra_FO4259 | 329 | 1.000 | \boldsymbol{A}_{0}:=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO4260 | 329 | 0.998 | \lambda \in \sigma\left(\boldsymbol{A}_{0}+\boldsymbol{E}\right)=\sigma(\boldsymbol{E}) | | |
| lyche-numerical-linear-algebra_FO4261 | 329 | 0.998 | |\lambda| \leq\|\boldsymbol{E}\|_{\infty} | | |
| lyche-numerical-linear-algebra_FO4262 | 330 | 0.998 | \boldsymbol{A}_{0} | | |
| lyche-numerical-linear-algebra_FO4263 | 330 | 0.998 | \|\boldsymbol{E}\|_{\infty} | | |
| lyche-numerical-linear-algebra_FO4264 | 330 | 1.000 | \boldsymbol{A}_{1}+\boldsymbol{E} | | |
| lyche-numerical-linear-algebra_FO4265 | 330 | 1.000 | \pi(\lambda):=(-1)^{n}\left(\lambda^{n}-\epsilon\right) | | |
| lyche-numerical-linear-algebra_FO4266 | 330 | 1.000 | |\lambda|=\|\boldsymbol{E}\|_{\infty}^{1 / n} | | |
| lyche-numerical-linear-algebra_FO4267 | 330 | 1.000 | n=16 | | |
| lyche-numerical-linear-algebra_FO4268 | 330 | 0.978 | \epsilon=10^{-16} | | |
| lyche-numerical-linear-algebra_FO4269 | 330 | 1.000 | \|\boldsymbol{E}\|_{\infty}^{1 / n} | | |
| lyche-numerical-linear-algebra_FO4270 | 330 | 1.000 | \mu \in | | |
| lyche-numerical-linear-algebra_FO4271 | 330 | 1.000 | \sigma(\boldsymbol{A}+\boldsymbol{E}) | | |
| lyche-numerical-linear-algebra_FO4272 | 330 | 1.000 | \left\|\boldsymbol{u}_{1}\right\|_{2}=1 | | |
| lyche-numerical-linear-algebra_FO4273 | 330 | 1.000 | \|\boldsymbol{E}\|_{2}^{1 / n} | | |
| lyche-numerical-linear-algebra_FO4274 | 330 | 1.000 | \left[\begin{array}{cc}1 & 1 \\ \epsilon & 1\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO4275 | 330 | 1.000 | 1 \pm \sqrt{\epsilon} | | |
| lyche-numerical-linear-algebra_FO4276 | 330 | 1.000 | \epsilon=0 | | |
| lyche-numerical-linear-algebra_FO4277 | 331 | 1.000 | 1 / n | | |
| lyche-numerical-linear-algebra_FO4278 | 331 | 1.000 | \|\boldsymbol{E}\|_{2} | | |
| lyche-numerical-linear-algebra_FO4279 | 331 | 1.000 | \mu \in \mathbb{C} | | |
| lyche-numerical-linear-algebra_FO4280 | 331 | 0.999 | \boldsymbol{r}:=\boldsymbol{A} \boldsymbol{x}-\mu \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO4281 | 331 | 0.999 | K_{p}(\boldsymbol{X}):=\|\boldsymbol{X}\|_{p}\left\|\boldsymbol{X}^{-1}\right\|_{p} | | |
| lyche-numerical-linear-algebra_FO4282 | 331 | 0.902 | (\mu, \boldsymbol{x}) | | |
| lyche-numerical-linear-algebra_FO4283 | 331 | 1.000 | \mu \in \sigma(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO4284 | 331 | 1.000 | \lambda=\mu | | |
| lyche-numerical-linear-algebra_FO4285 | 331 | 1.000 | \mu \notin \sigma(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO4286 | 331 | 1.000 | \boldsymbol{X} \boldsymbol{D} \boldsymbol{X}^{-1} | | |
| lyche-numerical-linear-algebra_FO4287 | 331 | 1.000 | \left(\lambda_{j}, \boldsymbol{x}_{j}\right) | | |
| lyche-numerical-linear-algebra_FO4288 | 331 | 1.000 | \boldsymbol{D}_{1}:=\boldsymbol{D}-\mu \boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO4289 | 331 | 1.000 | \boldsymbol{D}_{1}^{-1}=\operatorname{diag}\left(\left(\lambda_{1}-\mu\right)^{-1}, \ldots,\left(\lambda_{n}-\mu\right)^{-1}\right) | | |
| lyche-numerical-linear-algebra_FO4290 | 331 | 0.658 | (\boldsymbol{A}+\boldsymbol{E}) \boldsymbol{x}=\mu \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO4291 | 331 | 0.658 | \mathbf{0}=\boldsymbol{A} \boldsymbol{x}-\mu \boldsymbol{x}+\boldsymbol{E} \boldsymbol{x}=\boldsymbol{r}+\boldsymbol{E} \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO4292 | 331 | 0.954 | \|\boldsymbol{r}\|_{p}=\|-\boldsymbol{E} \boldsymbol{x}\|_{p} \leq\|\boldsymbol{E}\|_{p} | | |
| lyche-numerical-linear-algebra_FO4293 | 331 | 1.000 | K_{p}(\boldsymbol{X}) | | |
| lyche-numerical-linear-algebra_FO4294 | 332 | 0.593 | K_{2}(\boldsymbol{X})=1 | | |
| lyche-numerical-linear-algebra_FO4295 | 332 | 1.000 | |\lambda-\mu| \leq\|\boldsymbol{E}\|_{2} | | |
| lyche-numerical-linear-algebra_FO4296 | 332 | 0.995 | \boldsymbol{A}=\operatorname{diag}\left(\lambda_{1}, \ldots, \lambda_{n}\right) | | |
| lyche-numerical-linear-algebra_FO4297 | 332 | 1.000 | \|\boldsymbol{A}\|_{p}=\rho(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO4298 | 332 | 1.000 | p<\infty | | |
| lyche-numerical-linear-algebra_FO4299 | 332 | 0.874 | \|\boldsymbol{A}\|_{p}=\max _{\boldsymbol{x} \neq \mathbf{0}} \frac{\|\boldsymbol{A}\|_{p}}{\|\boldsymbol{x}\|_{p}} \leq \rho(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO4300 | 332 | 0.784 | \mu, \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO4301 | 333 | 1.000 | \boldsymbol{B}:=\mu \boldsymbol{A}^{-1} | | |
| lyche-numerical-linear-algebra_FO4302 | 333 | 0.858 | \boldsymbol{F}:=-\boldsymbol{A}^{-1} \boldsymbol{E} | | |
| lyche-numerical-linear-algebra_FO4303 | 333 | 0.858 | \left(\lambda_{j}, \boldsymbol{x}\right) | | |
| lyche-numerical-linear-algebra_FO4304 | 333 | 0.858 | \left(\frac{\mu}{\lambda_{j}}, \boldsymbol{x}\right) | | |
| lyche-numerical-linear-algebra_FO4305 | 333 | 0.999 | (1, \boldsymbol{x}) | | |
| lyche-numerical-linear-algebra_FO4306 | 333 | 0.999 | \boldsymbol{B}+\boldsymbol{F} | | |
| lyche-numerical-linear-algebra_FO4307 | 333 | 1.000 | \left|\frac{\mu}{\lambda}-1\right| \leq K_{p}(\boldsymbol{X})\|\boldsymbol{F}\|_{p}=K_{p}(\boldsymbol{X})\left\|\boldsymbol{A}^{-1} \boldsymbol{E}\right\|_{p} | | |
| lyche-numerical-linear-algebra_FO4308 | 333 | 1.000 | 1,2, \ldots, i-2, i=3,4, \ldots, n | | |
| lyche-numerical-linear-algebra_FO4309 | 333 | 1.000 | \boldsymbol{A}_{1}=\boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO4310 | 333 | 1.000 | \boldsymbol{A}_{k+1}=\boldsymbol{H}_{k} \boldsymbol{A}_{k} \boldsymbol{H}_{k} | | |
| lyche-numerical-linear-algebra_FO4311 | 333 | 1.000 | k=1,2, \ldots, n-2 | | |
| lyche-numerical-linear-algebra_FO4312 | 333 | 1.000 | \boldsymbol{H}_{k} | | |
| lyche-numerical-linear-algebra_FO4313 | 333 | 1.000 | \boldsymbol{A}_{n-1} | | |
| lyche-numerical-linear-algebra_FO4314 | 333 | 1.000 | \boldsymbol{A}_{k}^{*}=\boldsymbol{A}_{k} | | |
| lyche-numerical-linear-algebra_FO4315 | 334 | 1.000 | \boldsymbol{B}_{k} \in \mathbb{C}^{k, k} | | |
| lyche-numerical-linear-algebra_FO4316 | 334 | 1.000 | \boldsymbol{D}_{k} \in | | |
| lyche-numerical-linear-algebra_FO4317 | 334 | 0.999 | \mathbb{C}^{n-k, k} | | |
| lyche-numerical-linear-algebra_FO4318 | 334 | 0.999 | \boldsymbol{D}_{k}=\left[\mathbf{0}, \mathbf{0}, \ldots, \mathbf{0}, \boldsymbol{d}_{k}\right] | | |
| lyche-numerical-linear-algebra_FO4319 | 334 | 0.999 | \boldsymbol{V}_{k}=\boldsymbol{I}-\boldsymbol{v}_{k} \boldsymbol{v}_{k}^{*} \in \mathbb{C}^{n-k, n-k} | | |
| lyche-numerical-linear-algebra_FO4320 | 334 | 0.999 | \boldsymbol{V}_{k} \boldsymbol{d}_{k}=\alpha_{k} \boldsymbol{e}_{1} | | |
| lyche-numerical-linear-algebra_FO4321 | 334 | 1.000 | \boldsymbol{V}_{k} \boldsymbol{D}_{k}=\left[\boldsymbol{V}_{k} \mathbf{0}, \ldots, \boldsymbol{V}_{k} \mathbf{0}, \boldsymbol{V}_{k} \boldsymbol{d}_{k}\right]=\left(\mathbf{0}, \ldots, \mathbf{0}, \alpha_{k} \boldsymbol{e}_{1}\right) | | |
| lyche-numerical-linear-algebra_FO4322 | 334 | 1.000 | (k+1) \times(k+1) | | |
| lyche-numerical-linear-algebra_FO4323 | 334 | 1.000 | \boldsymbol{v}_{k} | | |
| lyche-numerical-linear-algebra_FO4324 | 334 | 1.000 | \boldsymbol{L}(k+1: n, k)=\boldsymbol{v}_{k} | | |
| lyche-numerical-linear-algebra_FO4325 | 334 | 0.994 | \boldsymbol{B} . \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO4326 | 334 | 1.000 | \boldsymbol{B}=\boldsymbol{Q}^{*} \boldsymbol{A} \boldsymbol{Q} | | |
| lyche-numerical-linear-algebra_FO4327 | 335 | 1.000 | \boldsymbol{Q}^{*} \boldsymbol{A} \boldsymbol{Q} | | |
| lyche-numerical-linear-algebra_FO4328 | 335 | 0.950 | \boldsymbol{Q}=\boldsymbol{H}_{1} \boldsymbol{H}_{2} \cdots \boldsymbol{H}_{n-2} | | |
| lyche-numerical-linear-algebra_FO4329 | 335 | 0.950 | \boldsymbol{H}_{k}=\left[\begin{array}{cc}\boldsymbol{I} & \mathbf{0} \\ \mathbf{0} & \boldsymbol{I}-\boldsymbol{v}_{k} \boldsymbol{v}_{k}^{T}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO4330 | 335 | 0.950 | \boldsymbol{v}_{k} \in \mathbb{R}^{n-k} | | |
| lyche-numerical-linear-algebra_FO4331 | 335 | 0.950 | \boldsymbol{v}_{1} \in \mathbb{R}^{n-1} | | |
| lyche-numerical-linear-algebra_FO4332 | 335 | 1.000 | \boldsymbol{v}_{n-2} \in \mathbb{R}^{2} | | |
| lyche-numerical-linear-algebra_FO4333 | 335 | 1.000 | \boldsymbol{Q}_{k+1} | | |
| lyche-numerical-linear-algebra_FO4334 | 335 | 1.000 | \left[\begin{array}{cc}\boldsymbol{I}_{k} & \mathbf{0} \\ \mathbf{0} & \boldsymbol{U}_{k}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO4335 | 335 | 1.000 | \boldsymbol{U}_{k} \in \mathbb{R}^{n-k, n-k} | | |
| lyche-numerical-linear-algebra_FO4336 | 335 | 1.000 | \lambda_{1} \geq \lambda_{2} \geq \cdots \geq \lambda_{n} | | |
| lyche-numerical-linear-algebra_FO4337 | 335 | 1.000 | 1 \leq m \leq n | | |
| lyche-numerical-linear-algebra_FO4338 | 336 | 1.000 | c_{i}=0 | | |
| lyche-numerical-linear-algebra_FO4339 | 336 | 1.000 | \boldsymbol{A}=\left[\begin{array}{cc}\boldsymbol{A}_{1} & \mathbf{0} \\ \mathbf{0} & \boldsymbol{A}_{2}\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO4340 | 336 | 1.000 | c_{i} \neq 0 | | |
| lyche-numerical-linear-algebra_FO4341 | 336 | 1.000 | x \in \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO4342 | 336 | 1.000 | p_{k}(x):=\operatorname{det}\left(x \boldsymbol{I}_{k}-\boldsymbol{A}_{k}\right) | | |
| lyche-numerical-linear-algebra_FO4343 | 336 | 1.000 | p_{n} | | |
| lyche-numerical-linear-algebra_FO4344 | 336 | 1.000 | k \geq 2 | | |
| lyche-numerical-linear-algebra_FO4345 | 336 | 1.000 | p_{k+1}(x) | | |
| lyche-numerical-linear-algebra_FO4346 | 336 | 0.990 | p_{1}(x)=x-d_{1} | | |
| lyche-numerical-linear-algebra_FO4347 | 336 | 0.990 | p_{2}(x)=\left(x-d_{2}\right)\left(x-d_{1}\right)-c_{1}^{2} | | |
| lyche-numerical-linear-algebra_FO4348 | 336 | 0.990 | k=0,1 | | |
| lyche-numerical-linear-algebra_FO4349 | 336 | 1.000 | p_{-1}(x)=0 | | |
| lyche-numerical-linear-algebra_FO4350 | 336 | 1.000 | p_{0}(x)=1 | | |
| lyche-numerical-linear-algebra_FO4351 | 336 | 1.000 | y_{1} \in\left(-M, d_{1}\right) | | |
| lyche-numerical-linear-algebra_FO4352 | 336 | 1.000 | y_{2} \in\left(d_{1}, M\right) | | |
| lyche-numerical-linear-algebra_FO4353 | 336 | 1.000 | p_{2}\left(y_{1}\right)=p_{2}\left(y_{2}\right)=0 | | |
| lyche-numerical-linear-algebra_FO4354 | 336 | 1.000 | y_{1} | | |
| lyche-numerical-linear-algebra_FO4355 | 336 | 1.000 | y_{2} | | |
| lyche-numerical-linear-algebra_FO4356 | 337 | 1.000 | y_{1}<d_{1}<y_{2} | | |
| lyche-numerical-linear-algebra_FO4357 | 337 | 1.000 | x_{1}, x_{2}, x_{3} | | |
| lyche-numerical-linear-algebra_FO4358 | 337 | 1.000 | p_{3} | | |
| lyche-numerical-linear-algebra_FO4359 | 337 | 1.000 | z_{1}, \ldots, z_{k-1} | | |
| lyche-numerical-linear-algebra_FO4360 | 337 | 1.000 | p_{k-1} | | |
| lyche-numerical-linear-algebra_FO4361 | 337 | 1.000 | y_{1}, \ldots, y_{k} | | |
| lyche-numerical-linear-algebra_FO4362 | 337 | 1.000 | p_{k} | | |
| lyche-numerical-linear-algebra_FO4363 | 337 | 1.000 | y_{0}:=-M<y_{1}, y_{k+1}:=M>y_{k} | | |
| lyche-numerical-linear-algebra_FO4364 | 337 | 1.000 | j=0, k+1 | | |
| lyche-numerical-linear-algebra_FO4365 | 337 | 0.832 | x_{1}, \ldots, x_{k+1} | | |
| lyche-numerical-linear-algebra_FO4366 | 337 | 1.000 | \boldsymbol{A}=\boldsymbol{E}^{*} \boldsymbol{B} \boldsymbol{E} | | |
| lyche-numerical-linear-algebra_FO4367 | 337 | 1.000 | \boldsymbol{D}, \boldsymbol{U}^{*} \boldsymbol{A} \boldsymbol{U}=\boldsymbol{D} | | |
| lyche-numerical-linear-algebra_FO4368 | 337 | 1.000 | \pi(\boldsymbol{A}), \zeta(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO4369 | 337 | 1.000 | v(\boldsymbol{A}) | | |
| lyche-numerical-linear-algebra_FO4370 | 337 | 1.000 | \pi(\boldsymbol{A})+\zeta(\boldsymbol{A})+v(\boldsymbol{A})=n | | |
| lyche-numerical-linear-algebra_FO4371 | 337 | 1.000 | \pi(\boldsymbol{A})=\pi(\boldsymbol{B}), \zeta(\boldsymbol{A})=\zeta(\boldsymbol{B}) | | |
| lyche-numerical-linear-algebra_FO4372 | 337 | 1.000 | v(\boldsymbol{A})=v(\boldsymbol{B}) | | |
| lyche-numerical-linear-algebra_FO4373 | 337 | 1.000 | \pi(\boldsymbol{A})=k | | |
| lyche-numerical-linear-algebra_FO4374 | 337 | 1.000 | \pi(\boldsymbol{B})=m<k | | |
| lyche-numerical-linear-algebra_FO4375 | 337 | 1.000 | \boldsymbol{E}_{1} | | |
| lyche-numerical-linear-algebra_FO4376 | 337 | 1.000 | m \times k | | |
| lyche-numerical-linear-algebra_FO4377 | 337 | 1.000 | m<k | | |
| lyche-numerical-linear-algebra_FO4378 | 337 | 1.000 | \boldsymbol{E}_{1} \boldsymbol{x}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO4379 | 337 | 1.000 | \boldsymbol{y}^{T}= | | |
| lyche-numerical-linear-algebra_FO4380 | 337 | 0.996 | \left[\boldsymbol{x}^{T}, \mathbf{0}^{T}\right] \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO4381 | 337 | 0.996 | \boldsymbol{z}=\left[z_{1}, \ldots, z_{n}\right]^{T}=\boldsymbol{E} \boldsymbol{y} | | |
| lyche-numerical-linear-algebra_FO4382 | 337 | 0.996 | z_{i}=0 | | |
| lyche-numerical-linear-algebra_FO4383 | 337 | 0.996 | i=1,2, \ldots, m | | |
| lyche-numerical-linear-algebra_FO4384 | 337 | 1.000 | \mu_{i} \leq 0 | | |
| lyche-numerical-linear-algebra_FO4385 | 337 | 1.000 | i \geq m+1 | | |
| lyche-numerical-linear-algebra_FO4386 | 338 | 1.000 | \pi(\boldsymbol{A})=\pi(\boldsymbol{B}) | | |
| lyche-numerical-linear-algebra_FO4387 | 338 | 1.000 | v(\boldsymbol{A})= | | |
| lyche-numerical-linear-algebra_FO4388 | 338 | 1.000 | \pi(-\boldsymbol{A})=\pi(-\boldsymbol{B})=v(\boldsymbol{B}) | | |
| lyche-numerical-linear-algebra_FO4389 | 338 | 1.000 | \zeta(\boldsymbol{A})=n-\pi(\boldsymbol{A})-v(\boldsymbol{A})=n-\pi(\boldsymbol{B})-v(\boldsymbol{B})= | | |
| lyche-numerical-linear-algebra_FO4390 | 338 | 1.000 | \zeta(\boldsymbol{B}) | | |
| lyche-numerical-linear-algebra_FO4391 | 338 | 1.000 | \boldsymbol{U}_{1}^{*} \boldsymbol{A} \boldsymbol{U}_{1}=\boldsymbol{D}_{1} | | |
| lyche-numerical-linear-algebra_FO4392 | 338 | 0.999 | \boldsymbol{U}_{2}^{*} \boldsymbol{B} \boldsymbol{U}_{2}=\boldsymbol{D}_{2} | | |
| lyche-numerical-linear-algebra_FO4393 | 338 | 0.999 | \boldsymbol{D}_{1}=\boldsymbol{F}^{*} \boldsymbol{D}_{2} \boldsymbol{F} | | |
| lyche-numerical-linear-algebra_FO4394 | 338 | 0.999 | \boldsymbol{F}=\boldsymbol{U}_{2}^{*} \boldsymbol{E} \boldsymbol{U}_{1} | | |
| lyche-numerical-linear-algebra_FO4395 | 338 | 1.000 | \boldsymbol{D}_{1}, \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO4396 | 338 | 1.000 | \pi(\boldsymbol{A})=\pi\left(\boldsymbol{D}_{1}\right)=\pi\left(\boldsymbol{D}_{2}\right)=\pi(\boldsymbol{B}) | | |
| lyche-numerical-linear-algebra_FO4397 | 338 | 1.000 | \zeta | | |
| lyche-numerical-linear-algebra_FO4398 | 338 | 1.000 | \boldsymbol{A}=\operatorname{tridiag}\left(c_{i}, d_{i}, c_{i}\right) \in \mathbb{R}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO4399 | 338 | 1.000 | \alpha \in \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO4400 | 338 | 1.000 | \boldsymbol{A}-\alpha \boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO4401 | 338 | 0.999 | \boldsymbol{A}-\alpha \boldsymbol{I}=\boldsymbol{L} \boldsymbol{D} \boldsymbol{L}^{T} | | |
| lyche-numerical-linear-algebra_FO4402 | 338 | 1.000 | d_{1}(\alpha), \ldots, d_{n}(\alpha) | | |
| lyche-numerical-linear-algebra_FO4403 | 338 | 1.000 | \boldsymbol{L} \boldsymbol{D} \boldsymbol{L}^{T} | | |
| lyche-numerical-linear-algebra_FO4404 | 338 | 1.000 | v(\boldsymbol{A}-\alpha \boldsymbol{I})=v(\boldsymbol{D}) | | |
| lyche-numerical-linear-algebra_FO4405 | 338 | 0.988 | (\boldsymbol{A}-\alpha \boldsymbol{I}) \boldsymbol{x}=(\lambda-\alpha) \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO4406 | 338 | 0.988 | \lambda-\alpha | | |
| lyche-numerical-linear-algebra_FO4407 | 338 | 1.000 | v(\boldsymbol{A}-\alpha \boldsymbol{I}) | | |
| lyche-numerical-linear-algebra_FO4408 | 338 | 1.000 | \lambda_{1} \geq \lambda_{2} \geq | | |
| lyche-numerical-linear-algebra_FO4409 | 338 | 0.999 | \cdots \geq \lambda_{n} | | |
| lyche-numerical-linear-algebra_FO4410 | 338 | 0.999 | (a, b) | | |
| lyche-numerical-linear-algebra_FO4411 | 339 | 1.000 | \rho(a)= | | |
| lyche-numerical-linear-algebra_FO4412 | 339 | 1.000 | n, \rho(b)=0 | | |
| lyche-numerical-linear-algebra_FO4413 | 339 | 1.000 | \left\{\left[a_{j}, b_{j}\right]\right\} | | |
| lyche-numerical-linear-algebra_FO4414 | 339 | 1.000 | b_{j}-a_{j}= | | |
| lyche-numerical-linear-algebra_FO4415 | 339 | 1.000 | 2^{-j}(b-a) | | |
| lyche-numerical-linear-algebra_FO4416 | 339 | 1.000 | d_{k}(\alpha) | | |
| lyche-numerical-linear-algebra_FO4417 | 339 | 1.000 | \delta_{k}= | | |
| lyche-numerical-linear-algebra_FO4418 | 339 | 1.000 | c_{k} \epsilon_{M} | | |
| lyche-numerical-linear-algebra_FO4419 | 339 | 1.000 | \epsilon_{M} | | |
| lyche-numerical-linear-algebra_FO4420 | 339 | 1.000 | 2 \times 10^{-16} | | |
| lyche-numerical-linear-algebra_FO4421 | 339 | 1.000 | \left|d_{k}(\alpha)\right|<\left|\delta_{k}\right| | | |
| lyche-numerical-linear-algebra_FO4422 | 340 | 0.998 | \boldsymbol{A}\left(\left|a_{i, i}\right|>\right. | | |
| lyche-numerical-linear-algebra_FO4423 | 340 | 1.000 | \sum_{j \neq i}\left|a_{i, j}\right| | | |
| lyche-numerical-linear-algebra_FO4424 | 340 | 1.000 | R:=R_{1} \cup \cdots \cup R_{n} | | |
| lyche-numerical-linear-algebra_FO4425 | 340 | 1.000 | C=C_{1} \cup \cdots \cup C_{n} | | |
| lyche-numerical-linear-algebra_FO4426 | 340 | 0.987 | \boldsymbol{s}=\left[s_{1}, \ldots, s_{n}\right] \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO4427 | 340 | 1.000 | \boldsymbol{r}=\left[r_{1}, \ldots, r_{n}\right] \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO4428 | 340 | 1.000 | \boldsymbol{c}=\left[c_{1}, \ldots, c_{n}\right] \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO4429 | 340 | 1.000 | \boldsymbol{A}\left(t_{2}\right) | | |
| lyche-numerical-linear-algebra_FO4430 | 340 | 1.000 | \boldsymbol{A}=\boldsymbol{A}\left(t_{1}\right) | | |
| lyche-numerical-linear-algebra_FO4431 | 340 | 1.000 | \boldsymbol{E}=\boldsymbol{A}\left(t_{2}\right)-\boldsymbol{A}\left(t_{1}\right) | | |
| lyche-numerical-linear-algebra_FO4432 | 340 | 1.000 | \boldsymbol{A}\left(t_{1}\right) | | |
| lyche-numerical-linear-algebra_FO4433 | 340 | 1.000 | \|\boldsymbol{A}\|_{\infty}= | | |
| lyche-numerical-linear-algebra_FO4434 | 340 | 1.000 | \left[a_{k j}\right], \boldsymbol{E}=\left[e_{k j}\right] | | |
| lyche-numerical-linear-algebra_FO4435 | 340 | 1.000 | \boldsymbol{B}=\left[b_{k j}\right] | | |
| lyche-numerical-linear-algebra_FO4436 | 340 | 1.000 | \mathbb{R}^{n, n} | | |
| lyche-numerical-linear-algebra_FO4437 | 341 | 1.000 | \boldsymbol{B}=\boldsymbol{A}+\boldsymbol{E} | | |
| lyche-numerical-linear-algebra_FO4438 | 341 | 1.000 | \|\boldsymbol{A}\|_{2}=\|\boldsymbol{B}\|_{2}=1 | | |
| lyche-numerical-linear-algebra_FO4439 | 341 | 1.000 | \boldsymbol{A}, \boldsymbol{E}, \boldsymbol{B} | | |
| lyche-numerical-linear-algebra_FO4440 | 341 | 1.000 | \mu=\epsilon^{1 / n} | | |
| lyche-numerical-linear-algebra_FO4441 | 341 | 1.000 | \frac{10}{3} n^{3}=5 G_{n} | | |
| lyche-numerical-linear-algebra_FO4442 | 341 | 1.000 | \frac{4}{3} n^{3}=2 G_{n} | | |
| lyche-numerical-linear-algebra_FO4443 | 341 | 1.000 | \boldsymbol{V}_{k} \boldsymbol{E}_{k} \boldsymbol{V}_{k} | | |
| lyche-numerical-linear-algebra_FO4444 | 341 | 1.000 | \boldsymbol{E}_{k} | | |
| lyche-numerical-linear-algebra_FO4445 | 341 | 1.000 | \boldsymbol{G}=\left(\boldsymbol{I}-\boldsymbol{v} \boldsymbol{v}^{T}\right) \boldsymbol{E}\left(\boldsymbol{I}-\boldsymbol{v} \boldsymbol{v}^{T}\right) | | |
| lyche-numerical-linear-algebra_FO4446 | 341 | 1.000 | \boldsymbol{w}:=\boldsymbol{E} \boldsymbol{v}, \beta:=\frac{1}{2} \boldsymbol{v}^{T} \boldsymbol{w} | | |
| lyche-numerical-linear-algebra_FO4447 | 341 | 0.999 | \boldsymbol{z}:=\boldsymbol{w}-\beta \boldsymbol{v} | | |
| lyche-numerical-linear-algebra_FO4448 | 341 | 0.999 | \boldsymbol{G}=\boldsymbol{E}-\boldsymbol{v} \boldsymbol{z}^{T}-\boldsymbol{z} \boldsymbol{v}^{T} | | |
| lyche-numerical-linear-algebra_FO4449 | 341 | 1.000 | O\left(4 n^{3} / 3\right) | | |
| lyche-numerical-linear-algebra_FO4450 | 342 | 0.999 | d_{k} | | |
| lyche-numerical-linear-algebra_FO4451 | 342 | 0.999 | \boldsymbol{A}_{n} | | |
| lyche-numerical-linear-algebra_FO4452 | 342 | 1.000 | 5+\sqrt{24}<d_{k} \leq 10, k=1,2, \ldots, n | | |
| lyche-numerical-linear-algebra_FO4453 | 342 | 1.000 | D_{n}:=\operatorname{det}\left(\boldsymbol{A}_{n}\right)>(5+\sqrt{24})^{n} | | |
| lyche-numerical-linear-algebra_FO4454 | 342 | 1.000 | n_{0} \in \mathbb{N} | | |
| lyche-numerical-linear-algebra_FO4455 | 342 | 1.000 | D_{n_{0}} | | |
| lyche-numerical-linear-algebra_FO4456 | 342 | 0.935 | \boldsymbol{B}=\boldsymbol{U}^{T} \boldsymbol{D} \boldsymbol{U} | | |
| lyche-numerical-linear-algebra_FO4457 | 342 | 1.000 | \hat{\boldsymbol{A}}=\boldsymbol{D}^{-1 / 2} \boldsymbol{U} \boldsymbol{A} \boldsymbol{U}^{T} \boldsymbol{D}^{-1 / 2} | | |
| lyche-numerical-linear-algebra_FO4458 | 342 | 1.000 | \hat{\boldsymbol{A}} | | |
| lyche-numerical-linear-algebra_FO4459 | 342 | 1.000 | \hat{\boldsymbol{A}}=\hat{\boldsymbol{U}}^{T} \hat{\boldsymbol{D}} \hat{\boldsymbol{U}} | | |
| lyche-numerical-linear-algebra_FO4460 | 342 | 1.000 | \hat{\boldsymbol{D}} | | |
| lyche-numerical-linear-algebra_FO4461 | 342 | 1.000 | \boldsymbol{E}= | | |
| lyche-numerical-linear-algebra_FO4462 | 342 | 1.000 | \boldsymbol{U}^{T} \boldsymbol{D}^{-1 / 2} \hat{\boldsymbol{U}}^{T} | | |
| lyche-numerical-linear-algebra_FO4463 | 342 | 0.999 | \boldsymbol{E}^{T} \boldsymbol{A} \boldsymbol{E}=\hat{\boldsymbol{D}}, \boldsymbol{E}^{T} \boldsymbol{B} \boldsymbol{E}=\boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO4464 | 342 | 1.000 | \boldsymbol{A}=\operatorname{tridiag}(\boldsymbol{c}, \boldsymbol{d}, \boldsymbol{c}) | | |
| lyche-numerical-linear-algebra_FO4465 | 342 | 1.000 | d_{1}, \ldots, d_{n} | | |
| lyche-numerical-linear-algebra_FO4466 | 342 | 1.000 | c_{1}, \ldots, c_{n-1} | | |
| lyche-numerical-linear-algebra_FO4467 | 342 | 0.998 | \mathrm{k}=\operatorname{counting}(\mathrm{c}, \mathrm{d}, \mathrm{x}) | | |
| lyche-numerical-linear-algebra_FO4468 | 342 | 1.000 | d_{j}(x) | | |
| lyche-numerical-linear-algebra_FO4469 | 342 | 0.907 | (a, b] | | |
| lyche-numerical-linear-algebra_FO4470 | 342 | 0.901 | \left\{\left(a_{j}, b_{j}\right]\right\} | | |
| lyche-numerical-linear-algebra_FO4471 | 342 | 0.901 | b_{j}-a_{j} \leq(b-a) \epsilon_{M} | | |
| lyche-numerical-linear-algebra_FO4472 | 342 | 1.000 | \epsilon_{M} \approx 2.22 \times 10^{-16} | | |
| lyche-numerical-linear-algebra_FO4473 | 342 | 0.795 | \boldsymbol{T}:=\operatorname{tridiag}(-1,2,-1) | | |
| lyche-numerical-linear-algebra_FO4474 | 342 | 1.000 | \lambda_{5} | | |
| lyche-numerical-linear-algebra_FO4475 | 343 | 1.000 | x \in \mathbb{C} | | |
| lyche-numerical-linear-algebra_FO4476 | 343 | 1.000 | f(x)= | | |
| lyche-numerical-linear-algebra_FO4477 | 343 | 1.000 | \operatorname{det}(\boldsymbol{A}-x \boldsymbol{I}) | | |
| lyche-numerical-linear-algebra_FO4478 | 343 | 1.000 | \mu \in \sigma(\boldsymbol{A}+\boldsymbol{E}) | | |
| lyche-numerical-linear-algebra_FO4479 | 345 | 0.604 | 2 z_{k} / 3^{k} | | |
| lyche-numerical-linear-algebra_FO4480 | 345 | 0.604 | [1,-1] | | |
| lyche-numerical-linear-algebra_FO4481 | 345 | 0.954 | \lambda=3 | | |
| lyche-numerical-linear-algebra_FO4482 | 345 | 0.954 | \left\{z_{k}^{T} \boldsymbol{A} z_{k} / z_{k}^{T} z_{k}\right\} | | |
| lyche-numerical-linear-algebra_FO4483 | 345 | 0.628 | \boldsymbol{z}_{0} | | |
| lyche-numerical-linear-algebra_FO4484 | 345 | 1.000 | \left(\lambda_{j}^{k}, \boldsymbol{v}_{j}\right), j=1,2 | | |
| lyche-numerical-linear-algebra_FO4485 | 345 | 0.749 | 3^{-k} z_{k}=c_{1} \boldsymbol{v}_{1}+3^{-k} c_{2} \boldsymbol{v}_{2} \rightarrow c_{1} \boldsymbol{v}_{1} | | |
| lyche-numerical-linear-algebra_FO4486 | 345 | 0.749 | c_{1} \neq 0 | | |
| lyche-numerical-linear-algebra_FO4487 | 345 | 1.000 | \left(\lambda_{j}, \boldsymbol{v}_{j}\right), j=1, \ldots, n | | |
| lyche-numerical-linear-algebra_FO4488 | 345 | 1.000 | \left|\lambda_{1}\right|>\left|\lambda_{2}\right| \geq \cdots \geq | | |
| lyche-numerical-linear-algebra_FO4489 | 345 | 1.000 | \left|\lambda_{n}\right| | | |
| lyche-numerical-linear-algebra_FO4490 | 345 | 0.905 | z_{0} \in \mathbb{C}^{n} | | |
| lyche-numerical-linear-algebra_FO4491 | 345 | 0.999 | z_{0} | | |
| lyche-numerical-linear-algebra_FO4492 | 345 | 0.522 | z_{0}=c_{1} \boldsymbol{v}_{1}+c_{2} \boldsymbol{v}_{2}+\cdots+c_{n} \boldsymbol{v}_{n} | | |
| lyche-numerical-linear-algebra_FO4493 | 345 | 1.000 | \boldsymbol{A}^{k} \boldsymbol{v}_{j}=\lambda_{j}^{k} \boldsymbol{v}_{j} | | |
| lyche-numerical-linear-algebra_FO4494 | 345 | 1.000 | \lambda_{1}^{k} | | |
| lyche-numerical-linear-algebra_FO4495 | 346 | 0.772 | \left(\lambda_{j} / \lambda_{1}\right)^{k} \rightarrow 0 | | |
| lyche-numerical-linear-algebra_FO4496 | 346 | 0.534 | (i) | | |
| lyche-numerical-linear-algebra_FO4497 | 346 | 0.989 | z_{k} | | |
| lyche-numerical-linear-algebra_FO4498 | 346 | 0.766 | \boldsymbol{x}_{0}=z_{0} /\left\|z_{0}\right\| | | |
| lyche-numerical-linear-algebra_FO4499 | 346 | 1.000 | k=1,2, \ldots | | |
| lyche-numerical-linear-algebra_FO4500 | 346 | 1.000 | \lambda_{1}>0 | | |
| lyche-numerical-linear-algebra_FO4501 | 346 | 1.000 | c_{1}>0 | | |
| lyche-numerical-linear-algebra_FO4502 | 346 | 1.000 | \boldsymbol{u}_{1}:=\boldsymbol{v}_{1} /\left\|\boldsymbol{v}_{1}\right\| | | |
| lyche-numerical-linear-algebra_FO4503 | 346 | 0.980 | \boldsymbol{x}_{k}=\boldsymbol{z}_{k} /\left\|\boldsymbol{z}_{k}\right\| | | |
| lyche-numerical-linear-algebra_FO4504 | 346 | 0.980 | \boldsymbol{z}_{k}= | | |
| lyche-numerical-linear-algebra_FO4505 | 346 | 0.999 | \boldsymbol{A}^{k} \boldsymbol{z}_{0} | | |
| lyche-numerical-linear-algebra_FO4506 | 346 | 0.999 | \boldsymbol{y}_{k}=\boldsymbol{A} \boldsymbol{x}_{k-1}= | | |
| lyche-numerical-linear-algebra_FO4507 | 346 | 1.000 | \boldsymbol{A} \boldsymbol{z}_{k-1} /\left\|\boldsymbol{z}_{k-1}\right\|=\boldsymbol{z}_{k} /\left\|\boldsymbol{z}_{k-1}\right\| | | |
| lyche-numerical-linear-algebra_FO4508 | 346 | 1.000 | \boldsymbol{x}_{k}=\left(\boldsymbol{z}_{k} /\left\|\boldsymbol{z}_{k-1}\right\|\right)\left(\left\|\boldsymbol{z}_{k-1}\right\| /\left\|\boldsymbol{z}_{k}\right\|\right)=\boldsymbol{z}_{k} /\left\|\boldsymbol{z}_{k}\right\| | | |
| lyche-numerical-linear-algebra_FO4509 | 346 | 1.000 | r(\lambda):=\boldsymbol{A} \boldsymbol{u}-\lambda \boldsymbol{u} | | |
| lyche-numerical-linear-algebra_FO4510 | 346 | 1.000 | \boldsymbol{u} \in \mathbb{C}^{n} \backslash\{\mathbf{0}\} | | |
| lyche-numerical-linear-algebra_FO4511 | 346 | 1.000 | \rho: \mathbb{C} \rightarrow \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO4512 | 346 | 1.000 | \rho(\lambda)=\|\boldsymbol{A} \boldsymbol{u}-\lambda \boldsymbol{u}\|_{2} | | |
| lyche-numerical-linear-algebra_FO4513 | 346 | 0.906 | \lambda:=\frac{\boldsymbol{u}^{*} \boldsymbol{A} \boldsymbol{u}}{\boldsymbol{u}^{*} \boldsymbol{u}} | | |
| lyche-numerical-linear-algebra_FO4514 | 347 | 1.000 | \{\boldsymbol{u}, \boldsymbol{U}\} | | |
| lyche-numerical-linear-algebra_FO4515 | 347 | 1.000 | \boldsymbol{U}^{*} \boldsymbol{u}=\mathbf{0} | | |
| lyche-numerical-linear-algebra_FO4516 | 347 | 1.000 | \lambda=\boldsymbol{u}^{*} \boldsymbol{A} \boldsymbol{u} | | |
| lyche-numerical-linear-algebra_FO4517 | 347 | 0.909 | (\mu, \boldsymbol{u}) | | |
| lyche-numerical-linear-algebra_FO4518 | 347 | 0.909 | \|\boldsymbol{u}\|_{2}=1 | | |
| lyche-numerical-linear-algebra_FO4519 | 347 | 0.860 | l, \boldsymbol{x} | | |
| lyche-numerical-linear-algebra_FO4520 | 347 | 0.767 | \|\boldsymbol{A} \boldsymbol{x}-l \boldsymbol{x}\|_{2} /\|\boldsymbol{A}\|_{F}<t o l | | |
| lyche-numerical-linear-algebra_FO4521 | 347 | 0.877 | =K+1 | | |
| lyche-numerical-linear-algebra_FO4522 | 348 | 0.900 | \boldsymbol{z}=[0.6602,0.3420] | | |
| lyche-numerical-linear-algebra_FO4523 | 348 | 0.900 | =10^{-6} | | |
| lyche-numerical-linear-algebra_FO4524 | 348 | 1.000 | \frac{\left|\lambda_{2}\right|}{\left|\lambda_{1}\right|} | | |
| lyche-numerical-linear-algebra_FO4525 | 348 | 1.000 | \lambda_{1}=5.3723 | | |
| lyche-numerical-linear-algebra_FO4526 | 348 | 1.000 | \lambda_{2}=-0.3723 | | |
| lyche-numerical-linear-algebra_FO4527 | 348 | 1.000 | \lambda_{2}=1.9 | | |
| lyche-numerical-linear-algebra_FO4528 | 348 | 1.000 | \boldsymbol{A}-s \boldsymbol{I} | | |
| lyche-numerical-linear-algebra_FO4529 | 348 | 1.000 | \lambda-s | | |
| lyche-numerical-linear-algebra_FO4530 | 348 | 1.000 | (\boldsymbol{A}-s \boldsymbol{I})^{-1} | | |
| lyche-numerical-linear-algebra_FO4531 | 348 | 1.000 | \lim _{s \rightarrow \lambda_{1}}\left|\mu_{1}(s)\right|=\infty | | |
| lyche-numerical-linear-algebra_FO4532 | 348 | 1.000 | \lim _{s \rightarrow \lambda_{1}} \mu_{j}(s)=\left(\lambda_{j}-\lambda_{1}\right)^{-1}<\infty | | |
| lyche-numerical-linear-algebra_FO4533 | 348 | 1.000 | j=2, \ldots, n | | |
| lyche-numerical-linear-algebra_FO4534 | 349 | 0.987 | s_{k-1} | | |
| lyche-numerical-linear-algebra_FO4535 | 349 | 1.000 | \boldsymbol{A} \boldsymbol{x}_{k} | | |
| lyche-numerical-linear-algebra_FO4536 | 350 | 0.913 | i i | | |
| lyche-numerical-linear-algebra_FO4537 | 350 | 0.995 | (s, \boldsymbol{x}) | | |
| lyche-numerical-linear-algebra_FO4538 | 350 | 0.995 | (\lambda, \boldsymbol{v}) | | |
| lyche-numerical-linear-algebra_FO4539 | 350 | 1.000 | n r | | |
| lyche-numerical-linear-algebra_FO4540 | 350 | 1.000 | \lambda_{1}=(5-\sqrt{33}) / 2 \approx-0.37 | | |
| lyche-numerical-linear-algebra_FO4541 | 350 | 1.000 | s= | | |
| lyche-numerical-linear-algebra_FO4545 | 351 | 0.960 | \boldsymbol{Q} \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO4546 | 351 | 1.000 | \boldsymbol{R} \in \mathbb{C}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO4547 | 351 | 1.000 | \boldsymbol{R}_{k} \boldsymbol{Q}_{k} | | |
| lyche-numerical-linear-algebra_FO4548 | 351 | 1.000 | \boldsymbol{R}_{k}=\boldsymbol{Q}_{k}^{*} \boldsymbol{A}_{k} | | |
| lyche-numerical-linear-algebra_FO4549 | 352 | 1.000 | \boldsymbol{A}_{10} | | |
| lyche-numerical-linear-algebra_FO4550 | 352 | 1.000 | \lambda_{1}=3 | | |
| lyche-numerical-linear-algebra_FO4551 | 352 | 0.999 | \lambda_{2}=1 | | |
| lyche-numerical-linear-algebra_FO4552 | 352 | 1.000 | \lambda_{1}, \ldots, \lambda_{4} | | |
| lyche-numerical-linear-algebra_FO4553 | 352 | 0.835 | \boldsymbol{A}_{14} | | |
| lyche-numerical-linear-algebra_FO4554 | 352 | 1.000 | \lambda_{1} \approx 2.323 | | |
| lyche-numerical-linear-algebra_FO4555 | 352 | 1.000 | \lambda_{4} \approx 0.2275 | | |
| lyche-numerical-linear-algebra_FO4556 | 352 | 0.813 | \lambda_{2} \approx 0.0914+0.4586 i | | |
| lyche-numerical-linear-algebra_FO4557 | 352 | 0.813 | \lambda_{3} \approx | | |
| lyche-numerical-linear-algebra_FO4558 | 352 | 1.000 | \left(\boldsymbol{A}_{k}\right)_{k} | | |
| lyche-numerical-linear-algebra_FO4559 | 352 | 0.674 | k=1,2,3 \ldots | | |
| lyche-numerical-linear-algebra_FO4560 | 352 | 0.674 | A^{k} | | |
| lyche-numerical-linear-algebra_FO4561 | 352 | 1.000 | \boldsymbol{A}^{k}=\tilde{\boldsymbol{Q}}_{k} \tilde{\boldsymbol{R}}_{k} | | |
| lyche-numerical-linear-algebra_FO4562 | 352 | 1.000 | \boldsymbol{Q}_{1}, \ldots, \boldsymbol{Q}_{k}, \boldsymbol{R}_{1}, \ldots, \boldsymbol{R}_{k} | | |
| lyche-numerical-linear-algebra_FO4563 | 353 | 1.000 | \tilde{\boldsymbol{Q}}_{1} \tilde{\boldsymbol{R}}_{1}=\boldsymbol{Q}_{1} \boldsymbol{R}_{1}=\boldsymbol{A}_{1} | | |
| lyche-numerical-linear-algebra_FO4564 | 353 | 1.000 | \tilde{\boldsymbol{Q}}_{k-1} \tilde{\boldsymbol{R}}_{k-1}=\boldsymbol{A}^{k-1} | | |
| lyche-numerical-linear-algebra_FO4565 | 353 | 1.000 | \boldsymbol{Q}_{k} \boldsymbol{R}_{k}=\boldsymbol{A}_{k} | | |
| lyche-numerical-linear-algebra_FO4566 | 353 | 1.000 | \tilde{\boldsymbol{R}}_{k} | | |
| lyche-numerical-linear-algebra_FO4567 | 353 | 1.000 | \left\|\tilde{\boldsymbol{q}}_{1}^{(k)}\right\|_{2}=1 | | |
| lyche-numerical-linear-algebra_FO4568 | 353 | 1.000 | \tilde{\boldsymbol{Q}}_{k} | | |
| lyche-numerical-linear-algebra_FO4569 | 353 | 1.000 | \boldsymbol{x}_{0}=\boldsymbol{e}_{1} | | |
| lyche-numerical-linear-algebra_FO4570 | 353 | 1.000 | \lambda_{1} \boldsymbol{e}_{1} | | |
| lyche-numerical-linear-algebra_FO4571 | 353 | 1.000 | \left(\tilde{\boldsymbol{Q}}_{k}^{*} \boldsymbol{A} \tilde{\boldsymbol{Q}}_{k}\right)_{k} | | |
| lyche-numerical-linear-algebra_FO4572 | 353 | 1.000 | \boldsymbol{H}_{1} | | |
| lyche-numerical-linear-algebra_FO4573 | 353 | 1.000 | \boldsymbol{P}_{i, i+1}, i=1, \ldots, n-1 | | |
| lyche-numerical-linear-algebra_FO4574 | 353 | 1.000 | \boldsymbol{P}_{n-1, n} \cdots \boldsymbol{P}_{1,2} \boldsymbol{H}_{1}=\boldsymbol{R}_{1} | | |
| lyche-numerical-linear-algebra_FO4575 | 353 | 1.000 | \boldsymbol{H}_{1}=\boldsymbol{Q}_{1} \boldsymbol{R}_{1} | | |
| lyche-numerical-linear-algebra_FO4576 | 353 | 0.999 | \boldsymbol{Q}_{1}=\boldsymbol{P}_{1,2}^{*} \cdots \boldsymbol{P}_{n-1, n}^{*} | | |
| lyche-numerical-linear-algebra_FO4577 | 353 | 0.996 | \boldsymbol{R}_{1} \boldsymbol{Q}_{1}=\boldsymbol{R}_{1} \boldsymbol{P}_{1,2}^{*} \cdots \boldsymbol{P}_{n-1, n}^{*} | | |
| lyche-numerical-linear-algebra_FO4578 | 353 | 0.999 | (i+1, i) | | |
| lyche-numerical-linear-algebra_FO4579 | 354 | 0.993 | \boldsymbol{R} \boldsymbol{P}_{1,2}^{*} \cdots \boldsymbol{P}_{n-1, n}^{*} | | |
| lyche-numerical-linear-algebra_FO4580 | 354 | 1.000 | a_{i+1, i} | | |
| lyche-numerical-linear-algebra_FO4581 | 354 | 1.000 | A(1: i, 1: i) | | |
| lyche-numerical-linear-algebra_FO4582 | 354 | 1.000 | A(i+1: n, i+1: n) | | |
| lyche-numerical-linear-algebra_FO4583 | 354 | 1.000 | \left|a_{i+1, i}^{(k)}\right| \leq \epsilon | | |
| lyche-numerical-linear-algebra_FO4584 | 354 | 0.992 | \hat{\boldsymbol{A}}_{k}:=\boldsymbol{A}_{k}-a_{i+1, i}^{(k)} \boldsymbol{e}_{i+1} \boldsymbol{e}_{i}^{T} | | |
| lyche-numerical-linear-algebra_FO4585 | 354 | 1.000 | \boldsymbol{A}_{k}=\tilde{\boldsymbol{Q}}_{k-1}^{*} \boldsymbol{A} \tilde{\boldsymbol{Q}}_{k-1} | | |
| lyche-numerical-linear-algebra_FO4586 | 354 | 1.000 | \tilde{\boldsymbol{Q}}_{k-1} | | |
| lyche-numerical-linear-algebra_FO4587 | 354 | 1.000 | \|\boldsymbol{E}\|_{F}=\left\|a_{i+1, i}^{(k)} \boldsymbol{e}_{i+1} \boldsymbol{e}_{i}^{T}\right\|_{F}=\left|a_{i+1, i}^{(k)}\right| \leq \epsilon | | |
| lyche-numerical-linear-algebra_FO4588 | 354 | 1.000 | a_{i+1, i}^{(k)}=0 | | |
| lyche-numerical-linear-algebra_FO4589 | 355 | 1.000 | \boldsymbol{R}_{k}=\boldsymbol{Q}_{k}^{*}\left(\boldsymbol{A}_{k}-s_{k} \boldsymbol{I}\right) | | |
| lyche-numerical-linear-algebra_FO4590 | 355 | 1.000 | \boldsymbol{Q}_{k} | | |
| lyche-numerical-linear-algebra_FO4591 | 355 | 0.998 | \boldsymbol{A}-s_{k} \boldsymbol{I}=\boldsymbol{Q}_{k} \boldsymbol{R}_{k} | | |
| lyche-numerical-linear-algebra_FO4592 | 355 | 0.998 | \left(\boldsymbol{A}-s_{k} \boldsymbol{I}\right)^{*}=\boldsymbol{R}_{k}^{*} \boldsymbol{Q}_{k}^{*} | | |
| lyche-numerical-linear-algebra_FO4593 | 355 | 0.998 | \left(\boldsymbol{A}-s_{k} \boldsymbol{I}\right)^{*} \boldsymbol{Q}_{k}=\boldsymbol{R}_{k}^{*} | | |
| lyche-numerical-linear-algebra_FO4594 | 355 | 1.000 | \boldsymbol{R}_{k}^{*} | | |
| lyche-numerical-linear-algebra_FO4595 | 355 | 1.000 | n, n | | |
| lyche-numerical-linear-algebra_FO4596 | 355 | 1.000 | \bar{r}_{n n}^{(k)} | | |
| lyche-numerical-linear-algebra_FO4597 | 355 | 1.000 | \left(\boldsymbol{A}-s_{k} \boldsymbol{I}\right)^{*} \boldsymbol{Q}_{k} \boldsymbol{e}_{n}= | | |
| lyche-numerical-linear-algebra_FO4598 | 355 | 1.000 | \boldsymbol{R}_{k}^{*} \boldsymbol{e}_{n}=\bar{r}_{n n}^{(k)} \boldsymbol{e}_{n} | | |
| lyche-numerical-linear-algebra_FO4599 | 355 | 1.000 | s_{k}:=\boldsymbol{e}_{n}^{T} \boldsymbol{A}_{k} \boldsymbol{e}_{n} | | |
| lyche-numerical-linear-algebra_FO4600 | 355 | 0.999 | \boldsymbol{A}:=?\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right] | | |
| lyche-numerical-linear-algebra_FO4601 | 355 | 0.999 | \boldsymbol{A}_{k}=\boldsymbol{A} | | |
| lyche-numerical-linear-algebra_FO4602 | 355 | 1.000 | \boldsymbol{A} \boldsymbol{u}-\lambda \boldsymbol{u} | | |
| lyche-numerical-linear-algebra_FO4603 | 355 | 1.000 | \lambda=\frac{\boldsymbol{u}^{*} \boldsymbol{A} \boldsymbol{u}}{\boldsymbol{u}^{*} \boldsymbol{u}} | | |
| lyche-numerical-linear-algebra_FO4604 | 358 | 1.000 | f: \mathbb{R}^{n} \rightarrow \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO4605 | 358 | 1.000 | \nabla f(\boldsymbol{x}) \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO4606 | 358 | 1.000 | \boldsymbol{H} f=\nabla \nabla^{T} f(\boldsymbol{x}) \in \mathbb{R}^{n \times n} | | |
| lyche-numerical-linear-algebra_FO4607 | 358 | 1.000 | \nabla^{T} f:=(\nabla f)^{T} | | |
| lyche-numerical-linear-algebra_FO4608 | 358 | 1.000 | \nabla \nabla^{T} | | |
| lyche-numerical-linear-algebra_FO4609 | 358 | 1.000 | \nabla^{T} \nabla | | |
| lyche-numerical-linear-algebra_FO4610 | 358 | 1.000 | \nabla^{T} \nabla f=D_{1}^{2} f+\cdots+D_{n}^{2} f=: \nabla^{2} | | |
| lyche-numerical-linear-algebra_FO4611 | 358 | 1.000 | f, g: \mathbb{R}^{n} \rightarrow \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO4612 | 358 | 0.982 | \nabla(f g)=f \nabla g+g \nabla f, \quad \nabla^{T}(f g)=f \nabla^{T} g+g \nabla^{T} f | | |
| lyche-numerical-linear-algebra_FO4613 | 358 | 1.000 | \nabla \nabla^{T}(f g)=\nabla f \nabla^{T} g+\nabla g \nabla^{T} f+f \nabla \nabla^{T} g+g \nabla \nabla^{T} f | | |
| lyche-numerical-linear-algebra_FO4614 | 358 | 1.000 | \nabla^{2}(f g)=2 \nabla^{T} f \nabla g+f \nabla^{2} g+g \nabla^{2} f | | |
| lyche-numerical-linear-algebra_FO4615 | 359 | 1.000 | \boldsymbol{f}=\left[f_{1}, \ldots f_{m}\right]^{T}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m} | | |
| lyche-numerical-linear-algebra_FO4616 | 359 | 1.000 | f(\boldsymbol{x})=f(x, y)=x^{2}-x y+y^{2} | | |
| lyche-numerical-linear-algebra_FO4617 | 359 | 1.000 | \boldsymbol{g}(x, y):=[f(x, y), x-y]^{T} | | |
| lyche-numerical-linear-algebra_FO4618 | 359 | 1.000 | f \in C^{2}(\Omega) | | |
| lyche-numerical-linear-algebra_FO4619 | 359 | 0.633 | \Omega \in \mathbb{R}^{n} | | |
| lyche-numerical-linear-algebra_FO4620 | 359 | 0.633 | \boldsymbol{x}, \boldsymbol{x}+\boldsymbol{h} \in \Omega | | |
| lyche-numerical-linear-algebra_FO4621 | 359 | 0.633 | L:=\{\boldsymbol{x}+t \boldsymbol{h}: | | |
| lyche-numerical-linear-algebra_FO4622 | 359 | 1.000 | t \in(0,1)\} \subset \Omega | | |
| lyche-numerical-linear-algebra_FO4623 | 359 | 1.000 | g:[0,1] \rightarrow \mathbb{R} | | |
| lyche-numerical-linear-algebra_FO4624 | 359 | 1.000 | g(t):=f(\boldsymbol{x}+t \boldsymbol{h}) | | |
| lyche-numerical-linear-algebra_FO4625 | 359 | 1.000 | g \in C^{2}[0,1] | | |
| lyche-numerical-linear-algebra_FO4626 | 359 | 1.000 | \boldsymbol{c}=\boldsymbol{x}+u \boldsymbol{h} | | |
| lyche-numerical-linear-algebra_FO4627 | 360 | 1.000 | m, n \in \mathbb{N}, \boldsymbol{B} \in \mathbb{R}^{n \times n}, \boldsymbol{C} \in | | |
| lyche-numerical-linear-algebra_FO4628 | 360 | 1.000 | \boldsymbol{x} \in \mathbb{R}^{n}, \boldsymbol{y} \in \mathbb{R}^{m} | | |
| lyche-numerical-linear-algebra_FO4629 | 360 | 1.000 | \nabla\left(\boldsymbol{y}^{T} \boldsymbol{C}\right)=\nabla^{T}(\boldsymbol{C} \boldsymbol{x})=\boldsymbol{C} | | |
| lyche-numerical-linear-algebra_FO4630 | 360 | 1.000 | \nabla\left(\boldsymbol{x}^{T} \boldsymbol{B} \boldsymbol{x}\right)=\left(\boldsymbol{B}+\boldsymbol{B}^{T}\right) \boldsymbol{x}, \quad \nabla^{T}\left(\boldsymbol{x}^{T} \boldsymbol{B} \boldsymbol{x}\right)=\boldsymbol{x}^{T}\left(\boldsymbol{B}+\boldsymbol{B}^{T}\right) | | |
| lyche-numerical-linear-algebra_FO4631 | 360 | 1.000 | \nabla \nabla^{T}\left(\boldsymbol{x}^{T} \boldsymbol{B} \boldsymbol{x}\right)=\boldsymbol{B}+\boldsymbol{B}^{T} | | |
| lyche-numerical-linear-algebra_FO4632 | 360 | 1.000 | D_{i}\left(\boldsymbol{y}^{T} \boldsymbol{C}\right)=\lim _{h \rightarrow 0} \frac{1}{h}\left(\left(\boldsymbol{y}+h \boldsymbol{e}_{i}\right)^{T} \boldsymbol{C}-\boldsymbol{y}^{T} \boldsymbol{C}\right)=\boldsymbol{e}_{i}^{T} \boldsymbol{C} | | |
| lyche-numerical-linear-algebra_FO4633 | 360 | 1.000 | D_{i}(\boldsymbol{C} \boldsymbol{x})= | | |
| lyche-numerical-linear-algebra_FO4634 | 360 | 1.000 | \lim _{h \rightarrow 0} \frac{1}{h}\left(\boldsymbol{C}\left(\boldsymbol{x}+h \boldsymbol{e}_{i}\right)-\boldsymbol{C} \boldsymbol{x}\right)=\boldsymbol{C} \boldsymbol{e}_{i} | | |
| lyche-numerical-linear-algebra_FO4635 | 363 | 0.783 | \boldsymbol{x}^{T} \boldsymbol{A y} | | |
| lyche-numerical-linear-algebra_FO4636 | 363 | 1.000 | \boldsymbol{T} \boldsymbol{H} \boldsymbol{x}=\boldsymbol{b}, 79 | | |
| lyche-numerical-linear-algebra_FO4637 | 365 | 0.995 | \boldsymbol{A}^{*} \boldsymbol{A}, 86 | | |
| lyche-numerical-linear-algebra_FO4638 | 374 | 1.000 | X X | |