| 1 | | 21 | \boldsymbol{x}=\left[\begin{array}{c} x_{1} \\ x_{2} \\ \vdots \\ x_{n} \end{array}\right], | |  |
| 2 | | 22 | \boldsymbol{x}+\boldsymbol{y}:=\left[\begin{array}{c} x_{1}+y_{1} \\ \vdots \\ x_{n}+y_{n} \end{array}\right], \quad a \boldsymbol{x}:=\left[\begin{array}{c} a x_{1} \\ \vdots \\ a x_{n} \end{array}\right], \quad \boldsymbol{x}, \boldsymbol{y} \in \mathbb{R}^{n}, \quad a \in \mathbb{R} . | |  |
| 3 | | 22 | \boldsymbol{A}=\left[\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \\ a_{21} & a_{22} & \cdots & a_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m 1} & a_{m 2} & \cdots & a_{m n} \end{array}\right] . | |  |
| 4 | | 22 | \boldsymbol{a}_{: j}:=\left[\begin{array}{c} a_{1 j} \\ a_{2 j} \\ \vdots \\ a_{m j} \end{array}\right], \quad \boldsymbol{a}_{i:}^{T}:=\left[a_{i 1}, a_{i 2}, \ldots, a_{i n}\right], \quad \boldsymbol{A}=\left[\boldsymbol{a}_{: 1}, \boldsymbol{a}_{: 2}, \ldots \boldsymbol{a}_{: n}\right]=\left[\begin{array}{c} \boldsymbol{a}_{1:}^{T} \\ \boldsymbol{a}_{2:}^{T} \\ \vdots \\ \boldsymbol{a}_{m:}^{T} \end{array}\right] | |  |
| 5 | | 22 | e^{x}=e^{a+i b}:=e^{a}(\cos b+i \sin b) . | |  |
| 6 | | 22 | e^{i \pi / 2}=i, \quad e^{i \pi}=-1, \quad e^{2 i \pi}=1 . | |  |
| 7 | | 23 | x=a+i b=r e^{i \theta}, \quad r=|x|=\sqrt{a^{2}+b^{2}}, \quad \cos \theta=\frac{a}{r}, \quad \sin \theta=\frac{b}{r} . | |  |
| 8 | | 23 | \boldsymbol{e}_{1}:=\left[\begin{array}{c} 1 \\ 0 \\ 0 \\ \vdots \\ 0 \end{array}\right], \quad \boldsymbol{e}_{2}:=\left[\begin{array}{c} 0 \\ 1 \\ 0 \\ \vdots \\ 0 \end{array}\right], \quad \boldsymbol{e}_{3}:=\left[\begin{array}{c} 0 \\ 0 \\ 1 \\ \vdots \\ 0 \end{array}\right], \quad \ldots, \quad \boldsymbol{e}_{n}:=\left[\begin{array}{c} 0 \\ 0 \\ 0 \\ \vdots \\ 1 \end{array}\right], | |  |
| 9 | 1.1 | 23 | \delta_{i j}:= \begin{cases}1 & \text { if } i=j \\ 0 & \text { otherwise }\end{cases} | |  |
| 10 | | 24 | \begin{gathered} \operatorname{diag}\left(d_{i}\right)=\operatorname{diag}\left(d_{1}, \ldots, d_{n}\right):=\left[\begin{array}{cccc} d_{1} & 0 & \cdots & 0 \\ 0 & d_{2} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & d_{n} \end{array}\right]=\left[\begin{array}{lll} d_{1} & & \\ & \ddots & \\ & & d_{n} \end{array}\right], \\ \boldsymbol{B}=\operatorname{tridiag}\left(a_{i}, d_{i}, c_{i}\right)=\operatorname{tridiag}(\boldsymbol{a}, \boldsymbol{d}, \boldsymbol{c}):=\left[\begin{array}{cccc} d_{1} & c_{1} & & \\ a_{1} & d_{2} & c_{2} & \\ & \ddots & \ddots & \ddots \\ & & a_{n-2} & d_{n-1} \\ & & & a_{n-1} \\ & & d_{n} \end{array}\right] . \end{gathered} | |  |
| 11 | | 24 | \boldsymbol{A}(\boldsymbol{i}, \boldsymbol{j}):=\boldsymbol{A}\left(\begin{array}{cccc} i_{1} & i_{2} & \cdots & i_{r} \\ j_{1} & j_{2} & \cdots & j_{c} \end{array}\right)=\left[\begin{array}{cccc} a_{i_{1}, j_{1}} & a_{i_{1}, j_{2}} & \cdots & a_{i_{1}, j_{c}} \\ a_{i_{2}, j_{1}} & a_{i_{2}, j_{2}} & \cdots & a_{i_{2}, j_{c}} \\ \vdots & \vdots & \ddots & \vdots \\ a_{i_{r}, j_{1}} & a_{i_{r}, j_{2}} & \cdots & a_{i_{r}, j_{c}} \end{array}\right] . | |  |
| 12 | | 24 | \boldsymbol{A}\left(r_{1}: r_{2}, c_{1}: c_{2}\right):=\left[\begin{array}{rrrr} a_{r_{1}, c_{1}} & a_{r_{1}, c_{1}+1} & \cdots & a_{r_{1}, c_{2}} \\ a_{r_{1}+1, c_{1}} & a_{r_{1}+1, c_{1}+1} & \cdots & a_{r_{1}+1, c_{2}} \\ \vdots & \vdots & \ddots & \vdots \\ a_{r_{2}, c_{1}} & a_{r_{2}, c_{1}+1} & \cdots & a_{r_{2}, c_{2}} \end{array}\right] . | |  |
| 13 | | 25 | (\boldsymbol{f}+\boldsymbol{g})(t):=\boldsymbol{f}(t)+\boldsymbol{g}(t), \quad(c \boldsymbol{f})(t):=c \boldsymbol{f}(t), \quad t \in \mathcal{D}, \quad c \in \mathbb{R} . | |  |
| 14 | 1.2 | 25 | p(t):=a_{0}+a_{1} t+a_{2} t^{2}+\cdots+a_{n} t^{n}, | |  |
| 15 | 1.3 | 26 | c_{1} \boldsymbol{x}_{1}+\cdots+c_{n} \boldsymbol{x}_{n}=\mathbf{0} \quad \Longrightarrow \quad c_{1}=\cdots=c_{n}=0 . | |  |
| 16 | 1.4 | 28 | \mathcal{S}+\mathcal{T}:=\{\boldsymbol{s}+\boldsymbol{t}: \boldsymbol{s} \in \mathcal{S} \text { and } \boldsymbol{t} \in \mathcal{T}\} . | |  |
| 17 | 1.5 | 28 | \mathcal{S} \cap \mathcal{T}:=\{\boldsymbol{x}: \boldsymbol{x} \in \mathcal{S} \text { and } \boldsymbol{x} \in \mathcal{T}\} . | |  |
| 18 | 1.6 | 28 | \mathcal{S} \cup \mathcal{T}:=\{\boldsymbol{x}: \boldsymbol{x} \in \mathcal{S} \text { or } \boldsymbol{x} \in \mathcal{T}\} . | |  |
| 19 | 1.7 | 29 | \operatorname{dim}(\mathcal{S}+\mathcal{T})=\operatorname{dim}(\mathcal{S})+\operatorname{dim}(\mathcal{T})-\operatorname{dim}(\mathcal{S} \cap \mathcal{T}) . | |  |
| 20 | 1.8 | 29 | \operatorname{dim}(\mathcal{S} \oplus \mathcal{T})=\operatorname{dim}(\mathcal{S})+\operatorname{dim}(\mathcal{T}) . | |  |
| 21 | | 29 | \left\{\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{p}, \boldsymbol{s}_{1}, \ldots, \boldsymbol{s}_{q}, \boldsymbol{t}_{1}, \ldots, \boldsymbol{t}_{r}\right\} | |  |
| 22 | | 29 | \left\{\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{p}, \boldsymbol{s}_{1}, \ldots, \boldsymbol{s}_{q}\right\} | |  |
| 23 | | 29 | \operatorname{dim}(\mathcal{S}+\mathcal{T})=p+q+r=(p+q)+(p+r)-p=\operatorname{dim}(\mathcal{S})+\operatorname{dim}(\mathcal{T})-\operatorname{dim}(\mathcal{S} \cap \mathcal{T}) | |  |
| 24 | 1.9 | 29 | \boldsymbol{s}_{j}=\sum_{i=1}^{m} a_{i j} \boldsymbol{v}_{i} \text { for } j=1, \ldots, n . | |  |
| 25 | | 30 | \sum_{i=1}^{m} b_{i} \boldsymbol{v}_{i}=\boldsymbol{x}=\sum_{j=1}^{n} c_{j} \boldsymbol{s}_{j} \stackrel{(1.9)}{=} \sum_{j=1}^{n} c_{j}\left(\sum_{i=1}^{m} a_{i j} \boldsymbol{v}_{i}\right)=\sum_{i=1}^{m}\left(\sum_{j=1}^{n} a_{i j} c_{j}\right) \boldsymbol{v}_{i} . | |  |
| 26 | | 30 | \mathcal{R}(\boldsymbol{X}):=\left\{\boldsymbol{X} \boldsymbol{c}: \boldsymbol{c} \in \mathbb{R}^{n}\right\}=\operatorname{span}(\mathcal{X}) . | |  |
| 27 | 1.10 | 31 | \langle\boldsymbol{x}, \boldsymbol{y}\rangle:=\boldsymbol{y}^{*} \boldsymbol{x}=\boldsymbol{x}^{T} \overline{\boldsymbol{y}}=\sum_{j=1}^{n} x_{j} \overline{y_{j}} . | |  |
| 28 | 1.11 | 31 | \|x\|_{2}:=\left(\sum_{j=1}^{n}\left|x_{j}\right|^{2}\right)^{1 / 2}=\sqrt{x^{*} x} . | |  |
| 29 | | 31 | \begin{array}{ccc} a_{11} x_{1}+a_{12} x_{2}+\cdots+ & a_{1 n} x_{n}=b_{1} \\ a_{21} x_{1}+a_{22} x_{2}+\cdots+ & a_{2 n} x_{n}= & b_{2} \\ \vdots & \vdots & \vdots \\ a_{m 1} x_{1}+a_{m 2} x_{2}+\cdots+ & \vdots \\ a_{m n} x_{n} & =b_{m} \end{array} | |  |
| 30 | | 31 | x_{1} \boldsymbol{a}_{1}+x_{2} \boldsymbol{a}_{2}+\cdots+x_{n} \boldsymbol{a}_{n}=\boldsymbol{b}, | |  |
| 31 | | 32 | \boldsymbol{A} \boldsymbol{x}=\left[\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \\ a_{21} & a_{22} & \cdots & a_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m 1} & a_{m 2} & \cdots & a_{m n} \end{array}\right]\left[\begin{array}{c} x_{1} \\ x_{2} \\ \vdots \\ x_{n} \end{array}\right]=\left[\begin{array}{c} b_{1} \\ b_{2} \\ \vdots \\ b_{m} \end{array}\right]=\boldsymbol{b} . | |  |
| 32 | | 32 | \boldsymbol{B} z=\left[\begin{array}{ll} \boldsymbol{A} & \boldsymbol{b} \end{array}\right]\left[\begin{array}{c} \tilde{\boldsymbol{z}} \\ z_{n+1} \end{array}\right]=\boldsymbol{A} \tilde{\boldsymbol{z}}+z_{n+1} \boldsymbol{b}=\mathbf{0} . | |  |
| 33 | | 33 | \boldsymbol{A} \boldsymbol{x}=-\boldsymbol{A}\left(\frac{\tilde{\boldsymbol{z}}}{z_{n+1}}\right)=-\frac{1}{z_{n+1}} \boldsymbol{A} \tilde{\boldsymbol{z}}=-\frac{1}{z_{n+1}}\left(-z_{n+1} \boldsymbol{b}\right)=\boldsymbol{b}, | |  |
| 34 | | 33 | C=C I=C(A B)=(C A) B=I B=B | |  |
| 35 | 1.12 | 35 | \operatorname{det}(\boldsymbol{A})=\sum_{\sigma \in S_{n}} \operatorname{sign}(\sigma) a_{\sigma(1), 1} a_{\sigma(2), 2} \cdots a_{\sigma(n), n} | |  |
| 36 | | 35 | \left|\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \\ a_{21} & a_{22} & \cdots & a_{2 n} \\ \vdots & \vdots & & \vdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n} \end{array}\right| . | |  |
| 37 | | 35 | \left|\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right|=a_{11} a_{22}-a_{21} a_{12} . | |  |
| 38 | | 35 | \begin{aligned} &\left|\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right|=a_{11} a_{22} a_{33}-a_{11} a_{32} a_{23}-a_{21} a_{12} a_{33} \\ &+a_{21} a_{32} a_{13}+a_{31} a_{12} a_{23}-a_{31} a_{22} a_{13} \end{aligned} | |  |
| 39 | 1.13 | 36 | \operatorname{det}(\boldsymbol{B})=\operatorname{det}(\boldsymbol{A}) . | |  |
| 40 | 1.14 | 36 | \begin{aligned} \operatorname{det}(\boldsymbol{A}) & =\sum_{j=1}^{n}(-1)^{i+j} a_{i j} \operatorname{det}\left(\boldsymbol{A}_{i j}\right) \text { for } i=1, \ldots, n, \text { row } \\ \operatorname{det}(\boldsymbol{A}) & =\sum_{i=1}^{n}(-1)^{i+j} a_{i j} \operatorname{det}\left(\boldsymbol{A}_{i j}\right) \text { for } j=1, \ldots, n, \text { column. } \end{aligned} | |  |
| 41 | | 36 | \operatorname{det}(\boldsymbol{A}):=\left|\begin{array}{ccc} 1 & x & y \\ 1 & x_{1} & y_{1} \\ 1 & x_{2} & y_{2} \end{array}\right|=0 | |  |
| 42 | | 36 | \begin{aligned} \left|\begin{array}{ccc} 1 & x & y \\ 1 & x_{1} & y_{1} \\ 1 & x_{2} & y_{2} \end{array}\right| & =\left|\begin{array}{ccc} 1 & x & y \\ 0 & x_{1}-x & y_{1}-y \\ 0 & x_{2}-x_{1} & y_{2}-y_{1} \end{array}\right| \\ & =\left|\begin{array}{cc} x_{1}-x & y_{1}-y \\ x_{2}-x_{1} & y_{2}-y_{1} \end{array}\right|=\left(x_{1}-x\right)\left(y_{2}-y_{1}\right)-\left(y_{1}-y\right)\left(x_{2}-x_{1}\right) \end{aligned} | |  |
| 43 | | 36 | y-y_{1}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\left(x-x_{1}\right) | |  |
| 44 | | 37 | x_{j}=\frac{\operatorname{det}\left(\boldsymbol{A}_{j}(\boldsymbol{b})\right)}{\operatorname{det}(\boldsymbol{A})}, \quad j=1,2, \ldots, n, | |  |
| 45 | | 37 | \boldsymbol{A}^{-1}=\frac{1}{\operatorname{det}(\boldsymbol{A})} \operatorname{adj}(\boldsymbol{A}), | |  |
| 46 | | 37 | \left[\begin{array}{ccc} c_{i_{1}, j_{1}} & \cdots & c_{i_{1}, j_{r}} \\ \vdots & & \vdots \\ c_{i_{r}, j_{1}} & \cdots & c_{i_{r}, j_{r}} \end{array}\right]=\sum_{\boldsymbol{k}}\left[\begin{array}{ccc} a_{i_{1}, k_{1}} & \cdots & a_{i_{1}, k_{r}} \\ \vdots & & \vdots \\ a_{i_{r} k_{1}} & \cdots & a_{i_{r}, k_{r}} \end{array}\right]\left[\begin{array}{ccc} b_{k_{1}, j_{1}} & \cdots & b_{k_{1}, j_{r}} \\ \vdots & & \vdots \\ b_{k_{r}, j_{1}} & \cdots & b_{k_{r}, j_{r}} \end{array}\right], | |  |
| 47 | 1.15 | 37 | \operatorname{det}(\boldsymbol{C}(\boldsymbol{i}, \boldsymbol{j}))=\sum_{\boldsymbol{k}} \operatorname{det}(\boldsymbol{A}(\boldsymbol{i}, \boldsymbol{k})) \operatorname{det}(\boldsymbol{B}(\boldsymbol{k}, \boldsymbol{j})), | |  |
| 48 | | 38 | \operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})=\left|\begin{array}{ccc} a_{11}-\lambda & a_{12} & a_{13} \\ a_{21} & a_{22}-\lambda & a_{23} \\ a_{31} & a_{32} & a_{33}-\lambda \end{array}\right| . | |  |
| 49 | | 38 | \begin{aligned} \operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I}) & =\left(a_{11}-\lambda\right)\left|\begin{array}{cc} a_{22}-\lambda & a_{23} \\ a_{32} & a_{33}-\lambda \end{array}\right|-a_{21}\left|\begin{array}{cc} a_{12} & a_{13} \\ a_{32} & a_{33}-\lambda \end{array}\right| \\ & +a_{31}\left|\begin{array}{cc} a_{12} & a_{13} \\ a_{22}-\lambda & a_{23} \end{array}\right|=\left(a_{11}-\lambda\right)\left(a_{22}-\lambda\right)\left(a_{33}-\lambda\right)+r(\lambda) \end{aligned} | |  |
| 50 | 1.16 | 38 | \operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})=\left(a_{11}-\lambda\right)\left(a_{22}-\lambda\right) \cdots\left(a_{n n}-\lambda\right)+r(\lambda), | |  |
| 51 | 1.17 | 39 | \operatorname{trace}(\boldsymbol{A})=\lambda_{1}+\lambda_{2}+\cdots+\lambda_{n}, \quad \operatorname{det}(\boldsymbol{A})=\lambda_{1} \lambda_{2} \cdots \lambda_{n}, | |  |
| 52 | 1.18 | 39 | \operatorname{trace}(\boldsymbol{A}):=a_{11}+a_{22}+\cdots+a_{n n} . | |  |
| 53 | | 39 | \pi_{A}(\lambda)=(-1)^{n} \lambda^{n}+c_{n-1} \lambda^{n-1}+\cdots+c_{0}, | |  |
| 54 | | 39 | \pi_{A}(\lambda)=\left(\lambda_{1}-\lambda\right) \cdots\left(\lambda_{n}-\lambda\right)=(-1)^{n} \lambda^{n}+d_{n-1} \lambda^{n-1}+\cdots+d_{0}, | |  |
| 55 | 1.19 | 39 | \lambda^{2}-\operatorname{trace}(\boldsymbol{A}) \lambda+\operatorname{det}(\boldsymbol{A})=0 . | |  |
| 56 | | 40 | \left[\begin{array}{ll} W & X \\ Y & Z \end{array}\right]=\left[\begin{array}{ll} A & B \\ C & D \end{array}\right]\left[\begin{array}{ll} E & F \\ G & H \end{array}\right], | |  |
| 57 | | 40 | \begin{array}{ll} \mathbf{P}_{1}=(\boldsymbol{A}+\boldsymbol{D})(\boldsymbol{E}+\boldsymbol{H}), & \\ \mathbf{P}_{2}=(\boldsymbol{C}+\boldsymbol{D}) \boldsymbol{E}, & \mathbf{P}_{5}=(\boldsymbol{A}+\boldsymbol{B}) \boldsymbol{H}, \\ \mathbf{P}_{3}=\boldsymbol{A}(\boldsymbol{F}-\boldsymbol{H}), & \mathbf{P}_{6}=(\boldsymbol{C}-\boldsymbol{A})(\boldsymbol{E}+\boldsymbol{F}), \\ \mathbf{P}_{4}=\boldsymbol{D}(\boldsymbol{G}-\boldsymbol{E}), & \mathbf{P}_{7}=(\boldsymbol{B}-\boldsymbol{D})(\boldsymbol{G}+\boldsymbol{H}), \\ \boldsymbol{W}=\mathbf{P}_{1}+\mathbf{P}_{4}-\mathbf{P}_{5}+\mathbf{P}_{7}, & \boldsymbol{X}=\mathbf{P}_{3}+\mathbf{P}_{5}, \\ \boldsymbol{Y}=\mathbf{P}_{2}+\mathbf{P}_{4}, & \boldsymbol{Z}=\mathbf{P}_{1}+\mathbf{P}_{3}-\mathbf{P}_{2}+\mathbf{P}_{6} . \end{array} | |  |
| 58 | | 41 | \left[\begin{array}{ll} a & b \\ c & d \end{array}\right]^{-1}=\alpha\left[\begin{array}{cc} d & -b \\ -c & a \end{array}\right], \quad \alpha=\frac{1}{a d-b c} | |  |
| 59 | | 41 | A=\left[\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right] . | |  |
| 60 | | 41 | \left(\boldsymbol{A}+\boldsymbol{B} \boldsymbol{C}^{T}\right)^{-1}=\boldsymbol{A}^{-1}-\boldsymbol{A}^{-1} \boldsymbol{B}\left(\boldsymbol{I}+\boldsymbol{C}^{T} \boldsymbol{A}^{-1} \boldsymbol{B}\right)^{-1} \boldsymbol{C}^{T} \boldsymbol{A}^{-1} . | |  |
| 61 | 1.20 | 41 | \lambda:=\boldsymbol{a}^{T} \boldsymbol{C} \boldsymbol{e}_{i} \neq 0, | |  |
| 62 | 1.21 | 41 | \overline{\boldsymbol{C}}=\boldsymbol{C}\left(\boldsymbol{I}+\frac{1}{\lambda} \boldsymbol{e}_{i}\left(\boldsymbol{e}_{i}^{T}-\boldsymbol{a}^{T} \boldsymbol{C}\right)\right) | |  |
| 63 | | 42 | \left[\begin{array}{ll} 1 & 2 \\ 2 & 1 \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]=\left[\begin{array}{l} 3 \\ 6 \end{array}\right] | |  |
| 64 | | 42 | \boldsymbol{A}=\left[\begin{array}{ccc} 2 & -6 & 3 \\ 3 & -2 & -6 \\ 6 & 3 & 2 \end{array}\right], | |  |
| 65 | | 42 | \operatorname{adj}(\boldsymbol{A})=\left[\begin{array}{ccc} 14 & 21 & 42 \\ -42 & -14 & 21 \\ 21 & -42 & 14 \end{array}\right] . | |  |
| 66 | | 42 | \operatorname{adj}(\boldsymbol{A}) \boldsymbol{A}=\left[\begin{array}{ccc} 343 & 0 & 0 \\ 0 & 343 & 0 \\ 0 & 0 & 343 \end{array}\right]=\operatorname{det}(\boldsymbol{A}) \boldsymbol{I} . | |  |
| 67 | | 42 | \left|\begin{array}{llll} x & y & z & 1 \\ x_{1} & y_{1} & z_{1} & 1 \\ x_{2} & y_{2} & z_{2} & 1 \\ x_{3} & y_{3} & z_{3} & 1 \end{array}\right|=0 | |  |
| 68 | | 43 | A(T)=\frac{1}{2}\left|\begin{array}{ccc} 1 & 1 & 1 \\ x_{1} & x_{2} & x_{3} \\ y_{1} & y_{2} & y_{3} \end{array}\right| . | |  |
| 69 | | 43 | \left|\begin{array}{ccccc} 1 & x_{1} & x_{1}^{2} & \cdots & x_{1}^{n-1} \\ 1 & x_{2} & x_{2}^{2} & \cdots & x_{2}^{n-1} \\ \vdots & \vdots & \vdots & & \vdots \\ 1 & x_{n} & x_{n}^{2} & \cdots & x_{n}^{n-1} \end{array}\right|=\prod_{i>j}\left(x_{i}-x_{j}\right), | |  |
| 70 | | 43 | \operatorname{det}(\boldsymbol{A})=P g(\boldsymbol{\alpha}) g(\boldsymbol{\beta}) | |  |
| 71 | | 44 | g(\boldsymbol{\gamma})=\prod_{i=2}^{n}\left(\gamma_{i}-\gamma_{1}\right)\left(\gamma_{i}-\gamma_{2}\right) \cdots\left(\gamma_{i}-\gamma_{i-1}\right) | |  |
| 72 | | 44 | q(\boldsymbol{\alpha}, \boldsymbol{\beta})=k g(\boldsymbol{\alpha}) g(\boldsymbol{\beta}) | |  |
| 73 | | 44 | b_{j, k}=\left(\alpha_{k}+\beta_{j}\right) A_{k}\left(-\beta_{j}\right) B_{j}\left(-\alpha_{k}\right), | |  |
| 74 | | 44 | A_{k}(x)=\prod_{s \neq k}\left(\frac{\alpha_{s}-x}{\alpha_{s}-\alpha_{k}}\right), \quad B_{k}(x)=\prod_{s \neq k}\left(\frac{\beta_{s}-x}{\beta_{s}-\beta_{k}}\right) . | |  |
| 75 | | 44 | t_{i, j}^{n}=\frac{f(i) f(j)}{i+j-1}, | |  |
| 76 | | 44 | f(i+1)=\left(\frac{i^{2}-n^{2}}{i^{2}}\right) f(i), \quad i=1,2, \ldots, \quad f(1)=-n . | |  |
| 77 | 2.1 | 46 | x_{i}=a+\frac{i-1}{n}(b-a), \quad i=1,2, \ldots, n+1 | |  |
| 78 | 2.2 | 46 | g\left(x_{i}\right)=y_{i}, \text { for } i=1, \ldots, n+1 . | |  |
| 79 | 2.3 | 47 | g(x)=y_{1}+\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\left(x-x_{1}\right), | |  |
| 80 | | 47 | f(x)=\arctan (10 x)+\pi / 2, \quad x \in[-1,1] . | |  |
| 81 | 2.4 | 47 | g(x):= \begin{cases}p_{1}(x), & \text { if } x_{1} \leq x<x_{2}, \\ p_{2}(x), & \text { if } x_{2} \leq x<x_{3}, \\ \vdots & \\ p_{n-1}(x), & \text { if } x_{n-1} \leq x<x_{n}, \\ p_{n}(x), & \text { if } x_{n} \leq x \leq x_{n+1},\end{cases} | |  |
| 82 | 2.5 | 49 | g(x):= \begin{cases}p_{1}(x)=-\frac{1}{2} x+\frac{3}{2} x^{3}, & \text { if } 0 \leq x<1, \\ p_{2}(x)=1+4(x-1)+\frac{9}{2}(x-1)^{2}+\frac{13}{2}(x-1)^{3}, & \text { if } 1 \leq x \leq 2,\end{cases} | |  |
| 83 | | 49 | p_{i-1}^{(j)}\left(x_{i}\right)=p_{i}^{(j)}\left(x_{i}\right), \quad j=0,1,2, \quad i=2, \ldots, n . | |  |
| 84 | 2.6 | 50 | p_{i}\left(x_{i}\right)=y_{i}, \quad p_{i}\left(x_{i+1}\right)=y_{i+1}, \quad p_{i}^{\prime \prime}\left(x_{i}\right)=\mu_{i}, \quad p_{i}^{\prime \prime}\left(x_{i+1}\right)=\mu_{i+1} . | |  |
| 85 | 2.7 | 50 | p_{i}(x)=c_{i, 1}+c_{i, 2}\left(x-x_{i}\right)+c_{i, 3}\left(x-x_{i}\right)^{2}+c_{i, 4}\left(x-x_{i}\right)^{3} \quad i=1, \ldots, n, | |  |
| 86 | 2.8 | 50 | c_{i 1}=y_{i}, c_{i 2}=\frac{y_{i+1}-y_{i}}{h}-\frac{h}{3} \mu_{i}-\frac{h}{6} \mu_{i+1}, c_{i, 3}=\frac{\mu_{i}}{2}, c_{i, 4}=\frac{\mu_{i+1}-\mu_{i}}{6 h} . | |  |
| 87 | 2.9 | 51 | \mu_{i-1}+4 \mu_{i}+\mu_{i+1}=\frac{6}{h^{2}}\left(y_{i+1}-2 y_{i}+y_{i-1}\right), \quad i=2, \ldots, n, | |  |
| 88 | 2.10 | 51 | \begin{aligned} p_{i-1}^{\prime}\left(x_{i}\right) & =c_{i-1,2}+2 h c_{i-1,3}+3 h^{2} c_{i-1,4} \\ & =\frac{y_{i}-y_{i-1}}{h}-\frac{h}{3} \mu_{i-1}-\frac{h}{6} \mu_{i}+2 h \frac{\mu_{i-1}}{2}+3 h^{2} \frac{\mu_{i}-\mu_{i-1}}{6 h} \\ & =\frac{y_{i}-y_{i-1}}{h}+\frac{h}{6} \mu_{i-1}+\frac{h}{3} \mu_{i} \\ p_{i}^{\prime}\left(x_{i}\right) & =c_{i 2}=\frac{y_{i+1}-y_{i}}{h}-\frac{h}{3} \mu_{i}-\frac{h}{6} \mu_{i+1} . \end{aligned} | |  |
| 89 | 2.11 | 51 | \left[\begin{array}{ccccc} 4 & 1 & & & \\ 1 & 4 & 1 & & \\ & \ddots & \ddots & \ddots & \\ & & 1 & 4 & 1 \\ & & & 1 & 4 \end{array}\right]\left[\begin{array}{c} \mu_{2} \\ \mu_{3} \\ \vdots \\ \mu_{n-1} \\ \mu_{n} \end{array}\right]=\frac{6}{h^{2}}\left[\begin{array}{c} \delta^{2} y_{2}-\mu_{1} \\ \delta^{2} y_{3} \\ \vdots \\ \delta^{2} y_{n-1} \\ \delta^{2} y_{n}-\mu_{n+1} \end{array}\right], \delta^{2} y_{i}:=y_{i+1}-2 y_{i}+y_{i-1} . | |  |
| 90 | 2.12 | 52 | \left|a_{i i}\right|>\sum_{j \neq i}\left|a_{i j}\right|, i=1, \ldots, n . | |  |
| 91 | 2.13 | 52 | \max _{1 \leq i \leq n}\left|x_{i}\right| \leq \max _{1 \leq i \leq n}\left(\frac{\left|b_{i}\right|}{\sigma_{i}}\right), \text { where } \sigma_{i}:=\left|a_{i i}\right|-\sum_{j \neq i}\left|a_{i j}\right| . | |  |
| 92 | | 52 | \left|b_{k}\right|=\left|a_{k k} x_{k}+\sum_{j \neq k} a_{k j} x_{j}\right| \geq\left|a_{k k}\right|\left|x_{k}\right|-\sum_{j \neq k}\left|a_{k j}\right|\left|x_{j}\right| \geq\left|x_{k}\right|\left(\left|a_{k k}\right|-\sum_{j \neq k}\left|a_{k j}\right|\right), | |  |
| 93 | | 52 | \left[\begin{array}{lll} 4 & 1 & 0 \\ 1 & 4 & 1 \\ 0 & 1 & 4 \end{array}\right]\left[\begin{array}{l} \mu_{2} \\ \mu_{3} \\ \mu_{4} \end{array}\right]=\left[\begin{array}{c} 2 \\ -6 \\ 2 \end{array}\right] . | |  |
| 94 | 2.14 | 53 | g(x):= \begin{cases}p_{1}(x)=\frac{1}{6} x^{3}, & \text { if } 0 \leq x<1, \\ p_{2}(x)=\frac{1}{6}+\frac{1}{2}(x-1)+\frac{1}{2}(x-1)^{2}-\frac{1}{2}(x-1)^{3}, & \text { if } 1 \leq x<2, \\ p_{3}(x)=\frac{2}{3}-(x-2)^{2}+\frac{1}{2}(x-2)^{3}, & \text { if } 2 \leq x<3, \\ p_{4}(x)=\frac{1}{6}-\frac{1}{2}(x-3)+\frac{1}{2}(x-3)^{2}-\frac{1}{6}(x-3)^{3}, & \text { if } 3 \leq x \leq 4,\end{cases} | |  |
| 95 | 2.15 | 54 | \left[\begin{array}{ccccc} d_{1} & c_{1} & & & \\ a_{1} & d_{2} & c_{2} & & \\ & \ddots & \ddots & \ddots & \\ & & a_{n-2} & d_{n-1} & c_{n-1} \\ & & & a_{n-1} & d_{n} \end{array}\right]=\left[\begin{array}{cccc} 1 & & & \\ l_{1} & 1 & & \\ & \ddots & \ddots & \\ & & l_{n-1} & 1 \end{array}\right]\left[\begin{array}{cccc} u_{1} & c_{1} & & \\ & \ddots & \ddots & \\ & & u_{n-1} & c_{n-1} \\ & & & u_{n} \end{array}\right] | |  |
| 96 | 0 | 54 | \left[\begin{array}{lll} d_{1} & c_{1} & 0 \\ a_{1} & d_{2} & c_{2} \\ 0 & a_{2} & d_{3} \end{array}\right]=\left[\begin{array}{lll} 1 & 0 & 0 \\ l_{1} & 1 & 0 \\ 0 & l_{2} & 1 \end{array}\right]\left[\begin{array}{ccc} u_{1} & c_{1} & 0 \\ 0 & u_{2} & c_{2} \\ 0 & 0 & u_{3} \end{array}\right]=\left[\begin{array}{ccc} u_{1} & c_{1} & 0 \\ l_{1} u_{1} & l_{1} c_{1}+u_{2} & c_{2} \\ 0 & l_{2} u_{2} & l_{2} c_{2}+u_{3} \end{array}\right], | |  |
| 97 | | 54 | \left[\begin{array}{lll} 1 & 0 & 0 \\ l_{1} & 1 & 0 \\ 0 & l_{2} & 1 \end{array}\right]\left[\begin{array}{l} z_{1} \\ z_{2} \\ z_{3} \end{array}\right]=\left[\begin{array}{l} b_{1} \\ b_{2} \\ b_{3} \end{array}\right], \quad\left[\begin{array}{ccc} u_{1} & c_{1} & 0 \\ 0 & u_{2} & c_{2} \\ 0 & 0 & u_{3} \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]=\left[\begin{array}{l} z_{1} \\ z_{2} \\ z_{3} \end{array}\right] . | |  |
| 98 | | 54 | \begin{aligned} & u_{1}=d_{1}, \quad l_{1}=a_{1} / u_{1}, \quad u_{2}=d_{2}-l_{1} c_{1}, \quad l_{2}=a_{2} / u_{2}, \quad u_{3}=d_{3}-l_{2} c_{2} \\ & z_{1}=b_{1}, \quad z_{2}=b_{2}-l_{1} z_{1}, \quad z_{3}=b_{3}-l_{2} z_{2} \\ & x_{3}=z_{3} / u_{3}, \quad x_{2}=\left(z_{2}-c_{2} x_{3}\right) / u_{2}, \quad x_{1}=\left(z_{1}-c_{1} x_{2}\right) / u_{1} \end{aligned} | |  |
| 99 | 2.16 | 54 | u_{1}=d_{1}, \quad l_{k}=a_{k} / u_{k}, \quad u_{k+1}=d_{k+1}-l_{k} c_{k}, \quad k=1,2, \ldots, n-1, | |  |
| 100 | 2.17 | 55 | \left|u_{k}\right| \geq \sigma_{k}+\left|c_{k}\right|, \quad \text { where, } \sigma_{k}:=\left|d_{k}\right|-\left|a_{k-1}\right|-\left|c_{k}\right|>0, k=1, \ldots, n, | |  |
| 101 | 2.18 | 55 | \begin{aligned} \left|u_{k+1}\right| & =\left|d_{k+1}-l_{k} c_{k}\right|=\left|d_{k+1}-\frac{a_{k} c_{k}}{u_{k}}\right| \geq\left|d_{k+1}\right|-\frac{\left|a_{k}\right|\left|c_{k}\right|}{\left|u_{k}\right|} \\ & \geq\left|d_{k+1}\right|-\left|a_{k}\right|=\sigma_{k+1}+\left|c_{k+1}\right| \end{aligned} | |  |
| 102 | 2.19 | 56 | \left|l_{k}\right|=\frac{\left|a_{k}\right|}{\left|u_{k}\right|} \leq \frac{\left|a_{k}\right|}{\left|d_{k}\right|-\left|a_{k-1}\right|},\left|u_{k+1}\right| \leq\left|d_{k+1}\right|+\frac{\left|a_{k}\right|\left|c_{k}\right|}{\left|d_{k}\right|-\left|a_{k-1}\right|}, k=1, \ldots, n-1 . | |  |
| 103 | | 56 | \left|l_{k}\right|=\frac{\left|a_{k}\right|}{\left|u_{k}\right|} \leq \frac{\left|a_{k}\right|}{\left|d_{k}\right|-\left|a_{k-1}\right|},\left|u_{k+1}\right| \leq\left|d_{k+1}\right|+\left|l_{k}\right|\left|c_{k}\right| \leq\left|d_{k+1}\right|+\frac{\left|a_{k}\right|\left|c_{k}\right|}{\left|d_{k}\right|-\left|a_{k-1}\right|} . | |  |
| 104 | 2.20 | 56 | -u^{\prime \prime}(x)=f(x), \quad x \in[0,1], \quad u(0)=0, u(1)=0, | |  |
| 105 | | 56 | g^{\prime}(x)=\lim _{h \rightarrow 0} \frac{g\left(x+\frac{h}{2}\right)-g\left(x-\frac{h}{2}\right)}{h} | |  |
| 106 | | 57 | \begin{aligned} u^{\prime \prime}(x) & =\lim _{h \rightarrow 0} \frac{u^{\prime}\left(x+\frac{h}{2}\right)-u^{\prime}\left(x-\frac{h}{2}\right)}{h}=\lim _{h \rightarrow 0} \frac{\frac{u(x+h)-u(x)}{h}-\frac{u(x)-u(x-h)}{h}}{h} \\ & =\lim _{h \rightarrow 0} \frac{u(x+h)-2 u(x)+u(x-h)}{h^{2}} . \end{aligned} | |  |
| 107 | | 57 | \frac{-v_{j-1}+2 v_{j}-v_{j+1}}{h^{2}}=f(j h), \quad j=1, \ldots, m, \quad v_{0}=v_{m+1}=0 . | |  |
| 108 | 2.21 | 57 | \boldsymbol{T} \boldsymbol{v}=\left[\begin{array}{rrrrr} 2 & -1 & 0 & & \\ -1 & 2 & -1 & & \\ 0 & \ddots & \ddots & \ddots & \\ & & & & 0 \\ & & & -1 & 2-1 \\ & & & 0-1 & 2 \end{array}\right]\left[\begin{array}{c} v_{1} \\ v_{2} \\ \vdots \\ v_{m-1} \\ v_{m} \end{array}\right]=h^{2}\left[\begin{array}{c} f(h) \\ f(2 h) \\ \vdots \\ f((m-1) h) \\ f(m h) \end{array}\right]=: \boldsymbol{b} . | |  |
| 109 | 2.22 | 57 | \left|a_{i i}\right| \geq \sum_{j \neq i}\left|a_{i j}\right|, i=1, \ldots, n . | |  |
| 110 | | 58 | \boldsymbol{A}_{1}=\left[\begin{array}{lll} 1 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 1 \end{array}\right], \quad \boldsymbol{A}_{2}=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right], \quad \boldsymbol{A}_{3}=\left[\begin{array}{rrr} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{array}\right] . | |  |
| 111 | 2.23 | 58 | \left|u_{k+1}\right|=\left|d_{k+1}-l_{k} c_{k}\right|=\left|d_{k+1}-\frac{a_{k} c_{k}}{u_{k}}\right| \geq\left|d_{k+1}\right|-\frac{\left|a_{k}\right|\left|c_{k}\right|}{\left|u_{k}\right|}>\left|d_{k+1}\right|-\left|a_{k}\right| . | |  |
| 112 | 2.24 | 59 | R y^{\prime \prime}(x)=-F y(x), \quad y(0)=y(L)=0, | |  |
| 113 | 2.25 | 59 | u^{\prime \prime}(t)=-K u(t), \quad u(0)=u(1)=0, \quad K:=\frac{F L^{2}}{R} . | |  |
| 114 | | 59 | u^{\prime \prime}(j h) \approx \frac{u((j+1) h)-2 u(j h)+u((j-1) h)}{h^{2}}, \quad j=1, \ldots, m, | |  |
| 115 | | 59 | \frac{-v_{j-1}+2 v_{j}-v_{j+1}}{h^{2}}=K v_{j}, \quad j=1, \ldots, m, h=\frac{1}{m+1}, \quad v_{0}=v_{m+1}=0, | |  |
| 116 | 2.26 | 60 | \boldsymbol{T} \boldsymbol{v}=\lambda \boldsymbol{v}, \text { with } \boldsymbol{v}=\left[v_{1}, \ldots, v_{m}\right]^{T}, | |  |
| 117 | 2.27 | 60 | \boldsymbol{T}=\boldsymbol{T}_{m}:=\operatorname{tridiag}_{m}(-1,2,-1)-=\left[\begin{array}{rrrrr} 2 & -1 & 0 & & \\ -1 & 2 & -1 & & \\ 0 & \ddots & \ddots & \ddots & \\ & & & 0 \\ & & & -1 & 2-1 \\ & & & 0-1 & 2 \end{array}\right] \in \mathbb{R}^{m \times m} . | |  |
| 118 | | 60 | F=\frac{4 \sin ^{2}(\pi h / 2) R}{h^{2} L^{2}} . | |  |
| 119 | 2.28 | 60 | \boldsymbol{T}_{1}:=\operatorname{tridiag}(a, d, a) | |  |
| 120 | 2.29 | 60 | \boldsymbol{S}=\left[\sin \frac{j k \pi}{m+1}\right]_{j, k=1}^{m} \in \mathbb{R}^{m \times m} . | |  |
| 121 | 2.31 | 61 | \boldsymbol{S}=\left[\boldsymbol{s}_{1}, \boldsymbol{s}_{2}, \boldsymbol{s}_{3}\right]=\left[\begin{array}{c} \sin \frac{\pi}{4} \sin \frac{2 \pi}{4} \sin \frac{3 \pi}{4} \\ \sin \frac{2 \pi}{4} \sin \frac{4 \pi}{4} \sin \frac{6 \pi}{4} \\ \sin \frac{3 \pi}{4} \sin \frac{6 \pi}{4} \sin \frac{9 \pi}{4} \end{array}\right]=\left[\begin{array}{ccc} t & 1 & t \\ 1 & 0 & -1 \\ t & -1 & t \end{array}\right], \quad t:=\frac{1}{\sqrt{2}} . | |  |
| 122 | 2.30 | 61 | \begin{aligned} & \boldsymbol{s}_{j}=[\sin (j \pi h), \sin (2 j \pi h), \ldots, \sin (m j \pi h)]^{T}, \\ & \lambda_{j}=d+2 a \cos (j \pi h) . \end{aligned} | |  |
| 123 | 2.32 | 61 | \boldsymbol{s}_{j}^{T} \boldsymbol{s}_{k}=\frac{m+1}{2} \delta_{j, k}=\frac{1}{2 h} \delta_{j, k}, \quad j, k=1, \ldots, m . | |  |
| 124 | | 61 | \begin{aligned} \left(\boldsymbol{T}_{1} \boldsymbol{s}_{j}\right)_{k} & =\sum_{l=1}^{m} t_{k, l} \sin (l j \pi h) \\ & =a[\sin ((k-1) j \pi h)+\sin ((k+1) j \pi h)]+d \sin (k j \pi h) \\ & =2 a \cos (j \pi h) \sin (k j \pi h)+d \sin (k j \pi h)=\lambda_{j} s_{k, j} . \end{aligned} | |  |
| 125 | | 61 | \begin{aligned} \boldsymbol{s}_{j}^{T} \boldsymbol{s}_{j} & =\sum_{k=1}^{m} \sin ^{2}(k j \pi h)=\sum_{k=0}^{m} \sin ^{2}(k j \pi h)=\frac{1}{2} \sum_{k=0}^{m}(1-\cos (2 k j \pi h)) \\ & =\frac{m+1}{2}-\frac{1}{2} \sum_{k=0}^{m} \cos (2 k j \pi h)=\frac{m+1}{2}, \end{aligned} | |  |
| 126 | | 62 | \sum_{k=0}^{m} \cos (2 k j \pi h)+i \sum_{k=0}^{m} \sin (2 k j \pi h)=\sum_{k=0}^{m} e^{2 i k j \pi h}=\frac{e^{2 i(m+1) j \pi h}-1}{e^{2 i j \pi h}-1}=0, | |  |
| 127 | | 62 | \lambda \boldsymbol{y}^{*} \boldsymbol{x}=\boldsymbol{y}^{*} \boldsymbol{A} \boldsymbol{x}=\left(\boldsymbol{x}^{*} \boldsymbol{A}^{*} \boldsymbol{y}\right)^{*}=\left(\boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{y}\right)^{*}=\left(\mu \boldsymbol{x}^{*} \boldsymbol{y}\right)^{*}=\mu \boldsymbol{y}^{*} \boldsymbol{x}, | |  |
| 128 | | 62 | \boldsymbol{A}=\left[\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right] | |  |
| 129 | 89 | 63 | \boldsymbol{A}_{11}=[1], \quad \boldsymbol{A}_{12}=[2,3], \quad \boldsymbol{A}_{21}=\left[\begin{array}{l} 4 \\ 7 \end{array}\right], \quad \text { and } \boldsymbol{A}_{22}=\left[\begin{array}{ll} 5 & 6 \\ 8 & 9 \end{array}\right], | |  |
| 130 | | 63 | \boldsymbol{A} \boldsymbol{B}=\left[\boldsymbol{A} \boldsymbol{b}_{: 1}, \boldsymbol{A} \boldsymbol{b}_{: 2}, \ldots, \boldsymbol{A} \boldsymbol{b}_{: n}\right] . | |  |
| 131 | | 63 | \boldsymbol{A}=\boldsymbol{A} \boldsymbol{I}=\boldsymbol{A}\left[\boldsymbol{e}_{1}, \boldsymbol{e}_{2}, \ldots, \boldsymbol{e}_{p}\right]=\left[\boldsymbol{A} \boldsymbol{e}_{1}, \boldsymbol{A} \boldsymbol{e}_{2}, \ldots, \boldsymbol{A} \boldsymbol{e}_{p}\right] | |  |
| 132 | | 63 | \boldsymbol{A} \boldsymbol{B}=\left[\begin{array}{c} a_{1:}^{T} \\ a_{2:}^{T} \\ \vdots \\ a_{m:}^{T} \end{array}\right] \boldsymbol{B}=\left[\begin{array}{c} a_{1:}^{T} \boldsymbol{B} \\ a_{2:}^{T} \boldsymbol{B} \\ \vdots \\ a_{m:}^{T} \boldsymbol{B} \end{array}\right], | |  |
| 133 | | 63 | \boldsymbol{A} \boldsymbol{x}=x_{1} \boldsymbol{a}_{: 1}+x_{2} \boldsymbol{a}_{: 2}+\cdots+x_{p} \boldsymbol{a}_{: p} . | |  |
| 134 | | 64 | \boldsymbol{A}\left[\boldsymbol{B}_{1}, \boldsymbol{B}_{2}\right]=\left[\boldsymbol{A} \boldsymbol{B}_{1}, \boldsymbol{A} \boldsymbol{B}_{2}\right] . | |  |
| 135 | | 64 | \left[\begin{array}{l} A_{1} \\ A_{2} \end{array}\right] B=\left[\begin{array}{l} A_{1} B \\ A_{2} B \end{array}\right] . | |  |
| 136 | | 64 | \left[\boldsymbol{A}_{1}, \boldsymbol{A}_{2}\right]\left[\begin{array}{l} \boldsymbol{B}_{1} \\ \boldsymbol{B}_{2} \end{array}\right]=\left[\boldsymbol{A}_{1} \boldsymbol{B}_{1}+\boldsymbol{A}_{2} \boldsymbol{B}_{2}\right] . | |  |
| 137 | | 64 | \begin{aligned} (\boldsymbol{A} \boldsymbol{B})_{i j} & =\sum_{k=1}^{p} a_{i k} b_{k j}=\sum_{k=1}^{s} a_{i k} b_{k j}+\sum_{k=s+1}^{p} a_{i k} b_{k j} \\ & =\left(\boldsymbol{A}_{1} \boldsymbol{B}_{1}\right)_{i j}+\left(\boldsymbol{A}_{2} \boldsymbol{B}_{2}\right)_{i j}=\left(\boldsymbol{A}_{1} \boldsymbol{B}_{1}+\boldsymbol{A}_{2} \boldsymbol{B}_{2}\right)_{i j} . \end{aligned} | |  |
| 138 | | 64 | \left[\begin{array}{ll} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array}\right]\left[\begin{array}{ll} B_{11} & B_{12} \\ B_{21} & B_{22} \end{array}\right]=\left[\begin{array}{ll} A_{11} B_{11}+A_{12} B_{21} & A_{11} B_{12}+A_{12} B_{22} \\ A_{21} B_{11}+A_{22} B_{21} & A_{21} B_{12}+A_{22} B_{22} \end{array}\right], | |  |
| 139 | | 64 | \boldsymbol{A} \boldsymbol{B}=\left[\left[\begin{array}{ll} \boldsymbol{A}_{11} & \boldsymbol{A}_{12} \\ \boldsymbol{A}_{21} & \boldsymbol{A}_{22} \end{array}\right]\left[\begin{array}{l} \boldsymbol{B}_{11} \\ \boldsymbol{B}_{21} \end{array}\right],\left[\begin{array}{ll} \boldsymbol{A}_{11} & \boldsymbol{A}_{12} \\ \boldsymbol{A}_{21} & \boldsymbol{A}_{22} \end{array}\right]\left[\begin{array}{l} \boldsymbol{B}_{12} \\ \boldsymbol{B}_{22} \end{array}\right]\right] . | |  |
| 140 | | 65 | \boldsymbol{C}_{i j}=\sum_{k=1}^{s} \boldsymbol{A}_{i k} \boldsymbol{B}_{k j}, \quad i=1, \ldots, p, j=1, \ldots, q | |  |
| 141 | | 65 | \left[\begin{array}{ccc} \boldsymbol{A}_{11} & \cdots & \boldsymbol{A}_{1 s} \\ \vdots & & \vdots \\ \boldsymbol{A}_{p 1} & \cdots & \boldsymbol{A}_{p s} \end{array}\right]\left[\begin{array}{ccc} \boldsymbol{B}_{11} & \cdots & \boldsymbol{B}_{1 q} \\ \vdots & & \vdots \\ \boldsymbol{B}_{s 1} & \cdots & \boldsymbol{B}_{s q} \end{array}\right]=\left[\begin{array}{ccc} \boldsymbol{C}_{11} & \cdots & \boldsymbol{C}_{1 q} \\ \vdots & & \vdots \\ \boldsymbol{C}_{p 1} & \cdots & \boldsymbol{C}_{p q} \end{array}\right] . | |  |
| 142 | | 65 | \boldsymbol{A}=\left[\begin{array}{cc} \boldsymbol{A}_{11} & \boldsymbol{A}_{12} \\ \mathbf{0} & \boldsymbol{A}_{22} \end{array}\right] | |  |
| 143 | 2.33 | 65 | \boldsymbol{A}^{-1}=\left[\begin{array}{cc} \boldsymbol{A}_{11}^{-1} & \boldsymbol{C} \\ \mathbf{0} & \boldsymbol{A}_{22}^{-1} \end{array}\right], | |  |
| 144 | | 65 | \boldsymbol{B} \boldsymbol{A}=\left[\begin{array}{ll} \boldsymbol{B}_{11} & \boldsymbol{B}_{12} \\ \boldsymbol{B}_{21} & \boldsymbol{B}_{22} \end{array}\right]\left[\begin{array}{cc} \boldsymbol{A}_{11} & \boldsymbol{A}_{12} \\ \mathbf{0} & \boldsymbol{A}_{22} \end{array}\right]=\left[\begin{array}{cc} \boldsymbol{I} & \mathbf{0} \\ \mathbf{0} & \boldsymbol{I} \end{array}\right]=\boldsymbol{I} | |  |
| 145 | | 65 | \boldsymbol{B}_{11} \boldsymbol{A}_{11}=\boldsymbol{I}, \boldsymbol{B}_{21} \boldsymbol{A}_{11}=\mathbf{0}, \boldsymbol{B}_{21} \boldsymbol{A}_{12}+\boldsymbol{B}_{22} \boldsymbol{A}_{22}=\boldsymbol{I}, \boldsymbol{B}_{11} \boldsymbol{A}_{12}+\boldsymbol{B}_{12} \boldsymbol{A}_{22}=\mathbf{0} . | |  |
| 146 | | 66 | \boldsymbol{B}_{12}=\boldsymbol{C}=-\boldsymbol{A}_{11}^{-1} \boldsymbol{A}_{12} \boldsymbol{A}_{22}^{-1} . | |  |
| 147 | | 66 | \left[\begin{array}{cc} \boldsymbol{A}_{11}^{-1} & -\boldsymbol{A}_{11}^{-1} \boldsymbol{A}_{12} \boldsymbol{A}_{22}^{-1} \\ \mathbf{0} & \boldsymbol{A}_{22}^{-1} \end{array}\right]\left[\begin{array}{cc} \boldsymbol{A}_{11} & \boldsymbol{A}_{12} \\ \mathbf{0} & \boldsymbol{A}_{22} \end{array}\right]=\left[\begin{array}{ll} \boldsymbol{I} & \mathbf{0} \\ \mathbf{0} & \boldsymbol{I} \end{array}\right]=\boldsymbol{I} | |  |
| 148 | | 66 | \boldsymbol{A}=\left[\begin{array}{cc} \boldsymbol{A}_{k} & \boldsymbol{a}_{k} \\ \mathbf{0} & a_{k+1, k+1} \end{array}\right] | |  |
| 149 | | 66 | \boldsymbol{A}^{-1}=\left[\begin{array}{cc} \boldsymbol{A}_{k}^{-1} & \boldsymbol{c} \\ \mathbf{0} & a_{k+1, k+1}^{-1} \end{array}\right], | |  |
| 150 | | 67 | \max _{2 \leq j \leq n}\left|\mu_{j}\right| \leq \frac{3}{h^{2}} \max _{2 \leq i \leq n}\left|y_{i+1}-2 y_{i}+y_{i-1}\right| . | |  |
| 151 | 2.34 | 67 | \left[\begin{array}{ccccc} 2 & 1 & & & \\ 1 & 4 & 1 & & \\ & \ddots & \ddots & \ddots & \\ & & 1 & 4 & 1 \\ & & & 1 & 2 \end{array}\right]\left[\begin{array}{c} \mu_{1} \\ \mu_{2} \\ \vdots \\ \mu_{n} \\ \mu_{n+1} \end{array}\right]=\frac{6}{h^{2}}\left[\begin{array}{c} y_{2}-y_{1}-h s_{1} \\ \delta^{2} y_{2} \\ \delta^{2} y_{3} \\ \vdots \\ \delta^{2} y_{n-1} \\ \delta^{2} y_{n} \\ h s_{n+1}-y_{n+1}+y_{n} \end{array}\right], | |  |
| 152 | | 68 | \int_{a}^{b}\left(g^{\prime \prime}(x)\right)^{2} d x \leq \int_{a}^{b}\left(h^{\prime \prime}(x)\right)^{2} d x | |  |
| 153 | 2.35 | 68 | \int_{a}^{b} g^{\prime \prime} e^{\prime \prime}=0, \quad e:=h-g . | |  |
| 154 | | 68 | \int_{a}^{b} g^{\prime \prime \prime} e^{\prime}=\sum_{i=1}^{n} \int_{x_{i}}^{x_{i+1}} g^{\prime \prime \prime} e^{\prime}=\sum_{i=1}^{n} v_{i} \int_{x_{i}}^{x_{i+1}} e^{\prime}=\sum_{i=1}^{n} v_{i}\left(e\left(x_{i+1}\right)-e\left(x_{i}\right)\right)=0 . | |  |
| 155 | | 68 | \begin{aligned} \int_{a}^{b}\left(h^{\prime \prime}\right)^{2} & =\int_{a}^{b}\left(g^{\prime \prime}+e^{\prime \prime}\right)^{2} \\ & =\int_{a}^{b}\left(g^{\prime \prime}\right)^{2}+\int_{a}^{b}\left(e^{\prime \prime}\right)^{2}+2 \int_{a}^{b} g^{\prime \prime} e^{\prime \prime} \\ & =\int_{a}^{b}\left(g^{\prime \prime}\right)^{2}+\int_{a}^{b}\left(e^{\prime \prime}\right)^{2} \geq \int_{a}^{b}\left(g^{\prime \prime}\right)^{2} \end{aligned} | |  |
| 156 | | 71 | \delta^{2} f(x):=\frac{f(x+h)-2 f(x)+f(x-h)}{h^{2}}, \quad h>0, \quad f:[x-h, x+h] \rightarrow \mathbb{R} . | |  |
| 157 | | 71 | \delta^{2} f(x)=f^{\prime \prime}\left(\eta_{2}\right), \quad x-h<\eta_{2}<x+h . | |  |
| 158 | | 71 | \delta^{2} f(x)=f^{\prime \prime}(x)+\frac{h^{2}}{12} f^{(4)}\left(\eta_{4}\right), \quad x-h<\eta_{4}<x+h . | |  |
| 159 | 2.36 | 71 | \begin{aligned} -u^{\prime \prime}(x)+r(x) u^{\prime}(x)+q(x) u(x) & =f(x), \text { for } x \in[a, b], \\ u(a) & =g_{0}, \quad u(b)=g_{1} . \end{aligned} | |  |
| 160 | 2.37 | 71 | \frac{-v_{j-1}+2 v_{j}-v_{j+1}}{h^{2}}+r\left(x_{j}\right) \frac{v_{j+1}-v_{j-1}}{2 h}+q\left(x_{j}\right) v_{j}=f\left(x_{j}\right), \quad j=1, \ldots, m, | |  |
| 161 | | 71 | a_{j}=-1-\frac{h}{2} r\left(x_{j}\right), c_{j}=-1+\frac{h}{2} r\left(x_{j}\right), d_{j}=2+h^{2} q\left(x_{j}\right) \text {, } | |  |
| 162 | | 72 | b_{j}= \begin{cases}h^{2} f\left(x_{1}\right)-a_{1} g_{0}, & \text { if } j=1, \\ h^{2} f\left(x_{j}\right), & \text { if } 2 \leq j \leq m-1, \\ h^{2} f\left(x_{m}\right)-c_{m} g_{1}, & \text { if } j=m .\end{cases} | |  |
| 163 | | 72 | F=\frac{4 \sin ^{2}(\pi h / 2) R}{h^{2} L^{2}}=\frac{\pi^{2} R}{L^{2}}+O\left(h^{2}\right) . | |  |
| 164 | 2.38 | 73 | \boldsymbol{L}=\left[\begin{array}{ccccc} 1 & 0 & \cdots & \cdots & 0 \\ -\frac{1}{2} & 1 & \ddots & & \vdots \\ 0 & -\frac{2}{3} & 1 & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & -\frac{m-1}{m} & 1 \end{array}\right], \boldsymbol{U}=\left[\begin{array}{ccccc} 2 & -1 & 0 & \cdots & 0 \\ 0 & \frac{3}{2} & -1 & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ \vdots & \ddots & \frac{m}{m-1} & -1 \\ 0 & \cdots & \cdots & 0 & \frac{m+1}{m} \end{array}\right] . | |  |
| 165 | 2.39 | 73 | s_{i, j}=s_{j, i}=\frac{1}{m+1} j(m+1-i), \quad 1 \leq j \leq i \leq m . | |  |
| 166 | | 73 | \boldsymbol{A} \boldsymbol{B}^{T}=\boldsymbol{a}_{: 1} \boldsymbol{b}_{: 1}^{T}+\boldsymbol{a}_{: 2} \boldsymbol{b}_{: 2}^{T}+\cdots+\boldsymbol{a}_{: n} \boldsymbol{b}_{: n}^{T} . | |  |
| 167 | | 73 | \boldsymbol{A} \boldsymbol{X}=\boldsymbol{B} \quad \Longleftrightarrow \quad \boldsymbol{A} \boldsymbol{x}_{: j}=\boldsymbol{b}_{: j}, j=1, \ldots, p . | |  |
| 168 | | 74 | \boldsymbol{A}:=\left[\begin{array}{ll} \lambda & \boldsymbol{a}^{T} \\ \mathbf{0} & \boldsymbol{A}_{1} \end{array}\right], \quad \boldsymbol{B}:=\left[\begin{array}{ll} 1 & \mathbf{0}^{T} \\ \mathbf{0} & \boldsymbol{B}_{1} \end{array}\right], \quad \boldsymbol{C}:=\left[\begin{array}{ll} 1 & \mathbf{0}^{T} \\ \mathbf{0} & \boldsymbol{C}_{1} \end{array}\right], | |  |
| 169 | | 74 | \boldsymbol{C} \boldsymbol{A} \boldsymbol{B}=\left[\begin{array}{cc} \lambda & \boldsymbol{a}^{T} \boldsymbol{B}_{1} \\ \mathbf{0} & \boldsymbol{C}_{1} \boldsymbol{A}_{1} \boldsymbol{B}_{1} \end{array}\right] . | |  |
| 170 | | 75 | \begin{array}{ll} a_{11}^{(1)} x_{1}+a_{12}^{(1)} x_{2}+a_{13}^{(1)} x_{3}=b_{1}^{(1)}, & \mathrm{I} \\ a_{21}^{(1)} x_{1}+a_{22}^{(1)} x_{2}+a_{23}^{(1)} x_{3}=b_{2}^{(1)}, & \mathrm{II} \\ a_{31}^{(1)} x_{1}+a_{32}^{(1)} x_{2}+a_{33}^{(1)} x_{3}=b_{3}^{(1)} . & \mathrm{III} . \end{array} | |  |
| 171 | | 76 | \begin{aligned} a_{11}^{(1)} x_{1}+a_{12}^{(1)} x_{2}+a_{13}^{(1)} x_{3}=b_{1}^{(1)}, & \mathrm{I} \\ a_{22}^{(2)} x_{2}+a_{23}^{(2)} x_{3}=b_{2}^{(2)}, & \mathrm{II}^{\prime} \\ a_{32}^{(2)} x_{2}+a_{33}^{(2)} x_{3}=b_{3}^{(2)}, & \mathrm{III}^{\prime}, \end{aligned} | |  |
| 172 | | 76 | \begin{aligned} a_{11}^{(1)} x_{1}+a_{12}^{(1)} x_{2}+a_{13}^{(1)} x_{3}=b_{1}^{(1)}, & \mathrm{I} \\ a_{22}^{(2)} x_{2}+a_{23}^{(2)} x_{3}=b_{2}^{(2)}, & \mathrm{II}^{\prime} \\ a_{33}^{(3)} x_{3}=b_{3}^{(3)}, & \mathrm{III}^{\prime \prime}, \end{aligned} | |  |
| 173 | | 76 | \begin{aligned} & x_{3}=b_{3}^{(3)} / a_{33}^{(3)} \\ & x_{2}=\left(b_{2}^{(2)}-a_{23}^{(2)} x_{3}\right) / a_{22}^{(2)} \\ & x_{1}=\left(b_{1}^{(1)}-a_{12}^{(1)} x_{2}-a_{13}^{(1)} x_{3}\right) / a_{11}^{(1)} . \end{aligned} | |  |
| 174 | | 76 | \begin{aligned} \boldsymbol{L} \boldsymbol{U} & :=\left[\begin{array}{ccc} 1 & 0 & 0 \\ l_{21}^{(1)} & 1 & 0 \\ l_{31}^{(1)} & l_{32}^{(2)} & 1 \end{array}\right]\left[\begin{array}{ccc} a_{11}^{(1)} & a_{12}^{(1)} & a_{13}^{(1)} \\ 0 & a_{22}^{(2)} & a_{23}^{(2)} \\ 0 & 0 & a_{33}^{(3)} \end{array}\right] \\ = & {\left[\begin{array}{ccc} a_{11}^{(1)} & a_{12}^{(1)} & a_{13}^{(1)} \\ l_{21}^{(1)} a_{11}^{(1)} & l_{21}^{(1)} a_{12}^{(1)}+a_{22}^{(2)} & l_{21}^{(1)} a_{13}^{(1)}+a_{23}^{(2)} \\ l_{31}^{(1)} a_{11}^{(1)} & l_{31}^{(1)} a_{12}^{(1)}+l_{32}^{(2)} a_{22}^{(2)} & l_{31}^{(1)} a_{13}^{(1)}+l_{32}^{(2)} a_{23}^{(2)}+a_{33}^{(3)} \end{array}\right]=\left[\begin{array}{lll} a_{11}^{(1)} & a_{12}^{(1)} & a_{13}^{(1)} \\ a_{21}^{(1)} & a_{22}^{(1)} & a_{23}^{(1)} \\ a_{31}^{(1)} & a_{32}^{(1)} & a_{33}^{(1)} \end{array}\right]=\boldsymbol{A} . } \end{aligned} | |  |
| 175 | 3.1 | 77 | \boldsymbol{A}^{(k)}=\left[\begin{array}{ccc|ccccc} a_{1,1}^{(1)} & \cdots & a_{1, k-1}^{(1)} & a_{1, k}^{(1)} & \cdots & a_{1, j}^{(1)} & \cdots & a_{1, n}^{(1)} \\ & \ddots & \vdots & \vdots & & \vdots & & \vdots \\ & & a_{k-1, k-1}^{(k-1)} & a_{k-1, k}^{(k-1)} & \cdots & a_{k-1, j}^{(k-1)} & \cdots & a_{k-1, n}^{(k-1)} \\ \hline & & & a_{k, k}^{(k)} & \cdots & a_{k, j}^{(k)} & \cdots & a_{k, n}^{(k)} \\ & & & \vdots & & \vdots & & \vdots \\ & & & a_{i, k}^{(k)} & \cdots & a_{i, j}^{(k)} & \cdots & a_{i, n}^{(k)} \\ & & & \vdots & & \vdots & & \vdots \\ & & & a_{n, k}^{(k)} & \cdots & a_{n, j}^{(k)} & \cdots & a_{n, n}^{(k)} \end{array}\right] . | |  |
| 176 | | 77 | \text { for } \begin{aligned} i & =k+1: n \\ l_{i k}^{(k)} & =a_{i k}^{(k)} / a_{k k}^{(k)} \end{aligned} | |  |
| 177 | 3.2 | 77 | \text { for } j=k: n | |  |
| 178 | | 77 | a_{i j}^{(k+1)}=a_{i j}^{(k)}-l_{i k}^{(k)} a_{k j}^{(k)} | |  |
| 179 | | 78 | \boldsymbol{A}_{[k]}:=\boldsymbol{A}(1: k, 1: k)=\left[\begin{array}{ccc} a_{11} & \cdots & a_{k 1} \\ \vdots & & \vdots \\ a_{k 1} & \cdots & a_{k k} \end{array}\right] | |  |
| 180 | | 78 | b_{i, j}=a_{r_{i}, r_{j}}, \quad i, j=1, \ldots, k . | |  |
| 181 | | 78 | [1],[5],[9],\left[\begin{array}{ll} 1 & 2 \\ 4 & 5 \end{array}\right],\left[\begin{array}{ll} 1 & 3 \\ 7 & 9 \end{array}\right],\left[\begin{array}{ll} 5 & 6 \\ 8 & 9 \end{array}\right], \boldsymbol{A} . | |  |
| 182 | | 78 | [1],\left[\begin{array}{ll} 1 & 2 \\ 4 & 5 \end{array}\right], \boldsymbol{A} . | |  |
| 183 | 3.3 | 78 | \operatorname{det}\left(\boldsymbol{A}_{[k]}\right)=a_{11}^{(1)} a_{22}^{(2)} \cdots a_{k k}^{(k)}, \quad k=1, \ldots, n . | |  |
| 184 | 3.4 | 79 | \boldsymbol{L}=\left[\begin{array}{cccc} 1 & & & \\ l_{21}^{(1)} & 1 & & \\ \vdots & & \ddots & \\ l_{n 1}^{(1)} & l_{n 2}^{(2)} & \cdots & 1 \end{array}\right], \quad \boldsymbol{U}=\left[\begin{array}{ccc} a_{11}^{(1)} & \cdots & a_{1 n}^{(1)} \\ & \ddots & \vdots \\ & & a_{n n}^{(n)} \end{array}\right], | |  |
| 185 | | 79 | l_{i k}^{(k)} a_{k j}^{(k)}=a_{i j}^{(k)}-a_{i j}^{(k+1)} \text { for } k<\min (i, j) \text {, and } l_{i j}^{(k)} a_{j j}^{(j)}=a_{i j}^{(j)} \text { for } i>j . | |  |
| 186 | 3.5 | 79 | (\boldsymbol{L} \boldsymbol{U})_{i j}=\sum_{k=1}^{i-1} l_{i k}^{(k)} a_{k j}^{(k)}+a_{i j}^{(i)}=\sum_{k=1}^{i-1}\left(a_{i j}^{(k)}-a_{i j}^{(k+1)}\right)+a_{i j}^{(i)}=a_{i j}^{(1)}=a_{i j}, | |  |
| 187 | 3.6 | 79 | (\boldsymbol{L} \boldsymbol{U})_{i j}=\sum_{k=1}^{j-1} l_{i k}^{(k)} a_{k j}^{(k)}+l_{i j} a_{j j}^{(j)}=\sum_{k=1}^{j-1}\left(a_{i j}^{(k)}-a_{i j}^{(k+1)}\right)+a_{i j}^{(j)}=a_{i j} . | |  |
| 188 | | 80 | \left[\begin{array}{ccc} a_{11} & 0 & 0 \\ a_{21} & a_{22} & 0 \\ a_{31} & a_{32} & a_{33} \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]=\left[\begin{array}{l} b_{1} \\ b_{2} \\ b_{3} \end{array}\right] . | |  |
| 189 | 3.7 | 80 | x_{k}=\left(b_{k}-\sum_{j=1}^{k-1} a_{k, j} x_{j}\right) / a_{k k}, \quad k=1,2, \ldots, n . | |  |
| 190 | 3.8 | 80 | x_{k}=\left(b_{k}-\sum_{j=l_{k}}^{k-1} a_{k, j} x_{j}\right) / a_{k k}, \quad k=1,2, \ldots, n . | |  |
| 191 | | 80 | \left[\begin{array}{ccccc} a_{11} & 0 & 0 & 0 & 0 \\ a_{21} & a_{22} & 0 & 0 & 0 \\ 0 & a_{32} & a_{33} & 0 & 0 \\ 0 & 0 & a_{43} & a_{44} & 0 \\ 0 & 0 & 0 & a_{54} & a_{55} \end{array}\right], \quad\left[\begin{array}{ccccc} a_{11} & 0 & 0 & 0 & 0 \\ a_{21} & a_{22} & 0 & 0 & 0 \\ a_{31} & a_{32} & a_{33} & 0 & 0 \\ 0 & a_{42} & a_{43} & a_{44} & 0 \\ 0 & 0 & a_{53} & a_{54} & a_{55} \end{array}\right] | |  |
| 192 | | 81 | \left[\begin{array}{cccc} a_{k, k} & 0 & \cdots & 0 \\ a_{k+1, k} & a_{k+1, k+1} & \cdots & 0 \\ \vdots & & \ddots & \vdots \\ a_{n, k} & & \cdots & a_{n \times n} \end{array}\right]\left[\begin{array}{c} x_{k} \\ x_{k+1} \\ \vdots \\ x_{n} \end{array}\right]=\left[\begin{array}{c} b_{k} \\ b_{k+1} \\ \vdots \\ b_{n} \end{array}\right] . | |  |
| 193 | | 82 | \left[\begin{array}{cccc} a_{k+1, k+1} & 0 & \cdots & 0 \\ a_{k+2, k+1} & a_{k+2, k+2} & \cdots & 0 \\ \vdots & & \ddots & \vdots \\ a_{n, k+1} & & \cdots & a_{n, n} \end{array}\right]\left[\begin{array}{c} x_{k+1} \\ \vdots \\ x_{n} \end{array}\right]=\left[\begin{array}{c} b_{k+1} \\ \vdots \\ b_{n} \end{array}\right]-x_{k}\left[\begin{array}{c} a_{k+1, k} \\ \vdots \\ a_{n, k} \end{array}\right] . | |  |
| 194 | | 82 | b((k+1): n)=b((k+1): n)-x(k) * A((k+1): n, k) . | |  |
| 195 | 3.9 | 82 | \begin{figure} \captionsetup{labelformat=empty} \caption{Listing 3.3 cforwardsolve} \[ N_{L U}:=\frac{2}{3} n^{3}-\frac{1}{2} n^{2}-\frac{1}{6} n \] \end{figure} | |  |
| 196 | | 83 | M+D+A+S=\frac{2}{3} n(n-1)\left(n-\frac{1}{2}\right)+\frac{1}{2} n(n-1)=N_{L U} | |  |
| 197 | | 83 | M+S=2 \sum_{k=1}^{n-1}(n-k)^{2} \approx 2 \int_{1}^{n-1}(n-k)^{2} d k \approx 2 \int_{0}^{n}(n-k)^{2} d k=\frac{2}{3} n^{3} | |  |
| 198 | | 83 | N_{S}=2 \sum_{k=1}^{n}(2 k-1) \approx 2 \int_{1}^{n}(2 k-1) d k \approx 4 \int_{0}^{n} k d k=2 n^{2} | |  |
| 199 | | 84 | \boldsymbol{P}=\boldsymbol{I}(:, \boldsymbol{p})=\left[\boldsymbol{e}_{i_{1}}, \boldsymbol{e}_{i_{2}}, \ldots, \boldsymbol{e}_{i_{n}}\right] \in \mathbb{R}^{n \times n}, | |  |
| 200 | 3.10 | 84 | \boldsymbol{A} \boldsymbol{P}=\boldsymbol{A}(:, \boldsymbol{p}), \quad \boldsymbol{P}^{T} \boldsymbol{A}=\boldsymbol{A}(\boldsymbol{p},:) . | |  |
| 201 | 3.11 | 85 | \boldsymbol{p}:=\boldsymbol{p}_{n}, \text { where } \boldsymbol{p}_{1}:=[1,2, \ldots, n]^{T}, \text { and } \boldsymbol{p}_{k+1}:=\boldsymbol{I}_{r_{k}, k} \boldsymbol{p}_{k} \text { for } k=1, \ldots, n-1 . | |  |
| 202 | 3.12 | 85 | \boldsymbol{P}^{T}=\boldsymbol{P}_{n-1} \cdots \boldsymbol{P}_{1}=\boldsymbol{I}(\boldsymbol{p},:), \quad \boldsymbol{P}=\boldsymbol{P}_{1} \boldsymbol{P}_{2} \cdots \boldsymbol{P}_{n-1}=\boldsymbol{I}(:, \boldsymbol{p}) . | |  |
| 203 | 3.13 | 85 | \begin{aligned} & \boldsymbol{p}=[1, \ldots, n]^{T} ; \\ & \text { for } k=1: n-1 \\ & \text { choose } r_{k} \geq k \text { so that } a_{p_{r_{k}}, k}^{(k)} \neq 0 \text {. } \\ & \qquad \boldsymbol{p}=I_{r_{k}, k} \boldsymbol{p} \end{aligned} | |  |
| 204 | | 85 | \begin{aligned} & \text { for } i=k+1: n \\ & \qquad \begin{array}{l} a_{p_{i}, k}^{(k)}=a_{p_{i}, k}^{(k)} / a_{p_{k}, k}^{(k)} \\ \text { for } j=k: n \\ \qquad a_{p_{i}, j}^{(k+1)}=a_{p_{i}, j}^{(k)}-a_{p_{i}, k}^{(k)} a_{p_{k}, j}^{(k)} \end{array} \end{aligned} | |  |
| 205 | 3.14 | 86 | \boldsymbol{L}=\left[\begin{array}{ccc} 1 & & \\ a_{p_{2}, 1}^{(1)} & 1 & \\ \vdots & & \ddots \\ a_{p_{n}, 1}^{(1)} & a_{p_{n}, 2}^{(2)} & \cdots 1 \end{array}\right], \quad \boldsymbol{U}=\left[\begin{array}{ccc} a_{p_{1}, 1}^{(1)} & \cdots & a_{p_{1}, n}^{(1)} \\ & \ddots & \vdots \\ & & a_{p_{n}, n}^{(n)} \end{array}\right] . | |  |
| 206 | | 86 | a_{p_{i}, k}^{(k)} a_{p_{k}, j}^{(k)}=a_{p_{i}, j}^{(k)}-a_{p_{i}, j}^{(k+1)} \text { for } k<\min (i, j), \text { and } a_{p_{i}, j}^{(k)} a_{p_{j}, j}^{(j)}=a_{p_{i}, j}^{(j)} \text { for } i>j . | |  |
| 207 | | 86 | \begin{aligned} (\boldsymbol{L} \boldsymbol{U})_{i j} & =\sum_{k=1}^{n} l_{i, k} u_{k j}=\sum_{k=1}^{i-1} a_{p_{i}, k}^{(k)} a_{p_{k}, j}^{(k)}+a_{p_{i}, j}^{(i)} \\ & =\sum_{k=1}^{i-1}\left(a_{p_{i}, j}^{(k)}-a_{p_{i}, j}^{(k+1)}\right)+a_{p_{i}, j}^{(i)}=a_{p_{i}, j}^{(1)}=a_{p_{i}, j}=\left(\boldsymbol{P}^{T} \boldsymbol{A}\right)_{i j}, \end{aligned} | |  |
| 208 | | 86 | \begin{aligned} (\boldsymbol{L} \boldsymbol{U})_{i j} & =\sum_{k=1}^{n} l_{i k}^{(k)} u_{k j}=\sum_{k=1}^{j-1} a_{p_{i}, k}^{(k)} a_{p_{k}, j}^{(k)}+a_{p_{i}, j}^{(k)} a_{p_{j}, j}^{(j)} \\ & =\sum_{k=1}^{j-1}\left(a_{p_{i}, j}^{(k)}-a_{p_{i}, j}^{(k+1)}\right)+a_{p_{i}, j}^{(j)}=a_{p_{i}, j}^{(1)}=a_{p_{i}, j}=\left(\boldsymbol{P}^{T} \boldsymbol{A}\right)_{i j} . \end{aligned} | |  |
| 209 | | 87 | \left|a_{r_{k}, k}^{(k)}\right|:=\max \left\{\left|a_{i, k}^{(k)}\right|: k \leq i \leq n\right\} | |  |
| 210 | | 87 | \begin{array}{r} 10^{-4} x_{1}+2 x_{2}=4 \\ x_{1}+x_{2}=3 \end{array} | |  |
| 211 | | 87 | \begin{aligned} 10^{-4} x_{1}+2 x_{2} & =4 \\ \left(1-2 \times 10^{4}\right) x_{2} & =3-4 \times 10^{4} \end{aligned} | |  |
| 212 | | 87 | x_{2}=\frac{-39997}{-19999} \approx 2, \quad x_{1}=\frac{4-2 x_{2}}{10^{-4}}=\frac{20000}{19999} \approx 1 . | |  |
| 213 | | 87 | \mathrm{fl}\left(x_{2}\right)=2, \quad \mathrm{fl}\left(x_{1}\right)=0 . | |  |
| 214 | | 87 | \begin{aligned} x_{1}+x_{2} & =3 \\ 10^{-4} x_{1}+2 x_{2} & =4 \end{aligned} | |  |
| 215 | | 87 | \begin{aligned} x_{1}+x_{2} & =3 \\ \left(2-10^{-4}\right) x_{2} & =4-3 \times 10^{-4} \end{aligned} | |  |
| 216 | | 88 | x_{2}=\frac{3.9997}{1.9999} \approx 2, \quad x_{1}=3-x_{2} \approx 1 . | |  |
| 217 | | 88 | \frac{\left|a_{r_{k}, k}^{(k)}\right|}{s_{k}}:=\max \left\{\frac{\left|a_{i, k}^{(k)}\right|}{s_{k}}: k \leq i \leq n\right\}, \quad s_{k}:=\max _{1 \leq j \leq n}\left|a_{k j}\right| . | |  |
| 218 | | 88 | a_{r_{k}, s_{k}}^{(k)}:=\max \left\{\left|a_{i, j}^{(k)}\right|: k \leq i, j \leq n\right\} | |  |
| 219 | | 88 | \boldsymbol{L}=\left[\begin{array}{ccc} l_{1,1} & \cdots & 0 \\ \vdots & \ddots & \vdots \\ l_{n, 1} & \cdots & l_{n, n} \end{array}\right], \quad \boldsymbol{U}=\left[\begin{array}{ccc} u_{1,1} & \cdots & u_{1, n} \\ \vdots & \ddots & \vdots \\ 0 & \cdots & u_{n, n} \end{array}\right] . | |  |
| 220 | | 89 | \left[\begin{array}{ll} a & b \\ c & d \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ l_{1} & 1 \end{array}\right]\left[\begin{array}{ll} u_{1} & u_{2} \\ 0 & u_{3} \end{array}\right]=\left[\begin{array}{cc} u_{1} & u_{2} \\ u_{1} l_{1} & u_{2} l_{1}+u_{3} \end{array}\right] | |  |
| 221 | 3.15 | 89 | u_{1}=a, \quad u_{2}=b, \quad a l_{1}=c, \quad b l_{1}+u_{3}=d . | |  |
| 222 | | 89 | \boldsymbol{A}_{1}:=\left[\begin{array}{cc} 2 & -1 \\ -1 & 2 \end{array}\right], \quad \boldsymbol{A}_{2}:=\left[\begin{array}{ll} 0 & 1 \\ 1 & 1 \end{array}\right], \quad \boldsymbol{A}_{3}:=\left[\begin{array}{ll} 0 & 1 \\ 0 & 2 \end{array}\right], \quad \boldsymbol{A}_{4}:=\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right] . | |  |
| 223 | | 89 | \boldsymbol{A}_{[k]}:=\left[\begin{array}{ccc} a_{11} & \cdots & a_{k 1} \\ \vdots & & \vdots \\ a_{k 1} & \cdots & a_{k k} \end{array}\right] | |  |
| 224 | 3.16 | 90 | \left[\begin{array}{cc} \boldsymbol{A}_{[k]} & \boldsymbol{B}_{k} \\ \boldsymbol{C}_{k} & \boldsymbol{F}_{k} \end{array}\right]=\left[\begin{array}{cc} \boldsymbol{L}_{[k]} & \mathbf{0} \\ \boldsymbol{M}_{k} & \boldsymbol{N}_{k} \end{array}\right]\left[\begin{array}{cc} \boldsymbol{U}_{[k]} & \boldsymbol{S}_{k} \\ \mathbf{0} & \boldsymbol{T}_{k} \end{array}\right]=\left[\begin{array}{cc} \boldsymbol{L}_{[k]} \boldsymbol{U}_{[k]} & \boldsymbol{L}_{[k]} \boldsymbol{S}_{k} \\ \boldsymbol{M}_{k} \boldsymbol{U}_{[k]} & \boldsymbol{M}_{k} \boldsymbol{S}_{k}+\boldsymbol{N}_{k} \boldsymbol{T}_{k} \end{array}\right], | |  |
| 225 | 3.17 | 90 | \boldsymbol{A}=\left[\begin{array}{cc} \boldsymbol{A}_{[n-1]} & \boldsymbol{c}_{n} \\ \boldsymbol{r}_{n}^{T} & a_{n n} \end{array}\right]=\left[\begin{array}{cc} \boldsymbol{L}_{n-1} & \mathbf{0} \\ \boldsymbol{l}_{n}^{T} & 1 \end{array}\right]\left[\begin{array}{cc} \boldsymbol{U}_{n-1} & \boldsymbol{u}_{n} \\ 0 & u_{n n} \end{array}\right]=\left[\begin{array}{cc} \boldsymbol{L}_{n-1} \boldsymbol{U}_{n-1} & \boldsymbol{L}_{n-1} \boldsymbol{u}_{n} \\ \boldsymbol{l}_{n}^{T} \boldsymbol{U}_{n-1} & \boldsymbol{l}_{n}^{T} \boldsymbol{u}_{n}+u_{n n} \end{array}\right], | |  |
| 226 | 3.18 | 90 | \boldsymbol{U}_{n-1}^{T} \boldsymbol{l}_{n}=\boldsymbol{r}_{n}, \quad \boldsymbol{L}_{n-1} \boldsymbol{u}_{n}=\boldsymbol{c}_{n}, \quad u_{n n}=a_{n n}-\boldsymbol{l}_{n}^{T} \boldsymbol{u}_{n} . | |  |
| 227 | | 91 | \boldsymbol{U}_{k-1}^{T} \boldsymbol{l}_{k}=\boldsymbol{r}_{k}, \quad\left[\begin{array}{cc} \boldsymbol{L}_{k-1} & \mathbf{0} \\ \boldsymbol{l}_{k}^{T} & 1 \end{array}\right]\left[\begin{array}{c} \boldsymbol{u}_{k} \\ u_{k k} \end{array}\right]=\left[\begin{array}{c} \boldsymbol{c}_{k} \\ a_{k k} \end{array}\right] . | |  |
| 228 | 3.19 | 92 | \boldsymbol{A}:=\left[\begin{array}{ccc} \boldsymbol{A}_{11} & \cdots & \boldsymbol{A}_{1 m} \\ \vdots & & \vdots \\ \boldsymbol{A}_{m 1} & \cdots & \boldsymbol{A}_{m m} \end{array}\right], | |  |
| 229 | 3.20 | 92 | \boldsymbol{A}=\boldsymbol{L} \boldsymbol{U}=\left[\begin{array}{cccc} \boldsymbol{I} & & & \\ \boldsymbol{L}_{21} & \boldsymbol{I} & & \\ \vdots & & \ddots & \\ \boldsymbol{L}_{m 1} & \cdots & \boldsymbol{L}_{m, m-1} & \boldsymbol{I} \end{array}\right]\left[\begin{array}{cccc} \boldsymbol{U}_{11} & & \cdots & \boldsymbol{U}_{1 m} \\ & \boldsymbol{U}_{22} & \cdots & \boldsymbol{U}_{2 m} \\ & & \ddots & \vdots \\ & & & \boldsymbol{U}_{m m} \end{array}\right] | |  |
| 230 | | 92 | \boldsymbol{A}_{\{k\}}:=\left[\begin{array}{ccc} \boldsymbol{A}_{11} & \cdots & \boldsymbol{A}_{1 k} \\ \vdots & & \vdots \\ \boldsymbol{A}_{k 1} & \cdots & \boldsymbol{A}_{k k} \end{array}\right] | |  |
| 231 | 3.21 | 92 | \begin{aligned} \boldsymbol{A} & =\left[\begin{array}{cc} \boldsymbol{A}_{\{m-1\}} & \boldsymbol{B} \\ \boldsymbol{C}^{T} & \boldsymbol{A}_{m m} \end{array}\right] \\ & =\left[\begin{array}{cc} \boldsymbol{L}_{\{m-1\}} & \mathbf{0} \\ \boldsymbol{C}^{T} \boldsymbol{U}_{\{m-1\}}^{-1} & \boldsymbol{I} \end{array}\right]\left[\begin{array}{cc} \boldsymbol{U}_{\{m-1\}} & \boldsymbol{L}_{\{m-1\}}^{-1} \boldsymbol{B} \\ 0 & \boldsymbol{A}_{m m}-\boldsymbol{C}^{T} \boldsymbol{U}_{\{m-1\}}^{-1} \boldsymbol{L}_{\{m-1\}}^{-1} \boldsymbol{B} \end{array}\right], \end{aligned} | |  |
| 232 | 3.22 | 93 | \begin{figure} \captionsetup{labelformat=empty} \caption{Listing 3.5 cbacksolve} \[ \boldsymbol{A}((k+1): n,(k+1): n) \boldsymbol{b}_{k}((k+1): n)=-\boldsymbol{A}((k+1): n, k) b_{k}(k), \quad k=1, \ldots, n-1 . \] \end{figure} | |  |
| 233 | 3.23 | 94 | \boldsymbol{A}(1: k, 1: k) \boldsymbol{b}_{k}(1: k)=\boldsymbol{I}(1: k, k), \quad k=n, n-1, \ldots, 1, | |  |
| 234 | 3.25 | 94 | \begin{aligned} 1+2+\cdots+m & =\frac{1}{2} m(m+1), \\ 1^{2}+2^{2}+\cdots+m^{2} & =\frac{1}{3} m\left(m+\frac{1}{2}\right)(m+1), \\ 1+3+5+\cdots+2 m-1 & =m^{2}, \\ 1 * 2+2 * 3+3 * 4+\cdots+(m-1) m & =\frac{1}{3}(m-1) m(m+1) . \end{aligned} | |  |
| 235 | 3.28 | 95 | \boldsymbol{C}:=\boldsymbol{U}+\boldsymbol{v} \boldsymbol{e}_{1}^{T}, | |  |
| 236 | 3.29 | 95 | \boldsymbol{P}:=\boldsymbol{I}_{1,2} \boldsymbol{I}_{2,3} \cdots \boldsymbol{I}_{n-1, n}, | |  |
| 237 | 3.30 | 96 | \boldsymbol{B}:=\boldsymbol{A}+\boldsymbol{w} \boldsymbol{e}_{1}^{T} . | |  |
| 238 | | 96 | \operatorname{det}\left(\boldsymbol{A}_{[k]}\right)=u_{11} u_{22} \cdots u_{k k} \text { for } k=1, \ldots, n . | |  |
| 239 | 3.31 | 97 | u_{11}=a_{11}, \quad u_{k k}=\frac{\operatorname{det}\left(\boldsymbol{A}_{[k]}\right)}{\operatorname{det}\left(\boldsymbol{A}_{[k-1]}\right)}, \text { for } k=2, \ldots, n . | |  |
| 240 | 3.32 | 97 | \boldsymbol{L}=\left[\begin{array}{cc} 1 & 0 \\ \boldsymbol{\ell}_{1} & \boldsymbol{L}_{2,2} \end{array}\right], \quad \boldsymbol{U}=\left[\begin{array}{cc} u_{1,1} & \boldsymbol{u}_{1}^{T} \\ 0 & \boldsymbol{U}_{2,2} \end{array}\right], | |  |
| 241 | 3.33 | 97 | \boldsymbol{B}:=\left[\begin{array}{cc} \left(1+\boldsymbol{u}_{1}^{T}\right. & \left.\boldsymbol{B}_{2,2} \boldsymbol{\ell}_{1}\right) / u_{1,1} \\ -\boldsymbol{u}_{2,2}^{T} & \boldsymbol{B}_{2,2} / u_{1,1} \\ & \boldsymbol{B}_{2,2} \end{array}\right] . | |  |
| 242 | | 98 | \boldsymbol{B}:=\left[\boldsymbol{a}_{1}, \ldots, \boldsymbol{a}_{p-1}, \boldsymbol{a}_{p+1}, \ldots, \boldsymbol{a}_{n}, \boldsymbol{b}\right] \in \mathbb{R}^{n \times n} . | |  |
| 243 | | 98 | A:=\left[\begin{array}{rr} -3 & -2 \\ 4 & 2 \end{array}\right] . | |  |
| 244 | | 98 | \boldsymbol{P}:=\left[\boldsymbol{e}_{n}, \boldsymbol{e}_{n-1}, \ldots, \boldsymbol{e}_{1}\right] . | |  |
| 245 | | 100 | \left[\begin{array}{ll} a & \bar{b} \\ b & d \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ l_{1} & 1 \end{array}\right]\left[\begin{array}{cc} d_{1} & 0 \\ 0 & d_{2} \end{array}\right]\left[\begin{array}{cc} 1 & \overline{l_{1}} \\ 0 & 1 \end{array}\right]=\left[\begin{array}{cc} d_{1} & d_{1} \overline{l_{1}} \\ d_{1} l_{1} & d_{1}\left|l_{1}\right|^{2}+d_{2} \end{array}\right] | |  |
| 246 | 4.1 | 101 | d_{1}=a . \quad a l_{1}=b, \quad d_{2}=d-a\left|l_{1}\right|^{2} . | |  |
| 247 | 4.2 | 101 | \boldsymbol{A}=\left[\begin{array}{cc} \boldsymbol{A}_{[k]} & \boldsymbol{B}_{k}^{*} \\ \boldsymbol{B}_{k} & \boldsymbol{F}_{k} \end{array}\right]=\left[\begin{array}{cc} \boldsymbol{L}_{[k]} & \mathbf{0} \\ \boldsymbol{M}_{k} & \boldsymbol{N}_{k} \end{array}\right]\left[\begin{array}{cc} \boldsymbol{D}_{[k]} & \mathbf{0} \\ \mathbf{0} & \boldsymbol{E}_{k} \end{array}\right]\left[\begin{array}{cc} \boldsymbol{L}_{[k]}^{*} & \boldsymbol{M}_{k}^{*} \\ \mathbf{0} & \boldsymbol{N}_{k}^{*} \end{array}\right]=\boldsymbol{L} \boldsymbol{D} \boldsymbol{U}, | |  |
| 248 | 4.3 | 101 | \begin{aligned} \boldsymbol{A}=\left[\begin{array}{cc} \boldsymbol{A}_{[n-1]} & \boldsymbol{a}_{n} \\ \boldsymbol{a}_{n}^{*} & a_{n n} \end{array}\right] & =\left[\begin{array}{cc} \boldsymbol{L}_{n-1} & \mathbf{0} \\ \boldsymbol{l}_{n}^{*} & 1 \end{array}\right]\left[\begin{array}{cc} \boldsymbol{D}_{n-1} & \mathbf{0} \\ 0 & d_{n n} \end{array}\right]\left[\begin{array}{cc} \boldsymbol{L}_{n-1}^{*} & \boldsymbol{l}_{n} \\ \mathbf{0}^{*} & 1 \end{array}\right] \\ & =\left[\begin{array}{cc} \boldsymbol{L}_{n-1} \boldsymbol{D}_{n-1} \boldsymbol{L}_{n-1}^{*} & \boldsymbol{L}_{n-1} \boldsymbol{D}_{n-1} \boldsymbol{l}_{n} \\ \boldsymbol{l}_{n}^{*} \boldsymbol{D}_{n-1} \boldsymbol{L}_{n-1}^{*} & \boldsymbol{l}_{n}^{*} \boldsymbol{D}_{n-1} \boldsymbol{l}_{n}+d_{n n} \end{array}\right] \end{aligned} | |  |
| 249 | 4.4 | 102 | \boldsymbol{a}_{n}=\boldsymbol{L}_{n-1} \boldsymbol{D}_{n-1} \boldsymbol{l}_{n}, \quad a_{n n}=\boldsymbol{l}_{n}^{*} \boldsymbol{D}_{n-1} \boldsymbol{l}_{n}+d_{n n} . | |  |
| 250 | | 102 | \left[\begin{array}{ll} 3 & 1 \\ 1 & 3 \end{array}\right]=\left[\begin{array}{cc} 1 & 0 \\ 1 / 3 & 1 \end{array}\right]\left[\begin{array}{cc} 3 & 0 \\ 0 & 8 / 3 \end{array}\right]\left[\begin{array}{cc} 1 & 1 / 3 \\ 0 & 1 \end{array}\right] | |  |
| 251 | | 102 | f(\boldsymbol{x})=\boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{x}=\sum_{i=1}^{n} \sum_{j=1}^{n} a_{i j} \bar{x}_{i} x_{j} | |  |
| 252 | | 103 | \begin{aligned} z^{*} A z & =(x-i y)^{T} A(x+i y)=x^{T} A x-i y^{T} A x+i x^{T} A y-i^{2} y^{T} A y \\ & =x^{T} A x+y^{T} A y, \end{aligned} | |  |
| 253 | | 103 | \nabla f(\boldsymbol{x})=\left[\begin{array}{c} \frac{\partial f(\boldsymbol{x})}{\partial x_{1}} \\ \vdots \\ \frac{\partial f(\boldsymbol{x})}{\partial x_{n}} \end{array}\right] \in \mathbb{R}^{n}, \quad H f(\boldsymbol{x})=\left[\begin{array}{ccc} \frac{\partial^{2} f(\boldsymbol{x})}{\partial x_{1} \partial x_{1}} & \ldots & \frac{\partial^{2} f(\boldsymbol{x})}{\partial x_{1} \partial x_{n}} \\ \vdots & & \vdots \\ \frac{\partial^{2} f(\boldsymbol{x})}{\partial x_{n} \partial x_{1}} & \ldots & \frac{\partial^{2} f(\boldsymbol{x})}{\partial x_{n} \partial x_{n}} \end{array}\right] \in \mathbb{R}^{n \times n} . | |  |
| 254 | | 104 | \begin{aligned} \boldsymbol{x}^{T} \boldsymbol{T} \boldsymbol{x} & =2 \sum_{i=1}^{n} x_{i}^{2}-\sum_{i=1}^{n-1} x_{i} x_{i+1}-\sum_{i=2}^{n} x_{i-1} x_{i} \\ & =\sum_{i=1}^{n-1} x_{i}^{2}-2 \sum_{i=1}^{n-1} x_{i} x_{i+1}+\sum_{i=1}^{n-1} x_{i+1}^{2}+x_{1}^{2}+x_{n}^{2} \\ & =x_{1}^{2}+x_{n}^{2}+\sum_{i=1}^{n-1}\left(x_{i+1}-x_{i}\right)^{2} . \end{aligned} | |  |
| 255 | 4.5 | 104 | \boldsymbol{A}(\boldsymbol{r}, \boldsymbol{r})=\boldsymbol{X}^{*} \boldsymbol{A} \boldsymbol{X}, \quad \boldsymbol{X}:=\left[\boldsymbol{e}_{r_{1}}, \ldots, \boldsymbol{e}_{r_{k}}\right] \in \mathbb{C}^{n \times k} . | |  |
| 256 | 4.6 | 104 | \boldsymbol{y}^{*} \boldsymbol{B} \boldsymbol{y}=\boldsymbol{y}^{*} \boldsymbol{X}^{*} \boldsymbol{A} \boldsymbol{X} \boldsymbol{y}=\boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{x} \geq 0, \quad \boldsymbol{y} \in \mathbb{C}^{k}, \quad \boldsymbol{x}:=\boldsymbol{X} \boldsymbol{y} . | |  |
| 257 | 0 | 105 | \left[\begin{array}{cc} 2 & -1 \\ -1 & 2 \end{array}\right]=\left[\begin{array}{cc} 1 & 0 \\ -\frac{1}{2} & 1 \end{array}\right]\left[\begin{array}{ll} 2 & 0 \\ 0 & \frac{3}{2} \end{array}\right]\left[\begin{array}{cc} 1 & -\frac{1}{2} \\ 0 & 1 \end{array}\right]=\left[\begin{array}{cc} \sqrt{2} & 0 \\ -1 / \sqrt{2} & \sqrt{3 / 2} \end{array}\right]\left[\begin{array}{cc} \sqrt{2} & -1 / \sqrt{2} \\ 0 & \sqrt{3 / 2} \end{array}\right] . | |  |
| 258 | 4.7 | 106 | \begin{figure} \captionsetup{labelformat=empty} \caption{Listing 4.2 bandcholesky} \[ 0<(\leq)\left(\alpha \boldsymbol{e}_{i}+\beta \boldsymbol{e}_{j}\right)^{*} \boldsymbol{A}\left(\alpha \boldsymbol{e}_{i}+\beta \boldsymbol{e}_{j}\right)=|\alpha|^{2} a_{i i}+|\beta|^{2} a_{j j}+2 \operatorname{Re}\left(\bar{\alpha} \beta a_{i j}\right) . \] \end{figure} | |  |
| 259 | | 106 | 0<\left|a_{i j}\right|^{2} a_{i i}+a_{i i}^{2} a_{j j}-2\left|a_{i j}\right|^{2} a_{i i}=a_{i i}\left(a_{i i} a_{j j}-\left|a_{i j}\right|^{2}\right) . | |  |
| 260 | | 106 | \left|a_{i j}\right|=\left|b_{i j}\right|<\sqrt{b_{i i} b_{j j}}=\sqrt{\left(a_{i i}+\varepsilon\right)\left(a_{j j}+\varepsilon\right)}, \quad i \neq j . | |  |
| 261 | | 107 | \boldsymbol{A}_{1}=\left[\begin{array}{ll} 0 & 1 \\ 1 & 1 \end{array}\right], \quad \boldsymbol{A}_{2}=\left[\begin{array}{ll} 1 & 2 \\ 2 & 2 \end{array}\right], \quad \boldsymbol{A}_{3}=\left[\begin{array}{rr} -2 & 1 \\ 1 & 2 \end{array}\right] . | |  |
| 262 | | 107 | \boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{x}=z^{*} \boldsymbol{U}^{*} \boldsymbol{A} \boldsymbol{U} \boldsymbol{z}=z^{*} \operatorname{diag}\left(\lambda_{1}, \ldots, \lambda_{n}\right) \boldsymbol{z}=\sum_{j=1}^{n} \lambda_{j}\left|z_{j}\right|^{2} \geq 0 . | |  |
| 263 | | 108 | \boldsymbol{A}=\boldsymbol{B B}^{*}, \quad \boldsymbol{B}=\left[\begin{array}{cc} 1 & 0 \\ 1 / 3 & 1 \end{array}\right]\left[\begin{array}{cc} \sqrt{3} & 0 \\ 0 & \sqrt{8 / 3} \end{array}\right] . | |  |
| 264 | 4.8 | 109 | \boldsymbol{A}=\left[\begin{array}{ll} \alpha & \boldsymbol{v}^{*} \\ \boldsymbol{v} & \boldsymbol{B} \end{array}\right], \quad \alpha \in \mathbb{C}, \quad \boldsymbol{v} \in \mathbb{C}^{n-1}, \quad \boldsymbol{B} \in \mathbb{C}^{(n-1) \times(n-1)} . | |  |
| 265 | 4.9 | 109 | \begin{aligned} 0 \leq \boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{x} & =\left[-\boldsymbol{y}^{*} \boldsymbol{v} / \alpha, \boldsymbol{y}^{*}\right]\left[\begin{array}{ll} \alpha & \boldsymbol{v}^{*} \\ \boldsymbol{v} & \boldsymbol{B} \end{array}\right]\left[\begin{array}{c} -\boldsymbol{v}^{*} \boldsymbol{y} / \alpha \\ \boldsymbol{y} \end{array}\right] \\ & =\left[0,-\left(\boldsymbol{y}^{*} \boldsymbol{v}\right) \boldsymbol{v}^{*} / \alpha+\boldsymbol{y}^{*} \boldsymbol{B}\right]\left[\begin{array}{c} -\boldsymbol{v}^{*} \boldsymbol{y} / \alpha \\ \boldsymbol{y} \end{array}\right] \\ & =-\boldsymbol{y}^{*} \boldsymbol{v} \boldsymbol{v}^{*} \boldsymbol{y} / \alpha+\boldsymbol{y}^{*} \boldsymbol{B} \boldsymbol{y}=\boldsymbol{y}^{*} \boldsymbol{C} \boldsymbol{y} . \end{aligned} | |  |
| 266 | 4.10 | 109 | \boldsymbol{L}^{*}:=\left[\begin{array}{cc} \beta & \boldsymbol{v}^{*} / \beta \\ \mathbf{0} & \boldsymbol{L}_{1}^{*} \end{array}\right], \quad \beta:=\sqrt{\alpha}, | |  |
| 267 | | 109 | \boldsymbol{L} \boldsymbol{L}^{*}=\left[\begin{array}{cc} \beta & \mathbf{0} \\ \boldsymbol{v} / \beta & \boldsymbol{L}_{1} \end{array}\right]\left[\begin{array}{cc} \beta & \boldsymbol{v}^{*} / \beta \\ \mathbf{0} & \boldsymbol{L}_{1}^{*} \end{array}\right]=\left[\begin{array}{ll} \alpha & \boldsymbol{v}^{*} \\ \boldsymbol{v} & \boldsymbol{B} \end{array}\right]=\boldsymbol{A} | |  |
| 268 | | 110 | \boldsymbol{L} \boldsymbol{L}^{*}=\left[\begin{array}{ll} 0 & \mathbf{0}^{*} \\ \mathbf{0} & \boldsymbol{L}_{1} \end{array}\right]\left[\begin{array}{ll} 0 & \mathbf{0}^{*} \\ \mathbf{0} & \boldsymbol{L}_{1}^{*} \end{array}\right]=\left[\begin{array}{ll} 0 & \mathbf{0}^{*} \\ \mathbf{0} & \boldsymbol{B} \end{array}\right]=\boldsymbol{A} . | |  |
| 269 | 4.11 | 110 | \begin{aligned} & \text { if } A(1,1)>0 \\ & A(1,1)=\sqrt{A(1,1)} \\ & A(2: n, 1)=A(2: n, 1) / A(1,1) \\ & \text { for } j=2: n \\ & \quad A(j: n, j)=A(j: n, j)-A(j, 1) * A(j: n, 1) \end{aligned} | |  |
| 270 | | 113 | \boldsymbol{A}[a]:=\left[\begin{array}{cc} 2 & 2-a \\ a & 1 \end{array}\right], \quad a \in \mathbb{R} | |  |
| 271 | | 113 | \boldsymbol{x}^{T} \boldsymbol{A} \boldsymbol{x}=2 x_{1}^{2}+(2-a) x_{1} x_{2}+a x_{2} x_{1}+x_{2}^{2}=x_{1}^{2}+\left(x_{1}+x_{2}\right)^{2}>0 . | |  |
| 272 | | 113 | A=\left(\begin{array}{ccccc} 1 & -1 & 0 & \cdots & 0 \\ -1 & 2 & -1 & \ddots & \vdots \\ 0 & \ddots & \ddots & \ddots & 0 \\ \vdots & \ddots & -1 & 2 & -1 \\ 0 & \cdots & 0 & -1 & 2 \end{array}\right) . | |  |
| 273 | | 114 | \boldsymbol{A}=\boldsymbol{L}\left(\boldsymbol{I}+\boldsymbol{v} \boldsymbol{v}^{T}\right) \boldsymbol{L}^{T}, | |  |
| 274 | | 114 | \boldsymbol{A}_{+}=\left[\begin{array}{cc} \boldsymbol{A} & \boldsymbol{a} \\ \boldsymbol{a}^{T} & \alpha \end{array}\right], | |  |
| 275 | | 114 | \boldsymbol{L}_{+}=\left[\begin{array}{cc} \boldsymbol{L} & 0 \\ \boldsymbol{y}^{T} & \lambda \end{array}\right], | |  |
| 276 | | 117 | \langle\boldsymbol{x}, a \boldsymbol{y}+b \boldsymbol{z}\rangle=\overline{\langle a \boldsymbol{y}+b \boldsymbol{z}, \boldsymbol{x}\rangle}=\overline{a\langle\boldsymbol{y}, \boldsymbol{x}\rangle+b\langle\boldsymbol{z}, \boldsymbol{x}\rangle}=\bar{a} \overline{\langle\boldsymbol{y}, \boldsymbol{x}\rangle}+\overline{b\langle\boldsymbol{z}, \boldsymbol{x}\rangle} | |  |
| 277 | 5.1 | 117 | \langle\boldsymbol{x}, a \boldsymbol{y}+b \boldsymbol{z}\rangle=\bar{a}\langle\boldsymbol{x}, \boldsymbol{y}\rangle+\bar{b}\langle\boldsymbol{x}, \boldsymbol{z}\rangle, \quad\langle a \boldsymbol{x}, a \boldsymbol{y}\rangle=|a|^{2}\langle\boldsymbol{x}, \boldsymbol{y}\rangle . | |  |
| 278 | | 117 | \langle\boldsymbol{x}, \boldsymbol{y}\rangle=\langle\boldsymbol{y}, \boldsymbol{x}\rangle \quad \text { (symmetry). } | |  |
| 279 | | 117 | \langle\boldsymbol{x}, \boldsymbol{y}\rangle:=\boldsymbol{y}^{*} \boldsymbol{x}=\boldsymbol{x}^{T} \overline{\boldsymbol{y}}=\sum_{j=1}^{n} x_{j} \overline{y_{j}} . | |  |
| 280 | 5.2 | 117 | \|\cdot\|: \mathcal{V} \rightarrow \mathbb{R}, \quad \boldsymbol{x} \longmapsto\|\boldsymbol{x}\|:=\sqrt{\langle\boldsymbol{x}, \boldsymbol{x}\rangle} | |  |
| 281 | 5.3 | 118 | |\langle\boldsymbol{x}, \boldsymbol{y}\rangle| \leq\|\boldsymbol{x}\|\|\boldsymbol{y}\|, | |  |
| 282 | | 118 | \boldsymbol{z}:=\boldsymbol{x}-a \boldsymbol{y}, \quad a:=\frac{\langle\boldsymbol{x}, \boldsymbol{y}\rangle}{\langle\boldsymbol{y}, \boldsymbol{y}\rangle} . | |  |
| 283 | 5.4 | 118 | \langle a \boldsymbol{y}, \boldsymbol{z}\rangle+\langle\boldsymbol{z}, a \boldsymbol{y}\rangle=a \overline{\langle\boldsymbol{z}, \boldsymbol{y}\rangle}+\bar{a}\langle\boldsymbol{z}, \boldsymbol{y}\rangle=0 . | |  |
| 284 | | 118 | \begin{aligned} \|\boldsymbol{x}\|^{2} & =\langle\boldsymbol{x}, \boldsymbol{x}\rangle=\langle\boldsymbol{z}+a \boldsymbol{y}, \boldsymbol{z}+a \boldsymbol{y}\rangle \\ & \stackrel{(5.4)}{=}\langle\boldsymbol{z}, \boldsymbol{z}\rangle+\langle a \boldsymbol{y}, a \boldsymbol{y}\rangle \stackrel{(5.1)}{=}\|\boldsymbol{z}\|^{2}+|a|^{2}\|\boldsymbol{y}\|^{2} \\ & \geq|a|^{2}\|\boldsymbol{y}\|^{2}=\frac{|\langle\boldsymbol{x}, \boldsymbol{y}\rangle|^{2}}{\|\boldsymbol{y}\|^{2}} . \end{aligned} | |  |
| 285 | 5.5 | 119 | \|\boldsymbol{x}+a \boldsymbol{y}\|^{2}=\|\boldsymbol{x}\|^{2}+a\langle\boldsymbol{y}, \boldsymbol{x}\rangle+\bar{a}\langle\boldsymbol{x}, \boldsymbol{y}\rangle+|a|^{2}\|\boldsymbol{y}\|^{2}, \quad a \in \mathbb{C}, \quad \boldsymbol{x}, \boldsymbol{y} \in \mathcal{V} . | |  |
| 286 | | 119 | \|\boldsymbol{x}+\boldsymbol{y}\|^{2} \leq\|\boldsymbol{x}\|^{2}+2\|\boldsymbol{x}\|\|\boldsymbol{y}\|+\|\boldsymbol{y}\|^{2}=(\|\boldsymbol{x}\|+\|\boldsymbol{y}\|)^{2} . | |  |
| 287 | 5.6 | 119 | \cos \theta=\frac{\langle\boldsymbol{x}, \boldsymbol{y}\rangle}{\|\boldsymbol{x}\|\|\boldsymbol{y}\|} . | |  |
| 288 | 5.7 | 119 | \|\boldsymbol{x}+\boldsymbol{y}\|^{2}=\|\boldsymbol{x}\|^{2}+\|\boldsymbol{y}\|^{2}, \quad \text { if } \quad \boldsymbol{x} \perp \boldsymbol{y} . | |  |
| 289 | 5.8 | 120 | \boldsymbol{v}_{1}:=\boldsymbol{s}_{1}, \quad \boldsymbol{v}_{j}:=\boldsymbol{s}_{j}-\sum_{i=1}^{j-1} \frac{\left\langle\boldsymbol{s}_{j}, \boldsymbol{v}_{i}\right\rangle}{\left\langle\boldsymbol{v}_{i}, \boldsymbol{v}_{i}\right\rangle} \boldsymbol{v}_{i}, \quad j=2, \ldots, k . | |  |
| 290 | | 120 | \left\{\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{k}\right\}:=\left\{\frac{\boldsymbol{v}_{1}}{\left\|\boldsymbol{v}_{1}\right\|}, \ldots, \frac{\boldsymbol{v}_{k}}{\left\|\boldsymbol{v}_{k}\right\|}\right\} | |  |
| 291 | | 120 | \left\langle\boldsymbol{v}_{j}, \boldsymbol{v}_{l}\right\rangle=\left\langle\boldsymbol{s}_{j}, \boldsymbol{v}_{l}\right\rangle-\sum_{i=1}^{j-1} \frac{\left\langle\boldsymbol{s}_{j}, \boldsymbol{v}_{i}\right\rangle}{\left\langle\boldsymbol{v}_{i}, \boldsymbol{v}_{i}\right\rangle}\left\langle\boldsymbol{v}_{i}, \boldsymbol{v}_{l}\right\rangle=\left\langle\boldsymbol{s}_{j}, \boldsymbol{v}_{l}\right\rangle-\frac{\left\langle\boldsymbol{s}_{j}, \boldsymbol{v}_{l}\right\rangle}{\left\langle\boldsymbol{v}_{l}, \boldsymbol{v}_{l}\right\rangle}\left\langle\boldsymbol{v}_{l}, \boldsymbol{v}_{l}\right\rangle=0 | |  |
| 292 | | 121 | \operatorname{dim}(\mathcal{S} \oplus \mathcal{T})=\operatorname{dim}(\mathcal{S})+\operatorname{dim}(\mathcal{T}) . | |  |
| 293 | 5.9 | 122 | \left\langle\boldsymbol{s}_{0}, \boldsymbol{s}\right\rangle=\langle\boldsymbol{v}, \boldsymbol{s}\rangle, \quad \text { for all } \boldsymbol{s} \in \mathcal{S}, \quad\left\langle\boldsymbol{t}_{0}, \boldsymbol{t}\right\rangle=\langle\boldsymbol{v}, \boldsymbol{t}\rangle, \quad \text { for all } \boldsymbol{t} \in \mathcal{T} . | |  |
| 294 | 5.10 | 122 | \boldsymbol{s}_{0}=\sum_{i=1}^{k} \frac{\left\langle\boldsymbol{v}, \boldsymbol{v}_{i}\right\rangle}{\left\langle\boldsymbol{v}_{i}, \boldsymbol{v}_{i}\right\rangle} \boldsymbol{v}_{i} . | |  |
| 295 | | 122 | \left\langle\boldsymbol{s}_{0}, \boldsymbol{v}_{j}\right\rangle=\left\langle\sum_{i=1}^{k} \frac{\left\langle\boldsymbol{v}, \boldsymbol{v}_{i}\right\rangle}{\left\langle\boldsymbol{v}_{i}, \boldsymbol{v}_{i}\right\rangle} \boldsymbol{v}_{i}, \boldsymbol{v}_{j}\right\rangle=\sum_{i=1}^{k} \frac{\left\langle\boldsymbol{v}, \boldsymbol{v}_{i}\right\rangle}{\left\langle\boldsymbol{v}_{i}, \boldsymbol{v}_{i}\right\rangle}\left\langle\boldsymbol{v}_{i}, \boldsymbol{v}_{j}\right\rangle=\left\langle\boldsymbol{v}, \boldsymbol{v}_{j}\right\rangle . | |  |
| 296 | 5.11 | 123 | \left\|\boldsymbol{v}-\boldsymbol{s}_{0}\right\|<\|\boldsymbol{v}-\boldsymbol{s}\|, \text { for all } \boldsymbol{s} \in \mathcal{S}, \boldsymbol{s} \neq \boldsymbol{s}_{0} . | |  |
| 297 | | 123 | \|\boldsymbol{v}-\boldsymbol{s}\|^{2}=\left\|\boldsymbol{v}-\boldsymbol{s}_{0}+\boldsymbol{u}\right\|^{2}=\left\|\boldsymbol{v}-\boldsymbol{s}_{0}\right\|^{2}+\|\boldsymbol{u}\|^{2}>\left\|\boldsymbol{v}-\boldsymbol{s}_{0}\right\|^{2} . | |  |
| 298 | | 123 | \langle\boldsymbol{x}, \overline{\boldsymbol{A}} \boldsymbol{y}\rangle=(\overline{\boldsymbol{A}} \boldsymbol{y})^{*} \boldsymbol{x}=\boldsymbol{y}^{*} \overline{\boldsymbol{A}}^{*} \boldsymbol{x}=\boldsymbol{y}^{*} \boldsymbol{A}^{T} \boldsymbol{x}=\boldsymbol{y}^{*} \boldsymbol{A} \boldsymbol{x}=\langle\boldsymbol{A} \boldsymbol{x}, \boldsymbol{y}\rangle . | |  |
| 299 | | 123 | \boldsymbol{e}_{i}^{T} \boldsymbol{A} \boldsymbol{e}_{j}=\left\langle\boldsymbol{A} \boldsymbol{e}_{j}, \boldsymbol{e}_{i}\right\rangle=\left\langle\boldsymbol{e}_{j}, \overline{\boldsymbol{A}} \boldsymbol{e}_{i}\right\rangle=\boldsymbol{e}_{i}^{T} \boldsymbol{A}^{T} \boldsymbol{e}_{j} . | |  |
| 300 | | 124 | \langle U x, U y\rangle=(U y)^{*}(U x)=y^{*} U^{*} U x=y^{*} x=\langle x, y\rangle . | |  |
| 301 | | 124 | \left(\boldsymbol{U}^{*} \boldsymbol{U}\right)_{i, j}=\boldsymbol{e}_{i}^{*} \boldsymbol{U}^{*} \boldsymbol{U} \boldsymbol{e}_{j}=\left(\boldsymbol{U} \boldsymbol{e}_{i}\right)^{*}\left(\boldsymbol{U} \boldsymbol{e}_{j}\right)=\left\langle\boldsymbol{U} \boldsymbol{e}_{j}, \boldsymbol{U} \boldsymbol{e}_{i}\right\rangle=\left\langle\boldsymbol{e}_{j}, \boldsymbol{e}_{i}\right\rangle=\boldsymbol{e}_{i}^{*} \boldsymbol{e}_{j}, | |  |
| 302 | | 124 | \boldsymbol{H}:=\boldsymbol{I}-\boldsymbol{u} \boldsymbol{u}^{*}, \text { where } \boldsymbol{u} \in \mathbb{C}^{n} \text { and } \boldsymbol{u}^{*} \boldsymbol{u}=2 | |  |
| 303 | | 124 | \boldsymbol{H}=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]-\left[\begin{array}{l} u_{1} \\ u_{2} \end{array}\right]\left[\begin{array}{ll} u_{1} & u_{2} \end{array}\right]=\left[\begin{array}{cc} 1-u_{1}^{2} & -u_{1} u_{2} \\ -u_{2} u_{1} & 1-u_{2}^{2} \end{array}\right] . | |  |
| 304 | | 124 | \boldsymbol{H}^{*} \boldsymbol{H}=\boldsymbol{H}^{2}=\left(\boldsymbol{I}-\boldsymbol{u} \boldsymbol{u}^{*}\right)\left(\boldsymbol{I}-\boldsymbol{u} \boldsymbol{u}^{*}\right)=\boldsymbol{I}-2 \boldsymbol{u} \boldsymbol{u}^{*}+\boldsymbol{u}\left(\boldsymbol{u}^{*} \boldsymbol{u}\right) \boldsymbol{u}^{*}=\boldsymbol{I} . | |  |
| 305 | 5.12 | 124 | \boldsymbol{H}:=\boldsymbol{I}-2 \frac{\boldsymbol{v} \boldsymbol{v}^{*}}{\boldsymbol{v}^{*} \boldsymbol{v}} | |  |
| 306 | 5.13 | 125 | \boldsymbol{v}^{*} \boldsymbol{v}=(\boldsymbol{x}-\boldsymbol{y})^{*}(\boldsymbol{x}-\boldsymbol{y})=2 \boldsymbol{x}^{*} \boldsymbol{x}-2 \operatorname{Re}\left(\boldsymbol{y}^{*} \boldsymbol{x}\right)=2 \boldsymbol{v}^{*} \boldsymbol{x} . | |  |
| 307 | | 125 | \boldsymbol{H}=\boldsymbol{I}-\frac{2 \boldsymbol{v} \boldsymbol{v}^{*}}{\boldsymbol{v}^{*} \boldsymbol{v}}=\boldsymbol{P}-\frac{\boldsymbol{v} \boldsymbol{v}^{*}}{\boldsymbol{v}^{*} \boldsymbol{v}}, \text { where } \boldsymbol{P}:=\boldsymbol{I}-\frac{\boldsymbol{v} \boldsymbol{v}^{*}}{\boldsymbol{v}^{*} \boldsymbol{v}}, | |  |
| 308 | | 125 | \boldsymbol{P} \boldsymbol{x}=\boldsymbol{x}-\frac{\boldsymbol{v}^{*} \boldsymbol{x}}{\boldsymbol{v}^{*} \boldsymbol{v}} \boldsymbol{v} \stackrel{(5.13)}{=} \boldsymbol{x}-\frac{1}{2} \boldsymbol{v}=\frac{1}{2}(\boldsymbol{x}+\boldsymbol{y}) . | |  |
| 309 | | 126 | \begin{gathered} \boldsymbol{H}:=\boldsymbol{I}-\frac{2 \boldsymbol{v} \boldsymbol{v}^{T}}{\boldsymbol{v}^{T} \boldsymbol{v}}=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]-\frac{2}{4}\left[\begin{array}{l} 2 \\ 0 \\ 0 \end{array}\right]\left[\begin{array}{lll} 2 & 0 & 0 \end{array}\right]=\left[\begin{array}{ccc} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right], \\ \boldsymbol{P}:=\boldsymbol{I}-\frac{\boldsymbol{v} \boldsymbol{v}^{T}}{\boldsymbol{v}^{T} \boldsymbol{v}}=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]-\frac{1}{4}\left[\begin{array}{l} 2 \\ 0 \\ 0 \end{array}\right]\left[\begin{array}{lll} 2 & 0 & 0 \end{array}\right]=\left[\begin{array}{lll} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] . \end{gathered} | |  |
| 310 | | 126 | \mathcal{M}:=\left\{\boldsymbol{w} \in \mathbb{R}^{3}: \boldsymbol{w}^{T} \boldsymbol{v}=0\right\}=\left\{\left[\begin{array}{l} w_{1} \\ w_{2} \\ w_{3} \end{array}\right]: 2 w_{1}=0\right\} | |  |
| 311 | | 126 | \boldsymbol{H} \boldsymbol{x}=a \boldsymbol{e}_{1}, \quad a=-\rho\|\boldsymbol{x}\|_{2}, \quad \rho:= \begin{cases}x_{1} /\left|x_{1}\right|, & \text { if } x_{1} \neq 0, \\ 1, & \text { otherwise } .\end{cases} | |  |
| 312 | 5.14 | 126 | \boldsymbol{u}:=\frac{\boldsymbol{z}+\boldsymbol{e}_{1}}{\sqrt{1+z_{1}}}, \text { where } \boldsymbol{z}:=\bar{\rho} \boldsymbol{x} /\|\boldsymbol{x}\|_{2} . | |  |
| 313 | | 126 | \begin{aligned} \boldsymbol{H} \boldsymbol{x} & =\boldsymbol{x}-\left(\boldsymbol{u}^{*} \boldsymbol{x}\right) \boldsymbol{u}=\rho\|\boldsymbol{x}\|_{2}\left(\boldsymbol{z}-\left(\boldsymbol{u}^{*} \boldsymbol{z}\right) \boldsymbol{u}\right)=\rho\|\boldsymbol{x}\|_{2}\left(z-\frac{\left(z^{*}+\boldsymbol{e}_{1}^{*}\right) \boldsymbol{z}}{1+z_{1}}\left(z+\boldsymbol{e}_{1}\right)\right) \\ & =\rho\|\boldsymbol{x}\|_{2}\left(z-\left(z+\boldsymbol{e}_{1}\right)\right)=-\rho\|\boldsymbol{x}\|_{2} \boldsymbol{e}_{1}=a \boldsymbol{e}_{1} \end{aligned} | |  |
| 314 | | 127 | \boldsymbol{H} \boldsymbol{x}=\left[\begin{array}{c} \boldsymbol{y} \\ a \boldsymbol{e}_{1} \end{array}\right], \text { where } \boldsymbol{H}:=\boldsymbol{I}-\boldsymbol{u} \boldsymbol{u}^{*}=\left[\begin{array}{cc} \boldsymbol{I} & \mathbf{0} \\ \mathbf{0} & \boldsymbol{I} \end{array}\right]-\left[\begin{array}{c} \mathbf{0} \\ \hat{\boldsymbol{u}} \end{array}\right]\left[\begin{array}{ll} \mathbf{0} & \hat{\boldsymbol{u}}^{*} \end{array}\right]=\left[\begin{array}{cc} \boldsymbol{I} & \mathbf{0} \\ \mathbf{0} & \hat{\boldsymbol{H}} \end{array}\right], | |  |
| 315 | | 128 | \left[\begin{array}{cccc} x & x & x & x \\ 0 & x & x & x \\ 0 & 0 & x & x \end{array}\right], \quad\left[\begin{array}{cccc} x & x & x & x \\ 0 & x & x & x \\ 0 & 0 & x & x \\ 0 & 0 & 0 & x \end{array}\right], \quad\left[\begin{array}{ccc} x & x & x \\ 0 & x & x \\ 0 & 0 & x \\ 0 & 0 & 0 \end{array}\right] . | |  |
| 316 | | 128 | \boldsymbol{A}_{n+1}:=\boldsymbol{H}_{n} \boldsymbol{H}_{n-1} \cdots \boldsymbol{H}_{1} \boldsymbol{A}=\left[\begin{array}{c} \boldsymbol{R}_{1} \\ \mathbf{0} \end{array}\right]=\boldsymbol{R}, | |  |
| 317 | | 128 | \boldsymbol{A}_{1}:=\boldsymbol{A}, \quad \boldsymbol{A}_{k+1}=\boldsymbol{H}_{k} \boldsymbol{A}_{k}, \quad k=1,2, \ldots, n . | |  |
| 318 | 5.15 | 128 | \begin{aligned} \boldsymbol{A}_{k} & =\left[\begin{array}{ccc|ccccc} a_{1,1}^{(1)} & \cdots & a_{1, k-1}^{(1)} & a_{1, k}^{(1)} & \cdots & a_{1, j}^{(1)} & \cdots & a_{1, n}^{(1)} \\ & \ddots & \vdots & \vdots & & \vdots & & \vdots \\ & & a_{k-1, k-1}^{(k-1)} & a_{k-1, k}^{(k-1)} & \cdots & a_{k-1, j}^{(k-1)} & \cdots & a_{k-1, n}^{(k-1)} \\ \hline & & & a_{k, k}^{(k)} & \cdots & a_{k, j}^{(k)} & \cdots & a_{k, n}^{(k)} \\ & & & \vdots & & \vdots & & \vdots \\ & & & a_{i, k}^{(k)} & \cdots & a_{i, j}^{(k)} & \cdots & a_{i, n}^{(k)} \\ & & & \vdots & & \vdots & & \vdots \\ & & & a_{m, k}^{(k)} & \cdots & a_{m, j}^{(k)} & \cdots & a_{m, n}^{(k)} \end{array}\right] \\ & =\left[\begin{array}{cc} \boldsymbol{B}_{k} & \boldsymbol{C}_{k} \\ \mathbf{0} & \boldsymbol{D}_{k} \end{array}\right] . \end{aligned} | |  |
| 319 | | 129 | \boldsymbol{A}_{k+1}:=\boldsymbol{H}_{k} \boldsymbol{A}_{k}=\left[\begin{array}{cc} \boldsymbol{B}_{k} & \boldsymbol{C}_{k} \\ \mathbf{0} & \hat{\boldsymbol{H}}_{k} \boldsymbol{D}_{k} \end{array}\right]=\left[\begin{array}{cc} \boldsymbol{B}_{k+1} & \boldsymbol{C}_{k+1} \\ \mathbf{0} & \boldsymbol{D}_{k+1} \end{array}\right], | |  |
| 320 | | 129 | \boldsymbol{A}=\left[\begin{array}{lll} u_{11} & r_{12} & r_{13} \\ u_{21} & u_{22} & r_{23} \\ u_{31} & u_{32} & u_{33} \\ u_{41} & u_{42} & u_{43} \end{array}\right] \text { or } \boldsymbol{A}=\left[\begin{array}{lll} r_{11} & r_{12} & r_{13} \\ u_{21} & r_{22} & r_{23} \\ u_{31} & u_{32} & r_{33} \\ u_{41} & u_{42} & u_{43} \end{array}\right] . | |  |
| 321 | 5.16 | 130 | \int_{0}^{n} 4(m-k)(n+r-k) d k=2 m(n+r)^{2}-\frac{2}{3}(n+r)^{3} . | |  |
| 322 | | 131 | \boldsymbol{R}=\left[\begin{array}{c} \boldsymbol{R}_{1} \\ \mathbf{0}_{m-n, n} \end{array}\right] | |  |
| 323 | | 132 | \boldsymbol{A}=\left[\begin{array}{rrr} 1 & 3 & 1 \\ 1 & 3 & 7 \\ 1 & -1 & -4 \\ 1 & -1 & 2 \end{array}\right]=\frac{1}{2}\left[\begin{array}{rrrr} 1 & 1 & -1 & -1 \\ 1 & 1 & 1 & 1 \\ 1 & -1 & -1 & 1 \\ 1 & -1 & 1 & -1 \end{array}\right] \times\left[\begin{array}{lll} 2 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \\ 0 & 0 & 0 \end{array}\right]=\boldsymbol{Q} \boldsymbol{R} . | |  |
| 324 | | 132 | \boldsymbol{A}=\frac{1}{2}\left[\begin{array}{ccc} 1 & 1 & -1 \\ 1 & 1 & 1 \\ 1 & -1 & -1 \\ 1 & -1 & 1 \end{array}\right] \times\left[\begin{array}{lll} 2 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{array}\right]=\boldsymbol{Q}_{1} \boldsymbol{R}_{1} . | |  |
| 325 | 5.17 | 133 | \boldsymbol{v}_{1}=\boldsymbol{a}_{1}, \quad \boldsymbol{v}_{j}=\boldsymbol{a}_{j}-\sum_{i=1}^{j-1} \frac{\boldsymbol{a}_{j}^{T} \boldsymbol{v}_{i}}{\boldsymbol{v}_{i}^{T} \boldsymbol{v}_{i}} \boldsymbol{v}_{i}, \quad \text { for } j=2, \ldots, n . | |  |
| 326 | 5.18 | 133 | \begin{aligned} & \boldsymbol{Q}_{1}:=\left[\begin{array}{llll} \boldsymbol{q}_{1}, \ldots, \boldsymbol{q}_{n} \end{array}\right], \\ & \boldsymbol{R}_{1}:= \\ & \boldsymbol{q}_{j}:=\frac{\boldsymbol{v}_{j}}{\left\|\boldsymbol{v}_{j}\right\|_{2}}, \quad j=1, \ldots, n \text { and } \\ & {\left[\begin{array}{ccccc} \left\|\boldsymbol{v}_{1}\right\|_{2} & \boldsymbol{a}_{2}^{T} \boldsymbol{q}_{1} & \boldsymbol{a}_{3}^{T} \boldsymbol{q}_{1} & \cdots & \boldsymbol{a}_{n-1}^{T} \boldsymbol{q}_{1} \\ 0 & \left\|\boldsymbol{v}_{2}\right\|_{2} & \boldsymbol{a}_{3}^{T} \boldsymbol{q}_{2} & \cdots & \boldsymbol{a}_{n-1}^{T} \boldsymbol{q}_{2} \\ & 0 & \left\|\boldsymbol{v}_{3}\right\|_{2} & \cdots & \boldsymbol{a}_{n-1}^{T} \boldsymbol{q}_{2} \\ & & \ddots & \ddots & \boldsymbol{a}_{n}^{T} \boldsymbol{q}_{3} \\ & & & \ddots & \vdots \\ & & & \left\|\boldsymbol{v}_{n-1}\right\|_{2} & \boldsymbol{a}_{n}^{T} \boldsymbol{q}_{n-1} \\ & & & 0 & \left\|\boldsymbol{v}_{n}\right\|_{2} \end{array}\right] .} \end{aligned} | |  |
| 327 | | 133 | \boldsymbol{a}_{j}=\boldsymbol{v}_{j}+\sum_{i=1}^{j-1} \frac{\boldsymbol{a}_{j}^{T} \boldsymbol{v}_{i}}{\boldsymbol{v}_{i}^{T} \boldsymbol{v}_{i}} \boldsymbol{v}_{i}=r_{j j} \boldsymbol{q}_{j}+\sum_{i=1}^{j-1} \boldsymbol{q}_{i} r_{i j}=\boldsymbol{Q}_{1} \boldsymbol{R}_{1} \boldsymbol{e}_{j}, j=1, \ldots, n . | |  |
| 328 | | 133 | \boldsymbol{R}_{1}=\boldsymbol{R}=\left[\begin{array}{cc} \left\|\boldsymbol{v}_{1}\right\|_{2} & \boldsymbol{a}_{2}^{T} \boldsymbol{q}_{1} \\ 0 & \left\|\boldsymbol{v}_{2}\right\|_{2} \end{array}\right]=\frac{1}{\sqrt{5}}\left[\begin{array}{cc} 5 & -4 \\ 0 & 3 \end{array}\right] | |  |
| 329 | | 134 | \boldsymbol{A}=\left[\begin{array}{llll} x & x & x & x \\ x & x & x & x \\ 0 & x & x & x \\ 0 & 0 & x & x \end{array}\right] . | |  |
| 330 | | 134 | \boldsymbol{P}:=\left[\begin{array}{cc} c & s \\ -s & c \end{array}\right], \text { where } c^{2}+s^{2}=1 . | |  |
| 331 | | 135 | \boldsymbol{x}=\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right] \neq \mathbf{0}, \quad c:=\frac{x_{1}}{r}, \quad s:=\frac{x_{2}}{r}, \quad r:=\|\boldsymbol{x}\|_{2} . | |  |
| 332 | | 135 | \boldsymbol{P} \boldsymbol{x}=\frac{1}{r}\left[\begin{array}{rr} x_{1} & x_{2} \\ -x_{2} & x_{1} \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]=\frac{1}{r}\left[\begin{array}{c} x_{1}^{2}+x_{2}^{2} \\ 0 \end{array}\right]=\left[\begin{array}{l} r \\ 0 \end{array}\right], | |  |
| 333 | | 135 | \left[\begin{array}{cc} p_{i i} & p_{i j} \\ p_{j i} & p_{j j} \end{array}\right]=\left[\begin{array}{cc} c & s \\ -s & c \end{array}\right], \text { where } c^{2}+s^{2}=1 . | |  |
| 334 | | 135 | \boldsymbol{P}_{12}=\left[\begin{array}{cccc} c & s & 0 & 0 \\ -s & c & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right], \quad \boldsymbol{P}_{13}=\left[\begin{array}{cccc} c & 0 & s & 0 \\ 0 & 1 & 0 & 0 \\ -s & 0 & c & 0 \\ 0 & 0 & 0 & 1 \end{array}\right], \quad \boldsymbol{P}_{23}=\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & s & c & 0 \\ 0 & -s & c & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] . | |  |
| 335 | 5.19 | 135 | \left[\begin{array}{l} \boldsymbol{B}(i,:) \\ \boldsymbol{B}(j,:) \end{array}\right]=\left[\begin{array}{rc} c & s \\ -s c & \end{array}\right]\left[\begin{array}{l} \boldsymbol{A}(i,:) \\ \boldsymbol{A}(j,:) \end{array}\right],[\boldsymbol{C}(:, i) \boldsymbol{C}(:, j)]=[\boldsymbol{A}(:, i) \boldsymbol{A}(:, j)]\left[\begin{array}{rc} c & s \\ -s & c \end{array}\right] . | |  |
| 336 | | 136 | \boldsymbol{A}=\left[\begin{array}{llll} x & x & x & x \\ x & x & x & x \\ 0 & x & x & x \\ 0 & 0 & x & x \end{array}\right] \xrightarrow{\boldsymbol{P}_{12}}\left[\begin{array}{cccc} r_{11} & r_{12} & r_{13} & r_{14} \\ \mathbf{0} & x & x & x \\ 0 & x & x & x \\ 0 & 0 & x & x \end{array}\right] \xrightarrow{\boldsymbol{P}_{23}}\left[\begin{array}{cccc} r_{11} & r_{12} & r_{13} & r_{14} \\ 0 & r_{22} & r_{23} & r_{24} \\ 0 & \mathbf{0} & x & x \\ 0 & 0 & x & x \end{array}\right] \xrightarrow{\boldsymbol{P}_{34}}\left[\begin{array}{cccc} r_{11} & r_{12} & r_{13} & r_{14} \\ 0 & r_{22} & r_{23} & r_{24} \\ 0 & 0 & r_{33} & r_{34} \\ 0 & 0 & \mathbf{0} & r_{44} \end{array}\right] . | |  |
| 337 | 5.20 | 136 | \cos \theta=\frac{|\langle\boldsymbol{x}, \boldsymbol{y}\rangle|}{\|\boldsymbol{x}\|\|\boldsymbol{y}\|} . | |  |
| 338 | | 136 | \left|\boldsymbol{x}^{T} \boldsymbol{A y}\right|^{2} \leq \boldsymbol{x}^{T} \boldsymbol{A x} \boldsymbol{y}^{T} \boldsymbol{A y} | |  |
| 339 | | 137 | \boldsymbol{H}=\left[\begin{array}{cc} -\cos \phi & \sin \phi \\ \sin \phi & \cos \phi \end{array}\right] . | |  |
| 340 | | 137 | \boldsymbol{H} \boldsymbol{x}=\boldsymbol{y}, \quad \text { where } \quad \boldsymbol{H}:=\boldsymbol{I}-2 \frac{\boldsymbol{v} \boldsymbol{v}^{T}}{\boldsymbol{v}^{T} \boldsymbol{v}} . | |  |
| 341 | | 137 | \boldsymbol{B}:=\left[\begin{array}{llll} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \epsilon & 0 & 0 & 0 \end{array}\right], | |  |
| 342 | | 137 | \boldsymbol{A}=\left[\begin{array}{ll} 1 & 2 \\ 1 & 2 \\ 1 & 0 \\ 1 & 0 \end{array}\right], \quad \boldsymbol{Q}=\frac{1}{2}\left[\begin{array}{rrrr} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1 \\ 1 & -1 & 1 & -1 \end{array}\right], \quad \boldsymbol{R}=\left[\begin{array}{ll} 2 & 2 \\ 0 & 2 \\ 0 & 0 \\ 0 & 0 \end{array}\right] . | |  |
| 343 | | 137 | A:=\left[\begin{array}{ccc} 1 & 0 & 1 \\ -2 & -1 & 0 \\ 2 & 2 & 1 \end{array}\right] . | |  |
| 344 | 5.21 | 138 | |\operatorname{det}(\boldsymbol{A})| \leq \prod_{j=1}^{n}\left\|\boldsymbol{a}_{j}\right\|_{2} . | |  |
| 345 | | 139 | \boldsymbol{V}_{r}:=\boldsymbol{I}_{r}-2 \frac{\boldsymbol{v}_{r} \boldsymbol{v}_{r}^{*}}{\boldsymbol{v}_{r}^{*} \boldsymbol{v}_{r}}=\boldsymbol{I}_{r}-\boldsymbol{u}_{r} \boldsymbol{u}_{r}^{*}, \text { with } \boldsymbol{u}_{r}:=\sqrt{2} \frac{\boldsymbol{v}_{r}}{\left\|\boldsymbol{v}_{r}\right\|_{2}} . | |  |
| 346 | | 139 | A:=\left[\begin{array}{rr} 2 & 1 \\ 2 & -3 \\ -2 & -1 \\ -2 & 3 \end{array}\right] | |  |
| 347 | | 140 | \boldsymbol{G}_{3} \boldsymbol{G}_{2} \boldsymbol{G}_{1} \boldsymbol{H}=\boldsymbol{R} | |  |
| 348 | | 141 | \left[\begin{array}{ll} p_{i i} & p_{i j} \\ p_{j i} & p_{j j} \end{array}\right]=\left[\begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right], | |  |
| 349 | | 141 | \boldsymbol{P}=\left[\begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right] | |  |
| 350 | | 141 | \boldsymbol{P}_{1,2} \boldsymbol{P}_{2,3} \cdots \boldsymbol{P}_{m-2, m-1} \boldsymbol{P}_{m-1, m} \boldsymbol{w}=\left[\begin{array}{c} \alpha \\ 0 \\ \vdots \\ 0 \end{array}\right], | |  |
| 351 | | 141 | \left(a_{1,1}, a_{2,2}, \ldots, a_{n, n}\right) | |  |
| 352 | | 141 | \left(a_{2,1}, a_{3,2}, \ldots, a_{n, n-1}\right) . | |  |
| 353 | | 142 | \boldsymbol{P}_{m-1, m} \boldsymbol{P}_{m-2, m-1} \cdots \boldsymbol{P}_{2,3} \boldsymbol{P}_{1,2} \boldsymbol{H} | |  |
| 354 | | 142 | \left[\begin{array}{c} \boldsymbol{L}^{*} \\ \boldsymbol{z}^{*} \end{array}\right]=\boldsymbol{Q}\left[\begin{array}{c} \boldsymbol{R} \\ 0 \end{array}\right], | |  |
| 355 | 5.22 | 142 | P_{i_{1}, n+1} P_{i_{2}, n+1} \cdots P_{i_{n}, n+1}\left[\begin{array}{c} \boldsymbol{L}^{*} \\ \boldsymbol{z}^{*} \end{array}\right]=\left[\begin{array}{c} \boldsymbol{R}^{\prime} \\ 0 \end{array}\right], | |  |
| 356 | | 145 | \pi_{\boldsymbol{A}}(\lambda)=\operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I}) . | |  |
| 357 | | 146 | \sum_{j=1}^{m} c_{j} \boldsymbol{x}_{j}=\mathbf{0} \Rightarrow \sum_{j=1}^{m} c_{j} \boldsymbol{A} \boldsymbol{x}_{j}=\sum_{j=1}^{m} c_{j} \lambda_{j} \boldsymbol{x}_{j}=\mathbf{0} . | |  |
| 358 | | 146 | \boldsymbol{I}:=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right], \quad \boldsymbol{J}:=\left[\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right] . | |  |
| 359 | | 146 | \boldsymbol{x}=\sum_{j=1}^{n} c_{j} \boldsymbol{x}_{j} \text { for some scalars } c_{1}, \ldots, c_{n} | |  |
| 360 | | 146 | \boldsymbol{x}=\frac{x_{1}+x_{2}}{2}\left[\begin{array}{l} 1 \\ 1 \end{array}\right]+\frac{x_{1}-x_{2}}{2}\left[\begin{array}{c} 1 \\ -1 \end{array}\right] . | |  |
| 361 | | 147 | \begin{aligned} \pi_{\boldsymbol{B}}(\lambda) & =\operatorname{det}\left(\boldsymbol{S}^{-1} \boldsymbol{A} \boldsymbol{S}-\lambda \boldsymbol{I}\right)=\operatorname{det}\left(\boldsymbol{S}^{-1}(\boldsymbol{A}-\lambda \boldsymbol{I}) \boldsymbol{S}\right) \\ & =\operatorname{det}\left(\boldsymbol{S}^{-1}\right) \operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I}) \operatorname{det}(\boldsymbol{S})=\operatorname{det}\left(\boldsymbol{S}^{-1} \boldsymbol{S}\right) \operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})=\pi_{\boldsymbol{A}}(\lambda), \end{aligned} | |  |
| 362 | 6.1 | 147 | \pi_{A C}(\lambda)=\lambda^{m-n} \pi_{C A}(\lambda), \quad \lambda \in \mathbb{C} . | |  |
| 363 | | 147 | E:=\left[\begin{array}{cc} A C & 0 \\ C & 0 \end{array}\right], \quad F:=\left[\begin{array}{cc} 0 & 0 \\ C & C A \end{array}\right], \quad S=\left[\begin{array}{cc} I & A \\ 0 & I \end{array}\right] . | |  |
| 364 | 6.2 | 148 | \pi_{\boldsymbol{A}}(\lambda):=\operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})=\left(\lambda_{1}-\lambda\right)^{a_{1}} \cdots\left(\lambda_{k}-\lambda\right)^{a_{k}}, \quad \lambda_{i} \neq \lambda_{j}, i \neq j, \sum_{i=1}^{k} a_{i}=n . | |  |
| 365 | 6.3 | 148 | \mathcal{N}(\boldsymbol{A}-\lambda \boldsymbol{I}):=\left\{\boldsymbol{x} \in \mathbb{C}^{n}:(\boldsymbol{A}-\lambda \boldsymbol{I}) \boldsymbol{x}=\mathbf{0}\right\} | |  |
| 366 | 6.4 | 149 | \boldsymbol{J}_{m}(\lambda):=\left[\begin{array}{cccccc} \lambda & 1 & 0 & \cdots & 0 & 0 \\ 0 & \lambda & 1 & \cdots & 0 & 0 \\ 0 & 0 & \lambda & \cdots & 0 & 0 \\ \vdots & & & & & \vdots \\ 0 & 0 & 0 & \cdots & \lambda & \lambda \\ 0 & 0 & 0 & \cdots & 0 & \lambda \end{array}\right]=\lambda \boldsymbol{I}_{m}+\boldsymbol{E}_{m}, \quad \boldsymbol{E}_{m}:=\left[\begin{array}{cccccc} 0 & 1 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 1 & \cdots & 0 & 0 \\ 0 & 0 & 0 & \cdots & 0 & 0 \\ \vdots & & & & \vdots \\ 0 & 0 & 0 & \cdots & 0 & 1 \\ 0 & 0 & 0 & \cdots & 0 & 0 \end{array}\right] . | |  |
| 367 | 6.5 | 150 | \boldsymbol{J}:=\boldsymbol{S}^{-1} \boldsymbol{A} \boldsymbol{S}=\operatorname{diag}\left(\boldsymbol{U}_{1}, \ldots, \boldsymbol{U}_{k}\right), \text { with } \boldsymbol{U}_{i} \in \mathbb{C}^{a_{i} \times a_{i}}, | |  |
| 368 | 6.6 | 150 | \boldsymbol{U}_{i}=\operatorname{diag}\left(\boldsymbol{J}_{m_{i, 1}}\left(\lambda_{i}\right), \ldots, \boldsymbol{J}_{m_{i, g_{i}}}\left(\lambda_{i}\right)\right) . | |  |
| 369 | 6.7 | 150 | \boldsymbol{J}:=\operatorname{diag}\left(\boldsymbol{U}_{1}, \boldsymbol{U}_{2}\right)=\left[\begin{array}{ccccccc} 2 & 1 & 0 & & & & \\ 0 & 2 & 1 & & & & \\ 0 & 0 & 2 & & 1 & & \\ & & 0 & 1 & & \\ & & 0 & 2 & & & \\ & & & & & 0 & \\ & & & & & 0 & 1 \end{array}\right] \in \mathbb{R}^{8 \times 8} . | |  |
| 370 | | 150 | \begin{aligned} & \boldsymbol{A} \boldsymbol{s}_{1}=2 \boldsymbol{s}_{1}, \quad \boldsymbol{A} \boldsymbol{s}_{2}=\boldsymbol{s}_{1}+2 \boldsymbol{s}_{2}, \quad \boldsymbol{A} \boldsymbol{s}_{3}=\boldsymbol{s}_{2}+2 \boldsymbol{s}_{3}, \\ & \boldsymbol{A} \boldsymbol{s}_{4}=2 \boldsymbol{s}_{4}, \quad \boldsymbol{A} \boldsymbol{s}_{5}=\boldsymbol{s}_{4}+2 \boldsymbol{s}_{5} \\ & \boldsymbol{A} \boldsymbol{s}_{6}=2 \boldsymbol{s}_{6}, \\ & \boldsymbol{A} \boldsymbol{s}_{7}=3 \boldsymbol{s}_{7}, \quad \boldsymbol{A} \boldsymbol{s}_{8}=\boldsymbol{s}_{7}+3 \boldsymbol{s}_{8} . \end{aligned} | |  |
| 371 | | 151 | \boldsymbol{J}:=\left[\begin{array}{ccccccc} 3 & 1 & & & & & \\ 0 & 3 & & & & & \\ & & 2 & 1 & & & \\ & & 0 & 2 & & & \\ & & & & 2 & & \\ & & & & 0 & 1 & 0 \\ & & & & 0 & 2 & 1 \\ & & & & 0 & 2 \end{array}\right] | |  |
| 372 | | 152 | \boldsymbol{V}^{*} \boldsymbol{A} \boldsymbol{V} \boldsymbol{e}_{1}=\boldsymbol{V}^{*} \boldsymbol{A} \boldsymbol{v}_{1}=\lambda_{1} \boldsymbol{V}^{*} \boldsymbol{v}_{1}=\lambda_{1} \boldsymbol{e}_{1} . | |  |
| 373 | 6.8 | 152 | \boldsymbol{V}^{*} \boldsymbol{A} \boldsymbol{V}=\left[\begin{array}{c|c} \lambda_{1} & \boldsymbol{x}^{*} \\ \hline \mathbf{0} & \boldsymbol{M} \end{array}\right], \text { for some } \boldsymbol{M} \in \mathbb{C}^{k \times k} \text { and } \boldsymbol{x} \in \mathbb{C}^{k} . | |  |
| 374 | | 152 | \boldsymbol{W}=\left[\begin{array}{c|c} 1 & \mathbf{0}^{*} \\ \hline \mathbf{0} & \boldsymbol{W}_{1} \end{array}\right] \text { and } \boldsymbol{U}=\boldsymbol{V} \boldsymbol{W} . | |  |
| 375 | | 152 | \begin{aligned} \boldsymbol{U}^{*} \boldsymbol{A} \boldsymbol{U}=\boldsymbol{W}^{*}\left(\boldsymbol{V}^{*} \boldsymbol{A} \boldsymbol{V}\right) \boldsymbol{W} & =\left[\begin{array}{l|l} 1 & \mathbf{0}^{*} \\ \hline \mathbf{0} & \boldsymbol{W}_{1}^{*} \end{array}\right]\left[\begin{array}{l|l} \lambda_{1} & \boldsymbol{x}^{*} \\ \hline \mathbf{0} & \boldsymbol{M} \end{array}\right]\left[\begin{array}{l|l} 1 & \mathbf{0}^{*} \\ \hline \mathbf{0} & \boldsymbol{W}_{1} \end{array}\right] \\ & =\left[\begin{array}{l|l} \lambda_{1} & \boldsymbol{x}^{*} \boldsymbol{W}_{1} \\ \hline \mathbf{0} & \boldsymbol{W}_{1}^{*} \boldsymbol{M} \boldsymbol{W}_{1} \end{array}\right] \end{aligned} | |  |
| 376 | | 153 | \boldsymbol{V}=\left[\begin{array}{ccc} -\frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ 0 & 1 & 0 \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \end{array}\right] . | |  |
| 377 | | 153 | \boldsymbol{M}=\left[\begin{array}{cc} 2 & -\sqrt{2} \\ -\sqrt{2} & 2 \end{array}\right] . | |  |
| 378 | | 153 | \boldsymbol{A}^{*} \boldsymbol{A}=\operatorname{diag}\left(\overline{d_{1}} d_{1}, \ldots, \overline{d_{n}} d_{n}\right)=\operatorname{diag}\left(\left|d_{1}\right|^{2}, \ldots,\left|d_{n}\right|^{2}\right)=\boldsymbol{A} \boldsymbol{A}^{*}, | |  |
| 379 | | 154 | \begin{aligned} \boldsymbol{A} \boldsymbol{A}^{*} & =\left(\boldsymbol{U} \boldsymbol{B} \boldsymbol{U}^{*}\right)\left(\boldsymbol{U} \boldsymbol{B}^{*} \boldsymbol{U}^{*}\right)=\boldsymbol{U} \boldsymbol{B} \boldsymbol{B}^{*} \boldsymbol{U}^{*} \text { and } \\ \boldsymbol{A}^{*} \boldsymbol{A} & =\left(\boldsymbol{U} \boldsymbol{B}^{*} \boldsymbol{U}^{*}\right)\left(\boldsymbol{U} \boldsymbol{B} \boldsymbol{U}^{*}\right)=\boldsymbol{U} \boldsymbol{B}^{*} \boldsymbol{B} \boldsymbol{U}^{*} . \end{aligned} | |  |
| 380 | | 154 | \boldsymbol{B} \boldsymbol{B}^{*}=\boldsymbol{U}^{*} \boldsymbol{A} \boldsymbol{U} \boldsymbol{U}^{*} \boldsymbol{A}^{*} \boldsymbol{U}=\boldsymbol{U}^{*} \boldsymbol{A} \boldsymbol{A}^{*} \boldsymbol{U}=\boldsymbol{U}^{*} \boldsymbol{A}^{*} \boldsymbol{A} \boldsymbol{U}=\boldsymbol{B}^{*} \boldsymbol{B} . | |  |
| 381 | | 154 | e_{i i}=\sum_{k=1}^{n} \bar{b}_{k i} b_{k i}=\sum_{k=1}^{i}\left|b_{k i}\right|^{2} \text { and } f_{i i}=\sum_{k=1}^{n} b_{i k} \bar{b}_{i k}=\sum_{k=i}^{n}\left|b_{i k}\right|^{2} . | |  |
| 382 | | 154 | e_{i i}=\sum_{k=1}^{i}\left|b_{k i}\right|^{2}=\left|b_{i i}\right|^{2}=\sum_{k=i}^{n}\left|b_{i k}\right|^{2}=f_{i i} | |  |
| 383 | | 155 | R(x)=R_{A}(x):=\frac{x^{*} A x}{x^{*} x} | |  |
| 384 | 6.9 | 155 | R_{\boldsymbol{A}}(\boldsymbol{x})=\frac{\sum_{i=1}^{n} \lambda_{i}\left|c_{i}\right|^{2}}{\sum_{j=1}^{n}\left|c_{j}\right|^{2}}, \quad \boldsymbol{x} \neq \mathbf{0}, \quad \boldsymbol{x}=\sum_{j=1}^{n} c_{j} \boldsymbol{u}_{j} . | |  |
| 385 | 6.10 | 155 | \boldsymbol{M}=\left[\begin{array}{cc} \mu & v \\ -v & \mu \end{array}\right] | |  |
| 386 | | 156 | \boldsymbol{R}:=\left[\begin{array}{ccc} \boldsymbol{D}_{1} & \boldsymbol{R}_{1,2} & \boldsymbol{R}_{1,3} \\ \mathbf{0} & \boldsymbol{D}_{2} & \boldsymbol{R}_{2,3} \\ \mathbf{0} & \mathbf{0} & \boldsymbol{D}_{3} \end{array}\right], \boldsymbol{D}_{1}:=\left[\begin{array}{cc} 2 & 1 \\ -1 & 2 \end{array}\right], \boldsymbol{D}_{2}:=[1], \boldsymbol{D}_{3}:=\left[\begin{array}{cc} 3 & 2 \\ -1 & 1 \end{array}\right] . | |  |
| 387 | | 156 | \pi_{D_{1}}(\lambda)=\pi_{D_{3}}(\lambda)=\lambda^{2}-4 \lambda+5, \quad \pi_{D_{2}}(\lambda)=\lambda-1, | |  |
| 388 | | 156 | \boldsymbol{U}^{*} \boldsymbol{A} \boldsymbol{U}=\operatorname{diag}\left(\lambda_{1}, \ldots, \lambda_{n}\right) . | |  |
| 389 | | 156 | \boldsymbol{U}^{T} \boldsymbol{A} \boldsymbol{U}=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right) . | |  |
| 390 | 6.11 | 157 | \lambda_{k} \leq \max _{\substack{\boldsymbol{x} \in \mathcal{S} \\ \boldsymbol{x} \neq \mathbf{0}}} R(\boldsymbol{x}), | |  |
| 391 | | 157 | \operatorname{dim}\left(\mathcal{S} \cap \mathcal{S}^{\prime}\right)=\operatorname{dim}(\mathcal{S})+\operatorname{dim}\left(\mathcal{S}^{\prime}\right)-\operatorname{dim}\left(\mathcal{S}+\mathcal{S}^{\prime}\right) \geq(n-k+1)+k-n=1 . | |  |
| 392 | | 157 | \max _{\substack{x \in \mathcal{S} \\ \boldsymbol{x} \neq \mathbf{0}}} R(\boldsymbol{x}) \geq R(\boldsymbol{y})=\sum_{j=1}^{n} \lambda_{j}\left|c_{j}\right|^{2}=\sum_{j=1}^{k} \lambda_{j}\left|c_{j}\right|^{2} \geq \sum_{j=1}^{k} \lambda_{k}\left|c_{j}\right|^{2}=\lambda_{k}, | |  |
| 393 | 6.12 | 157 | \lambda_{k} \geq \min _{\substack{\boldsymbol{x} \in \mathcal{S} \\ \boldsymbol{x} \neq \mathbf{0}}} R(\boldsymbol{x}), | |  |
| 394 | 6.13 | 158 | \lambda_{k}=\min _{\operatorname{dim}(\mathcal{S})=n-k+1} \max _{\substack{\boldsymbol{x} \in \mathcal{S} \\ \boldsymbol{x} \neq \mathbf{0}}} R(\boldsymbol{x})=\max _{\operatorname{dim}(\mathcal{S})=k} \min _{\substack{\boldsymbol{x} \in \mathcal{S} \\ \boldsymbol{x} \neq \mathbf{0}}} R(\boldsymbol{x}), \quad k=1, \ldots, n . | |  |
| 395 | 6.14 | 158 | \alpha_{k}+\varepsilon_{n} \leq \beta_{k} \leq \alpha_{k}+\varepsilon_{1}, \text { for } k=1, \ldots, n, | |  |
| 396 | | 158 | \begin{aligned} \beta_{k} & \leq \max _{\substack{\boldsymbol{x} \in \mathcal{S} \\ \boldsymbol{x} \neq \mathbf{0}}} R_{\boldsymbol{B}}(\boldsymbol{x}) \leq \max _{\substack{\boldsymbol{x} \in \mathcal{S} \\ \boldsymbol{x} \neq \mathbf{0}}} R_{\boldsymbol{A}}(\boldsymbol{x})+\max _{\substack{\boldsymbol{x} \in S \\ \boldsymbol{x} \neq \mathbf{0}}} R_{\boldsymbol{E}}(\boldsymbol{x}) \\ & \leq \max _{\substack{\boldsymbol{x} \in \mathcal{S} \\ \boldsymbol{x} \neq \mathbf{0}}} R_{\boldsymbol{A}}(\boldsymbol{x})+\max _{\substack{\boldsymbol{x} \in \mathbb{C}^{n} \\ \boldsymbol{x} \neq \mathbf{0}}} R_{\boldsymbol{E}}(\boldsymbol{x})=\alpha_{k}+\varepsilon_{1}, \end{aligned} | |  |
| 397 | 6.15 | 159 | \sum_{j=1}^{n}\left|\mu_{i_{j}}-\lambda_{j}\right|^{2} \leq \sum_{i=1}^{n} \sum_{j=1}^{n}\left|a_{i j}-b_{i j}\right|^{2} | |  |
| 398 | 6.16 | 159 | \boldsymbol{v}=\sum_{j=1}^{n} \frac{\boldsymbol{y}_{j}^{*} \boldsymbol{v}}{\boldsymbol{y}_{j}^{*} \boldsymbol{x}_{j}} \boldsymbol{x}_{j}=\sum_{k=1}^{n} \frac{\boldsymbol{x}_{k}^{*} \boldsymbol{v}}{\boldsymbol{y}_{k}^{*} \boldsymbol{x}_{k}} \boldsymbol{y}_{k}, \quad \boldsymbol{v} \in \mathbb{C}^{n} | |  |
| 399 | | 160 | \boldsymbol{X}:=\left[\boldsymbol{x}_{1}, \ldots, \boldsymbol{x}_{n}\right], \quad \boldsymbol{Y}:=\left[\boldsymbol{y}_{1}, \ldots, \boldsymbol{y}_{n}\right], \quad \boldsymbol{D}:=\operatorname{diag}\left(\lambda_{1}, \ldots, \lambda_{n}\right) . | |  |
| 400 | 6.17 | 160 | \boldsymbol{v}=\sum_{j=1}^{n} \frac{\boldsymbol{x}_{j}^{*} \boldsymbol{v}}{\boldsymbol{x}_{j}^{*} \boldsymbol{x}_{j}} \boldsymbol{x}_{j} . | |  |
| 401 | | 160 | \boldsymbol{V}^{*} \boldsymbol{A} \boldsymbol{V}=\left[\begin{array}{c|c} \lambda & z^{*} \\ \hline \mathbf{0} & \boldsymbol{M} \end{array}\right], | |  |
| 402 | | 161 | \left(\boldsymbol{V}^{*} \boldsymbol{A}^{*} \boldsymbol{V}\right) \boldsymbol{u}=\boldsymbol{V}^{*} \boldsymbol{A}^{*} \boldsymbol{y}=\bar{\lambda} \boldsymbol{V}^{*} \boldsymbol{y}=\bar{\lambda} \boldsymbol{u}, | |  |
| 403 | | 161 | \boldsymbol{V}^{*} \boldsymbol{A}^{*} \boldsymbol{V} \boldsymbol{u}=\left[\begin{array}{c|c} \bar{\lambda} & \mathbf{0}^{*} \\ \hline \boldsymbol{z} & \boldsymbol{M}^{*} \end{array}\right]\left[\begin{array}{l} 0 \\ \boldsymbol{v} \end{array}\right]=\left[\begin{array}{c} 0 \\ \boldsymbol{M}^{*} \boldsymbol{v} \end{array}\right]=\bar{\lambda}\left[\begin{array}{l} 0 \\ \boldsymbol{v} \end{array}\right] | |  |
| 404 | | 161 | \left[\begin{array}{llllllll} 0 & 2 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right] \in \mathbb{R}^{8,8} ? | |  |
| 405 | 6.18 | 162 | \boldsymbol{A}^{k} \boldsymbol{x}=\sum_{j=1}^{n} c_{j} \lambda_{j}^{k} \boldsymbol{x}_{j} \text { for some scalars } c_{1}, \ldots, c_{n} | |  |
| 406 | | 162 | \boldsymbol{A}=\left[\begin{array}{ccccc} -q_{n-1} & -q_{n-2} & \cdots & -q_{1} & -q_{0} \\ 1 & 0 & \cdots & 0 & 0 \\ 0 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & 0 \end{array}\right] . | |  |
| 407 | | 162 | \boldsymbol{B}=\left[\begin{array}{ccccc} 0 & 0 & \cdots & 0 & -q_{0} \\ 1 & 0 & \cdots & 0 & -q_{1} \\ 0 & 1 & \cdots & 0 & -q_{2} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & -q_{n-1} \end{array}\right] . | |  |
| 408 | | 163 | \boldsymbol{A}=\frac{1}{6}\left[\begin{array}{cc} 3 & 4 \\ 4 & -3 \end{array}\right] | |  |
| 409 | | 163 | \boldsymbol{A}=\left[\begin{array}{cc} a & 1 \\ 0 & a \end{array}\right] | |  |
| 410 | 6.19 | 164 | \mu_{\boldsymbol{A}}(\lambda):=\prod_{i=1}^{k}\left(\lambda_{i}-\lambda\right)^{m_{i}} \text { where } m_{i}:=\max _{1 \leq j \leq g_{i}} m_{i, j} \text {, } | |  |
| 411 | | 164 | p(\boldsymbol{A}):=\sum_{j=0}^{r} b_{j} \boldsymbol{A}^{j}, | |  |
| 412 | 6.20 | 165 | \boldsymbol{x}=\sum_{j=1}^{n} c_{j} \boldsymbol{u}_{j}, \quad \boldsymbol{A} \boldsymbol{x}=\sum_{j=1}^{n} c_{j} \lambda_{j} \boldsymbol{u}_{j}, \text { where } c_{j}=\frac{\boldsymbol{u}_{j}^{*} \boldsymbol{x}}{\boldsymbol{u}_{j}^{*} \boldsymbol{u}_{j}} . | |  |
| 413 | | 165 | \lambda_{\min }=\min _{\boldsymbol{x} \neq 0} R(\boldsymbol{x})=\min _{\|\boldsymbol{x}\|_{2}=1} R(\boldsymbol{x}), | |  |
| 414 | | 165 | R(x):=\frac{x^{T} A x}{x^{T} x} . | |  |
| 415 | | 166 | R(\boldsymbol{x}-t \boldsymbol{y})=R(\boldsymbol{x})-2 t(\boldsymbol{A x}-R(\boldsymbol{x}) \boldsymbol{x})^{T} \boldsymbol{y}+\mathcal{O}\left(t^{2}\right), | |  |
| 416 | | 166 | B_{1}:=\left\{\boldsymbol{x} \in \mathbb{R}^{n} \mid\|\boldsymbol{x}\|_{2}=1\right\} . | |  |
| 417 | 7.1 | 168 | \boldsymbol{A}=\boldsymbol{U} \boldsymbol{D} \boldsymbol{U}^{*}, \text { where } \boldsymbol{U}^{*} \boldsymbol{U}=\boldsymbol{I} . | |  |
| 418 | 7.2 | 168 | \boldsymbol{A}=\frac{1}{15}\left[\begin{array}{cc} 14 & 2 \\ 4 & 22 \\ 16 & 13 \end{array}\right]=\frac{1}{3}\left[\begin{array}{rrr} 1 & 2 & 2 \\ 2 & -2 & 1 \\ 2 & 1 & -2 \end{array}\right]\left[\begin{array}{ll} 2 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}\right] \frac{1}{5}\left[\begin{array}{rr} 3 & 4 \\ 4 & -3 \end{array}\right]=\boldsymbol{U} \boldsymbol{\Sigma} \boldsymbol{V}^{*} . | |  |
| 419 | | 169 | \lambda_{1} \geq \cdots \geq \lambda_{r}>0=\lambda_{r+1}=\cdots=\lambda_{n} . | |  |
| 420 | | 169 | \lambda^{m} \pi_{\boldsymbol{A}^{*} \boldsymbol{A}}(\lambda)=\lambda^{n} \pi_{\boldsymbol{A} \boldsymbol{A}^{*}}(\lambda), \quad \lambda \in \mathbb{C}, | |  |
| 421 | 7.3 | 170 | \lambda=\frac{\boldsymbol{v}^{*} \boldsymbol{A}^{*} \boldsymbol{A} \boldsymbol{v}}{\boldsymbol{v}^{*} \boldsymbol{v}}=\frac{\|\boldsymbol{A} \boldsymbol{v}\|_{2}^{2}}{\|\boldsymbol{v}\|_{2}^{2}} \geq 0 . | |  |
| 422 | | 170 | \begin{aligned} \boldsymbol{U} \boldsymbol{\Sigma} & =\boldsymbol{U}\left[\sigma_{1} \boldsymbol{e}_{1}, \ldots, \sigma_{r} \boldsymbol{e}_{r}, 0, \ldots, 0\right] \\ & =\left[\sigma_{1} \boldsymbol{u}_{1}, \ldots, \sigma_{r} \boldsymbol{u}_{r}, 0, \ldots, 0\right] \\ & =\left[\boldsymbol{A} \boldsymbol{v}_{1}, \ldots, \boldsymbol{A} \boldsymbol{v}_{n}\right] . \end{aligned} | |  |
| 423 | | 171 | \boldsymbol{B}:=\boldsymbol{A}^{T} \boldsymbol{A}=\frac{1}{25}\left[\begin{array}{ll} 52 & 36 \\ 36 & 73 \end{array}\right] | |  |
| 424 | | 171 | \boldsymbol{B}\left[\begin{array}{l} 3 \\ 4 \end{array}\right]=4\left[\begin{array}{l} 3 \\ 4 \end{array}\right], \quad \boldsymbol{B}\left[\begin{array}{r} 4 \\ -3 \end{array}\right]=1\left[\begin{array}{r} 4 \\ -3 \end{array}\right] . | |  |
| 425 | 7.4 | 171 | \begin{aligned} & \boldsymbol{U}=\left[\boldsymbol{U}_{1}, \boldsymbol{U}_{2}\right] \in \mathbb{C}^{m \times m}, \quad \boldsymbol{U}_{1}:=\left[\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{r}\right], \quad \boldsymbol{U}_{2}:=\left[\boldsymbol{u}_{r+1}, \ldots, \boldsymbol{u}_{m}\right], \\ & \boldsymbol{V}=\left[\boldsymbol{V}_{1}, \boldsymbol{V}_{2}\right] \in \mathbb{C}^{n \times n}, \quad \boldsymbol{V}_{1}:=\left[\boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{r}\right], \quad \boldsymbol{V}_{2}:=\left[\boldsymbol{v}_{r+1}, \ldots, \boldsymbol{v}_{n}\right], \\ & \boldsymbol{\Sigma}=\left[\begin{array}{cc} \boldsymbol{\Sigma}_{1} & \mathbf{0}_{r, n-r} \\ \mathbf{0}_{m-r, r} & \mathbf{0}_{m-r, n-r} \end{array}\right] \in \mathbb{R}^{m \times n}, \text { where } \boldsymbol{\Sigma}_{1}:=\operatorname{diag}\left(\sigma_{1}, \ldots, \sigma_{r}\right) . \end{aligned} | |  |
| 426 | 7.5 | 172 | \boldsymbol{A}=\boldsymbol{U} \boldsymbol{\Sigma} \boldsymbol{V}^{*}=\boldsymbol{U}_{1} \boldsymbol{\Sigma}_{1} \boldsymbol{V}_{1}^{*} . | |  |
| 427 | 2 | 172 | \left[\begin{array}{ll} 1 & -1 \\ 1 & -1 \end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right]\left[\begin{array}{ll} 2 & 0 \\ 0 & 0 \end{array}\right] \frac{1}{\sqrt{2}}\left[\begin{array}{cc} 1 & -1 \\ 1 & 1 \end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}{l} 1 \\ 1 \end{array}\right][2] \frac{1}{\sqrt{2}}\left[\begin{array}{ll} 1 & -1 \end{array}\right] . | |  |
| 428 | 7.6 | 172 | \boldsymbol{A}=\sum_{j=1}^{r} \sigma_{j} \boldsymbol{u}_{j} \boldsymbol{v}_{j}^{*} . | |  |
| 429 | | 172 | \boldsymbol{A}=\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \\ 0 & 0 \end{array}\right] . | |  |
| 430 | | 172 | \boldsymbol{B}:=\boldsymbol{A}^{T} \boldsymbol{A}=\left[\begin{array}{ll} 2 & 2 \\ 2 & 2 \end{array}\right] | |  |
| 431 | | 173 | \boldsymbol{B}\left[\begin{array}{l} 1 \\ 1 \end{array}\right]=4\left[\begin{array}{l} 1 \\ 1 \end{array}\right], \quad \boldsymbol{B}\left[\begin{array}{r} 1 \\ -1 \end{array}\right]=0\left[\begin{array}{r} 1 \\ -1 \end{array}\right], | |  |
| 432 | | 173 | \boldsymbol{\Sigma}=\left[\begin{array}{cc} \boldsymbol{\Sigma}_{1} & 0 \\ 0 & 0 \\ 0 & 0 \end{array}\right], \quad \boldsymbol{\Sigma}_{1}=[2], \quad \boldsymbol{V}=\frac{1}{\sqrt{2}}\left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right] . | |  |
| 433 | | 173 | \boldsymbol{A}=\frac{1}{\sqrt{2}}\left[\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right][2] \frac{1}{\sqrt{2}}\left[\begin{array}{ll} 1 & 1 \end{array}\right] . | |  |
| 434 | | 173 | \boldsymbol{s}_{1}=\left[\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right], \quad \boldsymbol{s}_{2}=\left[\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right], \quad \boldsymbol{s}_{3}=\left[\begin{array}{l} 0 \\ 0 \\ 1 \end{array}\right], | |  |
| 435 | | 173 | \boldsymbol{U}:=\left[\begin{array}{ccc} 1 / \sqrt{2} & -1 / \sqrt{2} & 0 \\ 1 / \sqrt{2} & 1 / \sqrt{2} & 0 \\ 0 & 0 & 1 \end{array}\right] \in \mathbb{R}^{3,3}, \quad \boldsymbol{\Sigma}:=\left[\begin{array}{ll} 2 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}\right] \in \mathbb{R}^{3,2}, \quad \boldsymbol{V}:=\frac{1}{\sqrt{2}}\left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right] \in \mathbb{R}^{2,2} . | |  |
| 436 | 7.7 | 174 | \begin{aligned} \boldsymbol{A} \boldsymbol{v}_{i} & =\sigma_{i} \boldsymbol{u}_{i}, i=1, \ldots, r, \quad \boldsymbol{A} \boldsymbol{v}_{i}=0, i=r+1, \ldots, n, \\ \boldsymbol{A}^{*} \boldsymbol{u}_{i} & =\sigma_{i} \boldsymbol{v}_{i}, i=1, \ldots, r, \quad \boldsymbol{A}^{*} \boldsymbol{u}_{i}=0, i=r+1, \ldots, m . \end{aligned} | |  |
| 437 | | 174 | \mathcal{E}=\boldsymbol{U}_{1} \tilde{\mathcal{E}} \text { where } \tilde{\mathcal{E}}:=\left\{\boldsymbol{y}=\left[y_{1}, \ldots, y_{n}\right]^{T} \in \mathbb{R}^{n}: \frac{y_{1}^{2}}{\sigma_{1}^{2}}+\cdots+\frac{y_{n}^{2}}{\sigma_{n}^{2}}=1\right\} . | |  |
| 438 | | 175 | 1=\|z\|_{2}^{2}=\left\|\boldsymbol{V}_{1} \boldsymbol{\Sigma}_{1}^{-1} \boldsymbol{y}\right\|_{2}^{2}=\left\|\boldsymbol{\Sigma}_{1}^{-1} \boldsymbol{y}\right\|_{2}^{2}=\frac{y_{1}^{2}}{\sigma_{1}^{2}}+\cdots+\frac{y_{n}^{2}}{\sigma_{n}^{2}} . | |  |
| 439 | | 175 | A:=\frac{1}{25}\left[\begin{array}{ll} 11 & 48 \\ 48 & 39 \end{array}\right] | |  |
| 440 | | 176 | \frac{\left(\frac{3}{5} x_{1}+\frac{4}{5} x_{2}\right)^{2}}{9}+\frac{\left(-\frac{4}{5} x_{1}+\frac{3}{5} x_{2}\right)^{2}}{1}=1, | |  |
| 441 | 7.9 | 176 | \|\boldsymbol{A}\|_{F}:=\left(\sum_{i=1}^{m} \sum_{j=1}^{n}\left|a_{i j}\right|^{2}\right)^{1 / 2} . | |  |
| 442 | | 177 | \|\boldsymbol{A} \boldsymbol{B}\|_{F}^{2}=\sum_{i=1}^{m} \sum_{j=1}^{k}\left|\boldsymbol{a}_{i:}^{*} \boldsymbol{b}_{: j}\right|^{2} \leq \sum_{i=1}^{m} \sum_{j=1}^{k}\left\|\boldsymbol{a}_{i:}\right\|_{2}^{2}\left\|\boldsymbol{b}_{: j}\right\|_{2}^{2}=\|\boldsymbol{A}\|_{F}^{2}\|\boldsymbol{B}\|_{F}^{2} . | |  |
| 443 | | 177 | \|\boldsymbol{A}\|_{F} \stackrel{\text { 3. }}{=}\left\|\boldsymbol{U}^{*} \boldsymbol{A} \boldsymbol{V}\right\|_{F}=\|\boldsymbol{\Sigma}\|_{F}=\sqrt{\sigma_{1}^{2}+\cdots+\sigma_{n}^{2}} . | |  |
| 444 | | 177 | \left\|\boldsymbol{A}-\boldsymbol{A}^{\prime}\right\|_{F}=\left\|\boldsymbol{U}\left[\begin{array}{c} \boldsymbol{D}-\boldsymbol{D}^{\prime} \\ \mathbf{0} \end{array}\right] \boldsymbol{V}^{*}\right\|_{F}=\left\|\left[\begin{array}{c} \boldsymbol{D}-\boldsymbol{D}^{\prime} \\ \mathbf{0} \end{array}\right]\right\|_{F}=\sqrt{\sigma_{r+1}^{2}+\cdots+\sigma_{n}^{2}}<\epsilon . | |  |
| 445 | | 178 | \left\|\boldsymbol{A}-\boldsymbol{A}^{\prime}\right\|_{F}=\min _{\substack{\boldsymbol{B} \in \mathbb{R}^{m \times n} \\ \operatorname{rank}(\boldsymbol{B})=r}}\|\boldsymbol{A}-\boldsymbol{B}\|_{F}=\sqrt{\sigma_{r+1}^{2}+\cdots+\sigma_{n}^{2}} . | |  |
| 446 | 7.10 | 178 | \boldsymbol{A}:=\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right]\left[\begin{array}{ll} 2 & 0 \\ 0 & 0 \end{array}\right] \frac{1}{\sqrt{2}}\left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right]=\boldsymbol{U} \boldsymbol{D} \boldsymbol{U}^{T} | |  |
| 447 | 7.11 | 178 | \boldsymbol{A}:=\left[\begin{array}{ll} 1 & -1 \\ 1 & -1 \end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right]\left[\begin{array}{ll} 2 & 0 \\ 0 & 0 \end{array}\right] \frac{1}{\sqrt{2}}\left[\begin{array}{cc} 1 & -1 \\ 1 & 1 \end{array}\right]=: \boldsymbol{U} \boldsymbol{\Sigma} \boldsymbol{V}^{T} | |  |
| 448 | | 179 | A:=\frac{1}{15}\left[\begin{array}{ccc} 14 & 4 & 16 \\ 2 & 22 & 13 \end{array}\right] \in \mathbb{R}^{2 \times 3} . | |  |
| 449 | | 180 | \boldsymbol{C}:=\left[\begin{array}{cc} \mathbf{0} & \boldsymbol{A} \\ \boldsymbol{A}^{*} & \mathbf{0} \end{array}\right] \in \mathbb{R}^{(m+n) \times(m+n)} | |  |
| 450 | | 180 | \left\{\left(\sigma_{1}, \boldsymbol{p}_{1}\right), \ldots,\left(\sigma_{n}, \boldsymbol{p}_{n}\right),\left(-\sigma_{1}, \boldsymbol{q}_{1}\right), \ldots,\left(-\sigma_{n}, \boldsymbol{q}_{n}\right),\left(0, \boldsymbol{r}_{n+1}\right), \ldots,\left(0, \boldsymbol{r}_{m}\right)\right\}, | |  |
| 451 | | 180 | \boldsymbol{p}_{i}=\left[\begin{array}{c} \boldsymbol{u}_{i} \\ \boldsymbol{v}_{i} \end{array}\right], \quad \boldsymbol{q}_{i}=\left[\begin{array}{c} \boldsymbol{u}_{i} \\ -\boldsymbol{v}_{i} \end{array}\right], \quad \boldsymbol{r}_{j}=\left[\begin{array}{c} \boldsymbol{u}_{j} \\ \mathbf{0} \end{array}\right], \text { for } i=1, \ldots, n, j=n+1, \ldots, m . | |  |
| 452 | 7.12 | 180 | \boldsymbol{Q}:=\boldsymbol{U} \boldsymbol{V}^{T}, \quad \boldsymbol{P}:=\boldsymbol{V} \boldsymbol{\Sigma} \boldsymbol{V}^{T} | |  |
| 453 | 7.13 | 180 | A=Q P | |  |
| 454 | 7.14 | 181 | \boldsymbol{X}_{k+1}=\frac{1}{2}\left(\boldsymbol{X}_{k}+\boldsymbol{X}_{k}^{-T}\right), k=0,1,2, \ldots \text { with } \boldsymbol{X}_{0}=\boldsymbol{A}, | |  |
| 455 | 7.15 | 181 | \boldsymbol{X}_{k+1}-\boldsymbol{Q}=\frac{1}{2} \boldsymbol{X}_{k}^{-T}\left(\boldsymbol{X}_{k}^{T}-\boldsymbol{Q}^{T}\right)\left(\boldsymbol{X}_{k}-\boldsymbol{Q}\right) | |  |
| 456 | 7.16 | 181 | \left\|\boldsymbol{X}_{k+1}-\boldsymbol{Q}\right\|_{F} \leq \frac{1}{2}\left\|\boldsymbol{X}_{k}^{-1}\right\|_{F}\left\|\boldsymbol{X}_{k}-\boldsymbol{Q}\right\|_{F}^{2} . | |  |
| 457 | | 181 | \boldsymbol{A}=\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ -1 & 3 \end{array}\right] . | |  |
| 458 | | 181 | \boldsymbol{B}=\left[\begin{array}{ccc} 1 & 0 & -1 \\ 1 & 1 & 1 \end{array}\right] . | |  |
| 459 | | 182 | \begin{aligned} & x_{1}-x_{3}=4, \\ & x_{1}+x_{2}+x_{3}=12 . \end{aligned} | |  |
| 460 | 2 | 182 | \boldsymbol{A}:=\left[\begin{array}{rrr} 0 & 3 & 3 \\ 4 & 1 & -1 \\ 4 & 1 & -1 \\ 0 & 3 & 3 \end{array}\right]=\left[\begin{array}{rrrr} \frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} \\ \frac{1}{2} & -\frac{1}{2} & \frac{1}{2} & -\frac{1}{2} \end{array}\right]\left[\begin{array}{lll} 6 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\left[\begin{array}{rrr} \frac{2}{3} & \frac{2}{3} & \frac{1}{3} \\ \frac{2}{3} & -\frac{1}{3} & -\frac{2}{3} \\ \frac{1}{3} & -\frac{2}{3} & \frac{2}{3} \end{array}\right] | |  |
| 461 | | 182 | \boldsymbol{A}=\left[\begin{array}{rrrr} 1 & -1 & -1 & -1 \\ 0 & 1 & -1 & -1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 1 \end{array}\right] . | |  |
| 462 | | 183 | \boldsymbol{A}=\left[\begin{array}{cc} 3 & 1 \\ 2 & 3 \\ -1 & 5 \end{array}\right] . | |  |
| 463 | | 183 | \boldsymbol{T}=\left[\begin{array}{cc} 2 & -1 \\ -1 & 2 \end{array}\right] . | |  |
| 464 | | 183 | \left\|\boldsymbol{A}-\boldsymbol{A}^{\prime}\right\|_{F}^{2}=\sigma_{r+1}^{2}+\cdots+\sigma_{n}^{2} . | |  |
| 465 | 8.1 | 186 | \begin{aligned} \|\boldsymbol{x}\|_{p} & :=\left(\sum_{j=1}^{n}\left|x_{j}\right|^{p}\right)^{1 / p}, \\ \|\boldsymbol{x}\|_{\infty} & :=\max _{1 \leq j \leq n}\left|x_{j}\right| . \end{aligned} | |  |
| 466 | 8.3 | 186 | \sum_{j=1}^{n}\left|x_{j} y_{j}\right| \leq\|\boldsymbol{x}\|_{p}\|\boldsymbol{y}\|_{q}, \quad \frac{1}{p}+\frac{1}{q}=1, \quad \boldsymbol{x}, \boldsymbol{y} \in \mathbb{C}^{n} . | |  |
| 467 | 8.4 | 186 | \lim _{p \rightarrow \infty}\|\boldsymbol{x}\|_{p}=\|\boldsymbol{x}\|_{\infty} \text { for all } \boldsymbol{x} \in \mathbb{C}^{n} . | |  |
| 468 | 8.2 | 186 | \|\boldsymbol{x}\|_{p}:=\|\boldsymbol{x}\|_{\infty}\left(\sum_{j=1}^{n}\left(\frac{\left|x_{j}\right|}{\|\boldsymbol{x}\|_{\infty}}\right)^{p}\right)^{1 / p} . | |  |
| 469 | 8.5 | 186 | \|x\|_{\infty} \leq\|x\|_{p} \leq n^{1 / p}\|x\|_{\infty}, \quad p \geq 1 . | |  |
| 470 | 8.6 | 187 | \|\boldsymbol{x}\|_{p^{\prime}} \leq\|\boldsymbol{x}\|_{p} \leq n^{1 / p-1 / p^{\prime}}\|\boldsymbol{x}\|_{p^{\prime}}, \quad \boldsymbol{x} \in \mathbb{C}^{n}, \quad 1 \leq p \leq p^{\prime} \leq \infty . | |  |
| 471 | 8.7 | 187 | m\|\boldsymbol{x}\|^{\prime} \leq\|\boldsymbol{x}\| \leq M\|\boldsymbol{x}\|^{\prime} . | |  |
| 472 | | 187 | \mathcal{S}:=\left\{\boldsymbol{y} \in \mathcal{V}:\|\boldsymbol{y}\|^{\prime}=1\right\} . | |  |
| 473 | 8.8 | 187 | m \leq\|\boldsymbol{y}\| \leq M, \quad \boldsymbol{y} \in \mathcal{S} . | |  |
| 474 | | 188 | \|\boldsymbol{A}\|_{F}:=\left(\sum_{i=1}^{m} \sum_{j=1}^{n}\left|a_{i j}\right|^{2}\right)^{1 / 2} | |  |
| 475 | | 188 | \mu\|\boldsymbol{A}\| \leq\|\boldsymbol{A}\|^{\prime} \leq M\|\boldsymbol{A}\| | |  |
| 476 | 8.9 | 188 | \|\boldsymbol{A}\|_{S}:=\sum_{i=1}^{m} \sum_{j=1}^{n}\left|a_{i j}\right|, \quad\|\boldsymbol{A}\|_{F}:=\left(\sum_{i=1}^{m} \sum_{j=1}^{n}\left|a_{i j}\right|^{2}\right)^{1 / 2}, \quad\|\boldsymbol{A}\|_{M}:=\max _{i, j}\left|a_{i j}\right| . | |  |
| 477 | 8.10 | 189 | \left\|\boldsymbol{A}^{k}\right\| \leq\|\boldsymbol{A}\|^{k} \text { for } \boldsymbol{A} \in \mathbb{C}^{n \times n} \text { and } k \in \mathbb{N} \text {. } | |  |
| 478 | 8.11 | 189 | \|\boldsymbol{A} \boldsymbol{x}\| \leq\|\boldsymbol{A}\|\|\boldsymbol{x}\|, | |  |
| 479 | 8.12 | 190 | \|\boldsymbol{A}\|:=\max _{\boldsymbol{x} \neq 0} \frac{\|\boldsymbol{A} \boldsymbol{x}\|}{\|\boldsymbol{x}\|} . | |  |
| 480 | 8.13 | 190 | \|\boldsymbol{A}\|=\max _{\|\boldsymbol{x}\|=1}\|\boldsymbol{A} \boldsymbol{x}\| . | |  |
| 481 | 8.14 | 190 | \mathcal{S}:=\left\{x \in \mathbb{C}^{n}:\|\boldsymbol{x}\|=1\right\} | |  |
| 482 | | 190 | \max _{\boldsymbol{x} \neq 0} \frac{\|\boldsymbol{A} \boldsymbol{x}\|}{\|\boldsymbol{x}\|}=\max _{\boldsymbol{x} \neq 0}\left\|\boldsymbol{A}\left(\frac{\boldsymbol{x}}{\|\boldsymbol{x}\|}\right)\right\|=\max _{\|\boldsymbol{y}\|=1}\|\boldsymbol{A} \boldsymbol{y}\| . | |  |
| 483 | 8.15 | 190 | \|\boldsymbol{A} \boldsymbol{x}\| \leq\|\boldsymbol{A}\|\|\boldsymbol{x}\| \text { for all } \boldsymbol{A} \in \mathbb{C}^{m \times n} \text { and } \boldsymbol{x} \in \mathbb{C}^{n} . | |  |
| 484 | 8.16 | 190 | \|\boldsymbol{A}\|=\left\|\boldsymbol{A} \boldsymbol{x}^{*}\right\| \text { for some } \boldsymbol{x}^{*} \in \mathcal{S} . | |  |
| 485 | | 191 | \begin{aligned} & \|\boldsymbol{A} \boldsymbol{B}\|=\max _{\boldsymbol{x} \neq \mathbf{0}} \frac{\|\boldsymbol{A} \boldsymbol{B} \boldsymbol{x}\|}{\|\boldsymbol{x}\|}=\max _{\boldsymbol{B} \boldsymbol{x} \neq \mathbf{0}} \frac{\|\boldsymbol{A} \boldsymbol{B} \boldsymbol{x}\|}{\|\boldsymbol{x}\|}=\max _{\boldsymbol{B} \boldsymbol{x} \neq \mathbf{0}} \frac{\|\boldsymbol{A} \boldsymbol{B} \boldsymbol{x}\|}{\|\boldsymbol{B} \boldsymbol{x}\|} \frac{\|\boldsymbol{B} \boldsymbol{x}\|}{\|\boldsymbol{x}\|} \\ & \quad \leq \max _{\boldsymbol{y} \neq \mathbf{0}} \frac{\|\boldsymbol{A} \boldsymbol{y}\|}{\|\boldsymbol{y}\|} \max _{\boldsymbol{x} \neq \mathbf{0}} \frac{\|\boldsymbol{B} \boldsymbol{x}\|}{\|\boldsymbol{x}\|}=\|\boldsymbol{A}\|\|\boldsymbol{B}\| \end{aligned} | |  |
| 486 | | 191 | \|x\|_{p}:=\left(\sum_{j=1}^{n}\left|x_{j}\right|^{p}\right)^{1 / p}, p \geq 1, \quad\|x\|_{\infty}:=\max _{1 \leq j \leq n}\left|x_{j}\right| . | |  |
| 487 | 8.17 | 191 | \|\boldsymbol{A}\|_{p}:=\max _{\boldsymbol{x} \neq 0} \frac{\|\boldsymbol{A} \boldsymbol{x}\|_{p}}{\|\boldsymbol{x}\|_{p}}=\max _{\|\boldsymbol{y}\|_{p}=1}\|\boldsymbol{A} \boldsymbol{y}\|_{p} . | |  |
| 488 | 8.18 | 191 | \begin{aligned} & \|\boldsymbol{A}\|_{1}:=\max _{1 \leq j \leq n}\left\|\boldsymbol{A} \boldsymbol{e}_{j}\right\|_{1}=\max _{1 \leq j \leq n} \sum_{k=1}^{m}\left|a_{k, j}\right|, \quad \text { (max column sum) } \\ & \|\boldsymbol{A}\|_{2}:=\sigma_{1}, \quad \text { (largest singular value of } \boldsymbol{A} \text { ) } \\ & \|\boldsymbol{A}\|_{\infty}=\max _{1 \leq k \leq m}\left\|\boldsymbol{e}_{k}^{T} \boldsymbol{A}\right\|_{1}=\max _{1 \leq k \leq m} \sum_{j=1}^{n}\left|a_{k, j}\right| . \quad \text { (max row sum) } \end{aligned} | |  |
| 489 | | 192 | \|\boldsymbol{A} \boldsymbol{x}\|_{1}=\sum_{k=1}^{m}\left|\sum_{j=1}^{n} a_{k j} x_{j}\right| \leq \sum_{k=1}^{m} \sum_{j=1}^{n}\left|a_{k j}\right|\left|x_{j}\right|=\sum_{j=1}^{n}\left(\sum_{k=1}^{m}\left|a_{k j}\right|\right)\left|x_{j}\right| \leq K_{1} . | |  |
| 490 | | 192 | \|\boldsymbol{A} \boldsymbol{x}\|_{\infty}=\max _{1 \leq k \leq m}\left|\sum_{j=1}^{n} a_{k j} x_{j}\right| \leq \max _{1 \leq k \leq m} \sum_{j=1}^{n}\left|a_{k j}\right|\left|x_{j}\right| \leq K_{\infty} . | |  |
| 491 | | 192 | \|\boldsymbol{A}\|_{1}=\frac{29}{15}, \quad\|\boldsymbol{A}\|_{2}=2, \quad\|\boldsymbol{A}\|_{\infty}=\frac{37}{15}, \quad\|\boldsymbol{A}\|_{F}=\sqrt{5} . | |  |
| 492 | 8.20 | 193 | \begin{aligned} & \|\boldsymbol{A}\|_{2}=\sigma_{1} \text { and }\left\|\boldsymbol{A}^{-1}\right\|_{2}=\frac{1}{\sigma_{n}} \\ & \|\boldsymbol{A}\|_{2}=\lambda_{1} \text { and }\left\|\boldsymbol{A}^{-1}\right\|_{2}=\frac{1}{\lambda_{n}}, \text { if } \boldsymbol{A} \text { is positive definite, } \\ & \|\boldsymbol{A}\|_{2}=\left|\lambda_{1}\right| \text { and }\left\|\boldsymbol{A}^{-1}\right\|_{2}=\frac{1}{\left|\lambda_{n}\right|}, \text { if } \boldsymbol{A} \text { is normal. } \end{aligned} | |  |
| 493 | 8.21 | 193 | \|\boldsymbol{A}\|_{2}^{2}\|\boldsymbol{v}\|_{1}=\sigma^{2}\|\boldsymbol{v}\|_{1}=\left\|\sigma^{2} \boldsymbol{v}\right\|_{1}=\left\|\boldsymbol{A}^{*} \boldsymbol{A} \boldsymbol{v}\right\|_{1} \leq\left\|\boldsymbol{A}^{*}\right\|_{1}\|\boldsymbol{A}\|_{1}\|\boldsymbol{v}\|_{1} . | |  |
| 494 | | 194 | \left\|\boldsymbol{A}^{*}\right\|_{F}=\|\boldsymbol{A}\|_{F} \text { and }\left\|\boldsymbol{A}^{*}\right\|_{2}=\|\boldsymbol{A}\|_{2} . | |  |
| 495 | | 194 | \|\boldsymbol{U} \boldsymbol{A}\|_{2}=\max _{\|\boldsymbol{x}\|_{2}=1}\|\boldsymbol{U} \boldsymbol{A} \boldsymbol{x}\|_{2}=\max _{\|\boldsymbol{x}\|_{2}=1}\|\boldsymbol{A} \boldsymbol{x}\|_{2}=\|\boldsymbol{A}\|_{2} . | |  |
| 496 | | 194 | \|\boldsymbol{A} \boldsymbol{V}\|_{2}=\left\|(\boldsymbol{A} \boldsymbol{V})^{*}\right\|_{2}=\left\|\boldsymbol{V}^{*} \boldsymbol{A}^{*}\right\|_{2}=\left\|\boldsymbol{A}^{*}\right\|_{2}=\|\boldsymbol{A}\|_{2} . | |  |
| 497 | | 194 | \left|x_{i}\right| \leq\left|y_{i}\right|, i=1, \ldots, n \Longrightarrow\|\boldsymbol{x}\| \leq\|\boldsymbol{y}\| \text {, for all } \boldsymbol{x}, \boldsymbol{y} \in \mathbb{C}^{n} . | |  |
| 498 | | 194 | \begin{aligned} & x_{1}+\left(1-10^{-16}\right) x_{2}=20 \\ & x_{1}+\left(1-10^{-15}\right. \end{aligned} | |  |
| 499 | | 195 | x_{1}+\left(1+10^{-16}\right) x_{2}=20-10^{-15}, | |  |
| 500 | | 195 | \|\boldsymbol{A} \boldsymbol{B}\| \leq\|\boldsymbol{A}\|\|\boldsymbol{B}\| \text { and }\|\boldsymbol{A} \boldsymbol{x}\| \leq\|\boldsymbol{A}\|\|\boldsymbol{x}\| . | |  |
| 501 | 8.22 | 195 | \frac{1}{K(\boldsymbol{A})} \frac{\|\boldsymbol{e}\|}{\|\boldsymbol{b}\|} \leq \frac{\|\boldsymbol{y}-\boldsymbol{x}\|}{\|\boldsymbol{x}\|} \leq K(\boldsymbol{A}) \frac{\|\boldsymbol{e}\|}{\|\boldsymbol{b}\|}, | |  |
| 502 | 8.23 | 196 | \frac{1}{K(\boldsymbol{A})} \frac{\|\boldsymbol{r}(\boldsymbol{y})\|}{\|\boldsymbol{b}\|} \leq \frac{\|\boldsymbol{y}-\boldsymbol{x}\|}{\|\boldsymbol{x}\|} \leq K(\boldsymbol{A}) \frac{\|\boldsymbol{r}(\boldsymbol{y})\|}{\|\boldsymbol{b}\|} . | |  |
| 503 | 8.24 | 196 | \begin{aligned} & \frac{\|\boldsymbol{y}-\boldsymbol{x}\|}{\|\boldsymbol{y}\|} \leq r \leq K(\boldsymbol{A}) \frac{\|\boldsymbol{E}\|}{\|\boldsymbol{A}\|}, \\ & \frac{\|\boldsymbol{y}-\boldsymbol{x}\|}{\|\boldsymbol{x}\|} \leq \frac{r}{1-r} \leq \frac{K(\boldsymbol{A})}{1-r} \frac{\|\boldsymbol{E}\|}{\|\boldsymbol{A}\|} . \end{aligned} | |  |
| 504 | 8.26 | 197 | K_{2}(\boldsymbol{A})= \begin{cases}\lambda_{1} / \lambda_{n}, & \text { if } \boldsymbol{A} \text { is positive definite, } \\ \left|\lambda_{1}\right| /\left|\lambda_{n}\right|, & \text { if } \boldsymbol{A} \text { is normal, } \\ \sigma_{1} / \sigma_{n}, & \text { in general. }\end{cases} | |  |
| 505 | 8.27 | 198 | \frac{\left\|\boldsymbol{B}^{-1}-\boldsymbol{A}^{-1}\right\|}{\left\|\boldsymbol{B}^{-1}\right\|} \leq\left\|\boldsymbol{A}^{-1} \boldsymbol{E}\right\| \leq K(\boldsymbol{A}) \frac{\|\boldsymbol{E}\|}{\|\boldsymbol{A}\|}, | |  |
| 506 | 8.28 | 198 | \frac{1}{1+r} \leq \frac{\left\|\boldsymbol{B}^{-1}\right\|}{\left\|\boldsymbol{A}^{-1}\right\|} \leq \frac{1}{1-r} . | |  |
| 507 | 8.29 | 198 | \frac{\left\|\boldsymbol{B}^{-1}-\boldsymbol{A}^{-1}\right\|}{\left\|\boldsymbol{A}^{-1}\right\|} \leq \frac{r}{1-r} \leq \frac{K(\boldsymbol{A})}{1-r} \frac{\|\boldsymbol{E}\|}{\|\boldsymbol{A}\|} . | |  |
| 508 | 8.30 | 198 | \boldsymbol{B}^{-1}-\boldsymbol{A}^{-1}=-\boldsymbol{A}^{-1} \boldsymbol{E} \boldsymbol{B}^{-1}=-\boldsymbol{B}^{-1} \boldsymbol{E} \boldsymbol{A}^{-1} . | |  |
| 509 | | 198 | \left\|\boldsymbol{B}^{-1}-\boldsymbol{A}^{-1}\right\| \leq\left\|\boldsymbol{A}^{-1} \boldsymbol{E}\right\|\left\|\boldsymbol{B}^{-1}\right\| \leq K(A) \frac{\|\boldsymbol{E}\|}{\|\boldsymbol{A}\|}\left\|\boldsymbol{B}^{-1}\right\| . | |  |
| 510 | | 198 | \left\|\boldsymbol{B}^{-1}\right\| \leq\left\|\boldsymbol{A}^{-1}\right\|+\left\|\boldsymbol{A}^{-1} \boldsymbol{E} \boldsymbol{B}^{-1}\right\| \leq\left\|\boldsymbol{A}^{-1}\right\|+r\left\|\boldsymbol{B}^{-1}\right\| . | |  |
| 511 | | 199 | \|\boldsymbol{x}\|_{p}:=\left(\sum_{j=1}^{n}\left|x_{j}\right|^{p}\right)^{1 / p}, \quad\|\boldsymbol{x}\|_{\infty}:=\max _{1 \leq j \leq n}\left|x_{j}\right| . | |  |
| 512 | 8.31 | 199 | f\left((1-\lambda) x_{1}+\lambda x_{2}\right) \leq(1-\lambda) f\left(x_{1}\right)+\lambda f\left(x_{2}\right) | |  |
| 513 | | 200 | \begin{aligned} f(x) & =\frac{x_{2}-x}{x_{2}-x_{1}} f\left(x_{1}\right)+\frac{x-x_{1}}{x_{2}-x_{1}} f\left(x_{2}\right)+\left(x-x_{1}\right)\left(x-x_{2}\right) f^{\prime \prime}(c) / 2 \\ & =(1-\lambda) f\left(x_{1}\right)+\lambda f\left(x_{2}\right)+\left(x_{2}-x_{1}\right)^{2} \lambda(\lambda-1) f^{\prime \prime}(c) / 2, \quad \lambda:=\frac{x-x_{1}}{x_{2}-x_{1}} \end{aligned} | |  |
| 514 | | 200 | x=\frac{x_{2}-x}{x_{2}-x_{1}} x_{1}+\frac{x-x_{1}}{x_{2}-x_{1}} x_{2}=(1-\lambda) x_{1}+\lambda x_{2} | |  |
| 515 | | 200 | f\left(\sum_{j=1}^{n} \lambda_{j} z_{j}\right) \leq \sum_{j=1}^{n} \lambda_{j} f\left(z_{j}\right) . | |  |
| 516 | | 200 | f\left(\sum_{j=1}^{n} \lambda_{j} z_{j}\right)=f\left(\lambda_{1} z_{1}+\left(1-\lambda_{1}\right) u\right) \leq \lambda_{1} f\left(z_{1}\right)+\left(1-\lambda_{1}\right) f(u) \leq \sum_{j=1}^{n} \lambda_{j} f\left(z_{j}\right) | |  |
| 517 | 8.32 | 201 | a_{1}^{\lambda_{1}} a_{2}^{\lambda_{2}} \cdots a_{n}^{\lambda_{n}} \leq \sum_{j=1}^{n} \lambda_{j} a_{j}, | |  |
| 518 | | 201 | -\log \left(\sum_{j=1}^{n} \lambda_{j} a_{j}\right) \leq-\sum_{j=1}^{n} \lambda_{j} \log \left(a_{j}\right)=-\log \left(a_{1}^{\lambda_{1}} \cdots a_{n}^{\lambda_{n}}\right) | |  |
| 519 | 8.33 | 201 | \left(a_{1} a_{2} \cdots a_{n}\right)^{\frac{1}{n}} \leq \frac{1}{n} \sum_{j=1}^{n} a_{j} . | |  |
| 520 | | 201 | \sum_{j=1}^{n}\left|x_{j} y_{j}\right| \leq\|\boldsymbol{x}\|_{p}\|\boldsymbol{y}\|_{q}, \text { where } \frac{1}{p}+\frac{1}{q}=1 | |  |
| 521 | 8.34 | 201 | a^{\frac{1}{p}} b^{\frac{1}{q}} \leq \frac{1}{p} a+\frac{1}{q} b, \text { where } \frac{1}{p}+\frac{1}{q}=1 . | |  |
| 522 | | 201 | \frac{1}{\|\boldsymbol{x}\|_{p}\|\boldsymbol{y}\|_{q}} \sum_{j=1}^{n}\left|x_{j} y_{j}\right|=\sum_{j=1}^{n}\left(\frac{\left|x_{j}\right|^{p}}{\|\boldsymbol{x}\|_{p}^{p}}\right)^{\frac{1}{p}}\left(\frac{\left|y_{j}\right|^{q}}{\|\boldsymbol{y}\|_{q}^{q}}\right)^{\frac{1}{q}} \leq \sum_{j=1}^{n}\left(\frac{1}{p} \frac{\left|x_{j}\right|^{p}}{\|\boldsymbol{x}\|_{p}^{p}}+\frac{1}{q} \frac{\left|y_{j}\right|^{q}}{\|\boldsymbol{y}\|_{q}^{q}}\right)=1 | |  |
| 523 | | 202 | \|\boldsymbol{x}+\boldsymbol{y}\|_{p} \leq\|\boldsymbol{x}\|_{p}+\|\boldsymbol{y}\|_{p} . | |  |
| 524 | | 202 | \|\boldsymbol{x}+\boldsymbol{y}\|_{p}^{p}=\sum_{j=1}^{n}\left|x_{j}+y_{j}\right|^{p} \leq \sum_{j=1}^{n}\left|x_{j}\right|\left|x_{j}+y_{j}\right|^{p-1}+\sum_{j=1}^{n}\left|y_{j}\right|\left|x_{j}+y_{j}\right|^{p-1} . | |  |
| 525 | | 202 | \|\boldsymbol{x}+\boldsymbol{y}\|_{p}^{p} \leq\|\boldsymbol{x}\|_{p}\|\boldsymbol{x}+\boldsymbol{y}\|_{p}^{p / q}+\|\boldsymbol{y}\|_{p}\|\boldsymbol{x}+\boldsymbol{y}\|_{p}^{p / q}=\left(\|\boldsymbol{x}\|_{p}+\|\boldsymbol{y}\|_{p}\right)\|\boldsymbol{x}+\boldsymbol{y}\|_{p}^{p-1}, | |  |
| 526 | 8.35 | 202 | \|\boldsymbol{x}+\boldsymbol{y}\|^{2}+\|\boldsymbol{x}-\boldsymbol{y}\|^{2}=2\|\boldsymbol{x}\|^{2}+2\|\boldsymbol{y}\|^{2}, | |  |
| 527 | | 202 | \|\boldsymbol{x}+\boldsymbol{y}\|^{2}+\|\boldsymbol{x}-\boldsymbol{y}\|^{2}=\langle\boldsymbol{x}+\boldsymbol{y}, \boldsymbol{x}+\boldsymbol{y}\rangle+\langle\boldsymbol{x}-\boldsymbol{y}, \boldsymbol{x}-\boldsymbol{y}\rangle=2\|\boldsymbol{x}\|^{2}+2\|\boldsymbol{y}\|^{2} | |  |
| 528 | 8.36 | 202 | \langle\boldsymbol{x}, \boldsymbol{y}\rangle:=\frac{1}{4}\left(\|\boldsymbol{x}+\boldsymbol{y}\|^{2}-\|\boldsymbol{x}-\boldsymbol{y}\|^{2}\right), \quad \boldsymbol{x}, \boldsymbol{y} \in \mathcal{V} | |  |
| 529 | 8.37 | 203 | \begin{gathered} \langle\boldsymbol{x}, \boldsymbol{z}\rangle+\langle\boldsymbol{y}, \boldsymbol{z}\rangle=\langle\boldsymbol{x}+\boldsymbol{y}, \boldsymbol{z}\rangle, \quad \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z} \in \mathcal{V} \\ \langle a \boldsymbol{x}, \boldsymbol{y}\rangle=a\langle\boldsymbol{x}, \boldsymbol{y}\rangle, \quad a \in \mathbb{R}, \quad \boldsymbol{x}, \boldsymbol{y} \in \mathcal{V} . \end{gathered} | |  |
| 530 | 8.38 | 203 | \begin{aligned} & 4\langle x, z\rangle+4\langle y, z\rangle \stackrel{(8.36)}{=}\|x+z\|^{2}-\|x-z\|^{2}+\|y+z\|^{2}-\|y-z\|^{2} \\ & =\left\|\left(z+\frac{x+y}{2}\right)+\frac{x-y}{2}\right\|^{2}-\left\|\left(z-\frac{x+y}{2}\right)+\frac{y-x}{2}\right\|^{2} \\ & +\left\|\left(z+\frac{x+y}{2}\right)-\frac{x-y}{2}\right\|^{2}-\left\|\left(z-\frac{x+y}{2}\right)-\frac{y-x}{2}\right\|^{2} \\ & \stackrel{(8.35)}{=} 2\left\|z+\frac{x+y}{2}\right\|^{2}+2\left\|\frac{x-y}{2}\right\|^{2}-2\left\|z-\frac{x+y}{2}\right\|^{2}-2\left\|\frac{y-x}{2}\right\|^{2} \\ & \stackrel{(8.36)}{=} 8\left\langle\frac{x+y}{2}, z\right\rangle \end{aligned} | |  |
| 531 | | 203 | \langle\boldsymbol{x}, \boldsymbol{z}\rangle+\langle\boldsymbol{y}, \boldsymbol{z}\rangle=2\left\langle\frac{\boldsymbol{x}+\boldsymbol{y}}{2}, \boldsymbol{z}\right\rangle, \quad \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z} \in \mathcal{V} . | |  |
| 532 | 8.39 | 203 | \langle n \boldsymbol{x}, \boldsymbol{y}\rangle=\langle(n-1) \boldsymbol{x}+\boldsymbol{x}, \boldsymbol{y}\rangle \stackrel{\text { (8.37) }}{=}\langle(n-1) \boldsymbol{x}, \boldsymbol{y}\rangle+\langle\boldsymbol{x}, \boldsymbol{y}\rangle=n\langle\boldsymbol{x}, \boldsymbol{y}\rangle . | |  |
| 533 | | 203 | m^{2}\left\langle\frac{n}{m} \boldsymbol{x}, \boldsymbol{y}\right\rangle \stackrel{\text { (8.39) }}{=} m\langle n \boldsymbol{x}, \boldsymbol{y}\rangle \stackrel{\text { (8.39) }}{=} m n\langle\boldsymbol{x}, \boldsymbol{y}\rangle, | |  |
| 534 | | 203 | \left\langle\frac{n}{m} \boldsymbol{x}, \boldsymbol{y}\right\rangle=\frac{n}{m}\langle\boldsymbol{x}, \boldsymbol{y}\rangle . | |  |
| 535 | | 203 | a_{n}\langle\boldsymbol{x}, \boldsymbol{y}\rangle=\left\langle a_{n} \boldsymbol{x}, \boldsymbol{y}\right\rangle \stackrel{(8.36)}{=} \frac{1}{4}\left(\left\|a_{n} \boldsymbol{x}+\boldsymbol{y}\right\|^{2}-\left\|a_{n} \boldsymbol{x}-\boldsymbol{y}\right\|^{2}\right) . | |  |
| 536 | | 204 | (-a)\langle\boldsymbol{x}, \boldsymbol{y}\rangle=\langle(-a) \boldsymbol{x}, \boldsymbol{y}\rangle \stackrel{(8.36)}{=} \frac{1}{4}\left(\|-a \boldsymbol{x}+\boldsymbol{y}\|^{2}-\|-a \boldsymbol{x}-\boldsymbol{y}\|^{2}\right)=-\langle a \boldsymbol{x}, \boldsymbol{y}\rangle, | |  |
| 537 | | 204 | \|\boldsymbol{x}\|_{\boldsymbol{A}} \leq\|\boldsymbol{y}\|_{\boldsymbol{A}} \Longrightarrow\|\boldsymbol{x}\|_{2} \leq \sqrt{\frac{\lambda_{1}}{\lambda_{n}}}\|\boldsymbol{y}\|_{2}, | |  |
| 538 | | 204 | \|\boldsymbol{x}\|_{\boldsymbol{A}}:=\sqrt{\boldsymbol{x}^{T} \boldsymbol{A} \boldsymbol{x}}, \quad \boldsymbol{x} \in \mathbb{R}^{n} . | |  |
| 539 | | 204 | \tilde{\boldsymbol{b}}_{k}:=\boldsymbol{b}_{k}-\boldsymbol{B}_{k-1}\left(\boldsymbol{B}_{k-1}^{T} \boldsymbol{A} \boldsymbol{B}_{k-1}\right)^{-1} \boldsymbol{B}_{k-1}^{T} \boldsymbol{A} \boldsymbol{b}_{k}, \quad k=2, \ldots, n . | |  |
| 540 | | 205 | \left\|\boldsymbol{b}_{1}\right\|_{\boldsymbol{A}}\left\|\boldsymbol{b}_{2}\right\|_{\boldsymbol{A}} \cdots\left\|\boldsymbol{b}_{n}\right\|_{\boldsymbol{A}} \leq 2^{n(n-1) / 4} \sqrt{\operatorname{det}(\boldsymbol{A})} . | |  |
| 541 | | 205 | \|\boldsymbol{A}\|:=\sqrt{m n}\|\boldsymbol{A}\|_{M}, \quad \boldsymbol{A} \in \mathbb{C}^{m \times n} | |  |
| 542 | | 205 | \|\boldsymbol{A}\|_{2}=\max _{\|\boldsymbol{x}\|_{2}=\|\boldsymbol{y}\|_{2}=1}\left|\boldsymbol{y}^{*} \boldsymbol{A} \boldsymbol{x}\right| . | |  |
| 543 | | 206 | \left\|\boldsymbol{A}^{-1}\right\|_{2}=\max _{\boldsymbol{x} \neq \mathbf{0}} \frac{\|\boldsymbol{x}\|_{2}}{\|\boldsymbol{A} \boldsymbol{x}\|_{2}} . | |  |
| 544 | | 206 | \boldsymbol{A}=\left[\begin{array}{cc} 2 & -1 \\ -1 & 2 \end{array}\right] . | |  |
| 545 | | 207 | \boldsymbol{x}_{k+1}=\boldsymbol{x}_{k}+\frac{1}{\lambda_{k+1}} \boldsymbol{r}_{k}, \text { where } \boldsymbol{r}_{k}=\boldsymbol{b}-\boldsymbol{A} \boldsymbol{x}_{k} . | |  |
| 546 | | 207 | \boldsymbol{r}_{k}=\sum_{i=1}^{n} c_{i k} \boldsymbol{u}_{i}, | |  |
| 547 | | 207 | c_{i, k+1}= \begin{cases}0 & \text { if } \sigma_{i}=\lambda_{k+1}, \\ c_{i, k}\left(1-\frac{\sigma_{i}}{\lambda_{k+1}}\right) & \text { otherwise } .\end{cases} | |  |
| 548 | | 207 | \lambda_{j}=d+2 c \cos \left(\frac{j \pi}{n+1}\right), \quad j=1, \ldots, n . | |  |
| 549 | 8.40 | 207 | \boldsymbol{T} \boldsymbol{x}_{k+1}=\boldsymbol{b}-\boldsymbol{B} \boldsymbol{x}_{k} . | |  |
| 550 | | 207 | \min \left\{\max _{i} \sum_{j=1}^{n}\left|b_{i j}\right|, \max _{j} \sum_{i=1}^{n}\left|b_{i j}\right|\right\}<d-2 c . | |  |
| 551 | | 208 | \boldsymbol{A}:=\left[\begin{array}{cc} 1.1 & 1 \\ 1 & 1 \end{array}\right], \quad \boldsymbol{b}:=\left[\begin{array}{l} b_{1} \\ b_{2} \end{array}\right]=\left[\begin{array}{l} 2.1 \\ 2.0 \end{array}\right], \quad \boldsymbol{e}:=\left[\begin{array}{l} e_{1} \\ e_{2} \end{array}\right], \quad\|\boldsymbol{e}\|_{2}=0.1 . | |  |
| 552 | 8.41 | 208 | \frac{4}{\pi^{2}}(m+1)^{2}-2 / 3<\operatorname{cond}_{p}(\boldsymbol{T}) \leq \frac{1}{2}(m+1)^{2}, \quad p=1,2, \infty, | |  |
| 553 | 8.42 | 208 | \operatorname{cond}_{1}(\boldsymbol{T})=\operatorname{cond}_{\infty}(\boldsymbol{T})=\frac{1}{2} \begin{cases}h^{-2}, & m \text { odd } \\ h^{-2}-1, & m \text { even }\end{cases} | |  |
| 554 | | 208 | \operatorname{cond}_{2}(\boldsymbol{T})=\cot ^{2}\left(\frac{\pi h}{2}\right)=1 / \tan ^{2}\left(\frac{\pi h}{2}\right) . | |  |
| 555 | 8.43 | 209 | \frac{4}{\pi^{2}} h^{-2}-\frac{2}{3}<\operatorname{cond}_{2}(\boldsymbol{T})<\frac{4}{\pi^{2}} h^{-2} . | |  |
| 556 | | 209 | \frac{\left\|(\boldsymbol{I}-\boldsymbol{E})^{-1}-\boldsymbol{I}\right\|}{\left\|(\boldsymbol{I}-\boldsymbol{E})^{-1}\right\|} \leq\|\boldsymbol{E}\| | |  |
| 557 | | 209 | \frac{1}{1+\|\boldsymbol{E}\|} \leq\left\|(\boldsymbol{I}-\boldsymbol{E})^{-1}\right\| \leq \frac{1}{1-\|\boldsymbol{E}\|} | |  |
| 558 | | 209 | \left\|(\boldsymbol{I}-\boldsymbol{E})^{-1}-\boldsymbol{I}\right\| \leq \frac{\|\boldsymbol{E}\|}{1-\|\boldsymbol{E}\|} . | |  |
| 559 | | 209 | K(\boldsymbol{B})^{-1} \frac{\|\boldsymbol{E}\|}{\|\boldsymbol{A}\|} \leq \frac{\left\|\boldsymbol{B}^{-1}-\boldsymbol{A}^{-1}\right\|}{\left\|\boldsymbol{B}^{-1}\right\|} . | |  |
| 560 | | 209 | \frac{K(\boldsymbol{B})^{-1}}{1+r} \frac{\|\boldsymbol{E}\|}{\|\boldsymbol{A}\|} \leq \frac{\left\|\boldsymbol{B}^{-1}-\boldsymbol{A}^{-1}\right\|}{\left\|\boldsymbol{A}^{-1}\right\|} . | |  |
| 561 | | 209 | a=x_{0}<x_{1}<\cdots<x_{n}=b, | |  |
| 562 | | 210 | \begin{array}{ll} g\left(x_{i}\right)=y_{i}, & i=0,1, \ldots, n, \\ g^{\prime}(a)=g^{\prime}(b), & g^{\prime \prime}(a)=g^{\prime \prime}(b) . \end{array} | |  |
| 563 | 8.44 | 210 | \boldsymbol{A} \boldsymbol{s}=\boldsymbol{b}, | |  |
| 564 | | 210 | A:=\left[\begin{array}{cccccc} 2 & \mu_{1} & 0 & \cdots & 0 & \lambda_{1} \\ \lambda_{2} & 2 & \mu_{2} & \ddots & & 0 \\ 0 & \ddots & \ddots & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & \ddots & 0 \\ 0 & & \ddots & \lambda_{n-1} & 2 & \mu_{n-1} \\ \mu_{n} & 0 & \cdots & 0 & \lambda_{n} & 2 \end{array}\right], | |  |
| 565 | | 210 | \lambda_{i}:=\frac{h_{i}}{h_{i-1}+h_{i}}, \quad \mu_{i}:=\frac{h_{i-1}}{h_{i-1}+h_{i}}, \quad, i=1, \ldots, n, | |  |
| 566 | | 210 | h_{i}=x_{i+1}-x_{i}, \quad i=0, \ldots, n-1, \text { and } h_{n}=h_{0} . | |  |
| 567 | | 210 | \frac{1}{2} \leq \frac{h_{i}}{h_{i-1}} \leq 2, \quad i=1, \ldots, n . | |  |
| 568 | | 211 | \|\boldsymbol{A}\|_{\infty}=3 \quad \text { and that } \quad\|\boldsymbol{A}\|_{1} \leq \frac{10}{3} . | |  |
| 569 | | 211 | \boldsymbol{A} \hat{\boldsymbol{s}}=\boldsymbol{b}+\boldsymbol{e} . | |  |
| 570 | | 211 | \frac{\|\hat{\boldsymbol{s}}-\boldsymbol{s}\|_{\infty}}{\|\boldsymbol{s}\|_{\infty}} \quad \text { and } \quad \frac{\|\hat{\boldsymbol{s}}-\boldsymbol{s}\|_{1}}{\|\boldsymbol{s}\|_{1}} . | |  |
| 571 | 8.45 | 211 | \langle\boldsymbol{x}, \boldsymbol{y}\rangle:=\frac{1}{4}\left(\|\boldsymbol{x}+\boldsymbol{y}\|^{2}-\|\boldsymbol{x}-\boldsymbol{y}\|^{2}+i\|\boldsymbol{x}+i \boldsymbol{y}\|^{2}-i\|\boldsymbol{x}-i \boldsymbol{y}\|^{2}\right), | |  |
| 572 | | 212 | S_{p}=\left\{\boldsymbol{x} \in \mathbb{R}^{n}:\|\boldsymbol{x}\|_{p}=1\right\} | |  |
| 573 | | 212 | \|\boldsymbol{x}\|_{p} \leq\|\boldsymbol{x}\|_{q} \leq n^{1 / q-1 / p}\|\boldsymbol{x}\|_{p}, \quad \boldsymbol{x} \in \mathbb{C}^{n} . | |  |
| 574 | | 213 | E(\boldsymbol{x}):=\|\boldsymbol{A} \boldsymbol{x}-\boldsymbol{b}\|_{2}^{2}, | |  |
| 575 | | 213 | \begin{aligned} & x_{1}=1 \\ & x_{1}=1, \quad \boldsymbol{A}=\left[\begin{array}{l} 1 \\ 1 \\ x_{1}=2 \end{array}\right], \quad \boldsymbol{x}=\left[x_{1}\right], \quad \boldsymbol{b}=\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right] . \end{aligned} | |  |
| 576 | | 214 | \|\boldsymbol{A} \boldsymbol{x}-\boldsymbol{b}\|_{2}^{2}=\left(x_{1}-1\right)^{2}+\left(x_{1}-1\right)^{2}+\left(x_{1}-2\right)^{2}=3 x_{1}^{2}-8 x_{1}+6 . | |  |
| 577 | | 214 | \boldsymbol{A} \boldsymbol{x}=\left[\begin{array}{c} p\left(t_{1}\right) \\ \vdots \\ p\left(t_{m}\right) \end{array}\right]=\left[\begin{array}{ll} 1 & t_{1} \\ \vdots \\ 1 & t_{m} \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]=\left[\begin{array}{c} y_{1} \\ \vdots \\ y_{m} \end{array}\right]=\boldsymbol{b} . | |  |
| 578 | | 214 | c_{1}\left[\begin{array}{c} 1 \\ \vdots \\ 1 \end{array}\right]+c_{2}\left[\begin{array}{c} t_{1} \\ \vdots \\ t_{m} \end{array}\right]=\left[\begin{array}{c} 0 \\ \vdots \\ 0 \end{array}\right] \Longrightarrow\left[\begin{array}{cc} 1 & t_{i} \\ 1 & t_{j} \end{array}\right]\left[\begin{array}{c} c_{1} \\ c_{2} \end{array}\right]=\left[\begin{array}{c} 0 \\ 0 \end{array}\right] \Longrightarrow c_{1}=c_{2}=0 . | |  |
| 579 | | 215 | \begin{aligned} \boldsymbol{A}^{*} \boldsymbol{A} \boldsymbol{x} & =\left[\begin{array}{ccc} 1 & \cdots & 1 \\ t_{1} & \cdots & t_{m} \end{array}\right]\left[\begin{array}{c} 1 \\ \vdots \\ 1 \\ 1 \\ m \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]=\left[\begin{array}{cc} m & \sum t_{k} \\ \sum t_{k} & \sum t_{k}^{2} \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right], \\ & =\left[\begin{array}{ccc} 1 & \cdots & 1 \\ t_{1} & \cdots & t_{m} \end{array}\right]\left[\begin{array}{c} y_{1} \\ \vdots \\ y_{m} \end{array}\right]=\left[\begin{array}{c} \sum y_{k} \\ \sum t_{k} y_{k} \end{array}\right]=\boldsymbol{A}^{*} \boldsymbol{b}, \end{aligned} | |  |
| 580 | | 215 | \begin{array}{c|c|c|c|c} t & 1.0 & 2.0 & 3.0 & 4.0 \\ \hline y & 3.1 & 1.8 & 1.0 & 0.1 \end{array} | |  |
| 581 | | 215 | y=\boldsymbol{u}^{*} \boldsymbol{x}=\sum_{i=1}^{n} u_{i} x_{i}, | |  |
| 582 | | 216 | \begin{array}{c|c|c|c|c} \boldsymbol{u} & \boldsymbol{u}_{1} & \boldsymbol{u}_{2} & \cdots & \boldsymbol{u}_{m} \\ \hline \hline & y_{1} & y_{2} & \cdots & y_{m} \end{array} . | |  |
| 583 | | 216 | \boldsymbol{A} \boldsymbol{x}=\left[\begin{array}{c} \boldsymbol{u}_{1}^{*} \\ \boldsymbol{u}_{2}^{*} \\ \vdots \\ \boldsymbol{u}_{m}^{*} \end{array}\right] \boldsymbol{x}=\left[\begin{array}{c} y_{1} \\ y_{2} \\ \vdots \\ y_{m} \end{array}\right]=\boldsymbol{b} . | |  |
| 584 | 9.1 | 216 | \boldsymbol{A} \boldsymbol{x}=\left[\begin{array}{c} p\left(t_{1}\right) \\ \vdots \\ p\left(t_{m}\right) \end{array}\right]=\left[\begin{array}{ccc} \phi_{1}\left(t_{1}\right) & \cdots & \phi_{n}\left(t_{1}\right) \\ \vdots & & \vdots \\ \phi_{1}\left(t_{m}\right) & \cdots & \phi_{n}\left(t_{m}\right) \end{array}\right]\left[\begin{array}{c} x_{1} \\ \vdots \\ x_{n} \end{array}\right]=\left[\begin{array}{c} y_{1} \\ \vdots \\ y_{m} \end{array}\right]=: \boldsymbol{b} . | |  |
| 585 | 9.2 | 216 | E(\boldsymbol{x}):=\|\boldsymbol{A} \boldsymbol{x}-\boldsymbol{b}\|_{2}^{2}=\sum_{k=1}^{m}\left(\sum_{j=1}^{n} x_{j} \phi_{j}\left(t_{k}\right)-y_{k}\right)^{2} . | |  |
| 586 | | 217 | E(\boldsymbol{x}):=\sum_{k=1}^{m} w_{k}\left(\sum_{j=1}^{n} x_{j} \phi_{j}\left(t_{k}\right)-y_{k}\right)^{2} . | |  |
| 587 | 9.3 | 217 | p\left(t_{k}\right):=\sum_{j=1}^{n} x_{j} \phi_{j}\left(t_{k}\right)=0, \quad k=1, \ldots, m \Rightarrow x_{1}=\cdots=x_{n}=0 . | |  |
| 588 | | 217 | \boldsymbol{B}_{3} \boldsymbol{x}:=\left[\begin{array}{ccc} m & \sum t_{k} & \sum t_{k}^{2} \\ \sum t_{k} & \sum t_{k}^{2} & \sum t_{k}^{3} \\ \sum t_{k}^{2} & \sum t_{k}^{3} & \sum t_{k}^{4} \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]=\left[\begin{array}{c} \sum y_{k} \\ \sum t_{k} y_{k} \\ \sum t_{k}^{2} y_{k} \end{array}\right] . | |  |
| 589 | | 217 | \boldsymbol{H}_{3}=\left[\begin{array}{lll} 1 & \frac{1}{2} & \frac{1}{3} \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \end{array}\right] . | |  |
| 590 | | 218 | \begin{aligned} \frac{1}{m} b_{i, j} & =\frac{1}{m} \sum_{k=1}^{m} t_{k}^{i+j-2}=\frac{1}{m} \sum_{k=1}^{m}\left(\frac{k-1}{m-1}\right)^{i+j-2} \\ & \approx \int_{0}^{1} x^{i+j-2} d x=\frac{1}{i+j-1}=h_{i, j} \end{aligned} | |  |
| 591 | | 218 | E(\boldsymbol{x}):=\|\boldsymbol{A} \boldsymbol{x}-\boldsymbol{b}\|_{2}^{2}=(\boldsymbol{A} \boldsymbol{x}-\boldsymbol{b})^{*}(\boldsymbol{A} \boldsymbol{x}-\boldsymbol{b})=\boldsymbol{x}^{*} \boldsymbol{A}^{*} \boldsymbol{A} \boldsymbol{x}-2 \boldsymbol{x}^{*} \boldsymbol{A}^{*} \boldsymbol{b}+\boldsymbol{b}^{*} \boldsymbol{b} . | |  |
| 592 | 9.4 | 218 | \mathbb{C}^{m}=\mathcal{R}(\boldsymbol{A}) \stackrel{\perp}{\oplus} \mathcal{N}\left(\boldsymbol{A}^{*}\right) . | |  |
| 593 | | 218 | \|A x-b\|_{2}^{2}=\left\|\left(A x-b_{1}\right)-b_{2}\right\|_{2}^{2}=\left\|A x-b_{1}\right\|_{2}^{2}+\left\|b_{2}\right\|_{2}^{2} \geq\left\|b_{2}\right\|_{2}^{2} | |  |
| 594 | 9.5 | 219 | \left\{\boldsymbol{x} \in \mathbb{C}^{n}: \boldsymbol{A} \boldsymbol{x}=\boldsymbol{b}_{1}\right\} . | |  |
| 595 | | 219 | \left(\boldsymbol{A}^{*} \boldsymbol{b}\right)_{i}=\sum_{k=1}^{m} \bar{a}_{k, i} b_{k}, i=1, \ldots, n, | |  |
| 596 | 9.6 | 220 | K_{2}\left(\boldsymbol{A}^{*} \boldsymbol{A}\right):=\left\|\boldsymbol{A}^{*} \boldsymbol{A}\right\|_{2}\left\|\left(\boldsymbol{A}^{*} \boldsymbol{A}\right)^{-1}\right\|_{2}=\frac{\lambda_{1}}{\lambda_{n}}=\frac{\sigma_{1}^{2}}{\sigma_{n}^{2}}=K_{2}(\boldsymbol{A})^{2}, | |  |
| 597 | | 220 | \boldsymbol{A}^{*} \boldsymbol{A}=\boldsymbol{R}_{1}^{*} \boldsymbol{Q}_{1}^{*} \boldsymbol{Q}_{1} \boldsymbol{R}_{1}=\boldsymbol{R}_{1}^{*} \boldsymbol{R}_{1}, \quad \boldsymbol{A}^{*} \boldsymbol{b}=\boldsymbol{R}_{1}^{*} \boldsymbol{Q}_{1}^{*} \boldsymbol{b} . | |  |
| 598 | | 220 | \boldsymbol{A}^{*} \boldsymbol{A} \boldsymbol{x}=\boldsymbol{A}^{*} \boldsymbol{b} \Longrightarrow \boldsymbol{R}_{1} \boldsymbol{x}=\boldsymbol{c}_{1}, \quad \boldsymbol{c}_{1}:=\boldsymbol{Q}_{1}^{*} \boldsymbol{b} . | |  |
| 599 | | 221 | \boldsymbol{A}=\left[\begin{array}{ccc} 1 & 3 & 1 \\ 1 & 3 & 7 \\ 1 & -1 & -4 \\ 1 & -1 & 2 \end{array}\right] \text { and } \boldsymbol{b}=\left[\begin{array}{l} 1 \\ 1 \\ 1 \\ 1 \end{array}\right] . | |  |
| 600 | 0 | 221 | \left[\begin{array}{lll} 2 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]=\frac{1}{2}\left[\begin{array}{rrcr} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ -1 & 1 & -1 & 1 \end{array}\right] \times\left[\begin{array}{l} 1 \\ 1 \\ 1 \\ 1 \end{array}\right]=\left[\begin{array}{l} 2 \\ 0 \\ 0 \end{array}\right], | |  |
| 601 | 9.7 | 222 | \boldsymbol{A}=\boldsymbol{U} \boldsymbol{\Sigma} \boldsymbol{V}^{*}=\left[\boldsymbol{U}_{1}, \boldsymbol{U}_{2}\right]\left[\begin{array}{cc} \boldsymbol{\Sigma}_{1} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{array}\right]\left[\begin{array}{l} \boldsymbol{V}_{1}^{*} \\ \boldsymbol{V}_{2}^{*} \end{array}\right]=\boldsymbol{U}_{1} \boldsymbol{\Sigma}_{1} \boldsymbol{V}_{1}^{*}, \quad \boldsymbol{\Sigma}_{1}=\operatorname{diag}\left(\sigma_{1}, \ldots, \sigma_{r}\right), | |  |
| 602 | | 222 | \begin{array}{ll} \boldsymbol{U}_{1}=\left[\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{r}\right], & \boldsymbol{U}_{2}=\left[\boldsymbol{u}_{r+1}, \ldots, \boldsymbol{u}_{m}\right], \\ \boldsymbol{V}_{1}=\left[\boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{r}^{*}\right], & \boldsymbol{V}_{2}=\left[\boldsymbol{v}_{r+1}, \ldots, \boldsymbol{v}_{n}\right], \end{array} \quad \begin{aligned} & \boldsymbol{V}_{1}^{*} \boldsymbol{V}_{1}=\boldsymbol{I}, \quad \boldsymbol{V}_{2}^{*} \boldsymbol{V}_{2}=\boldsymbol{I}, \end{aligned} | |  |
| 603 | 9.8 | 222 | \boldsymbol{A} \boldsymbol{A}^{\dagger} \boldsymbol{A}=\boldsymbol{A}, \boldsymbol{A}^{\dagger} \boldsymbol{A} \boldsymbol{A}^{\dagger}=\boldsymbol{A}^{\dagger},\left(\boldsymbol{A}^{\dagger} \boldsymbol{A}\right)^{*}=\boldsymbol{A}^{\dagger} \boldsymbol{A},\left(\boldsymbol{A} \boldsymbol{A}^{\dagger}\right)^{*}=\boldsymbol{A} \boldsymbol{A}^{\dagger} . | |  |
| 604 | 9.9 | 222 | \boldsymbol{A}^{\dagger}=\boldsymbol{V}_{1} \boldsymbol{\Sigma}_{1}^{-1} \boldsymbol{U}_{1}^{*} . | |  |
| 605 | | 222 | A=U_{1} \Sigma_{1} V_{1}^{*}, \quad A^{\dagger}:=V_{1} \Sigma_{1}^{-1} U_{1}^{*} | |  |
| 606 | 9.10 | 222 | \begin{aligned} & A^{\dagger} A=V_{1} \Sigma_{1}^{-1} U_{1}^{*} U_{1} \Sigma_{1} V_{1}^{*}=V_{1} V_{1}^{*} \\ & A A^{\dagger}=U_{1} \Sigma_{1} V_{1}^{*} V_{1} \Sigma_{1}^{-1} U_{1}^{*}=U_{1} U_{1}^{*} \end{aligned} | |  |
| 607 | | 222 | \begin{aligned} A A^{\dagger} A & =U_{1} \Sigma_{1} V_{1}^{*} V_{1} V_{1}^{*}=U_{1} \Sigma_{1} V_{1}^{*}=A \\ A^{\dagger} A A^{\dagger} & =V_{1} \Sigma_{1}^{-1} U_{1}^{*} U_{1} U_{1}^{*}=V_{1} \Sigma^{-1} U_{1}^{*}=A^{\dagger} \end{aligned} | |  |
| 608 | 9.11 | 223 | \boldsymbol{A}^{\dagger}=\left(\boldsymbol{A}^{*} \boldsymbol{A}\right)^{-1} \boldsymbol{A}^{*} . | |  |
| 609 | 9.12 | 223 | \begin{gathered} \boldsymbol{P}_{\mathcal{R}(\boldsymbol{A})}=\boldsymbol{A} \boldsymbol{A}^{\dagger}=\boldsymbol{U}_{1} \boldsymbol{U}_{1}^{*}=\sum_{j=1}^{r} \boldsymbol{u}_{j} \boldsymbol{u}_{j}^{*} \in \mathbb{C}^{m \times m}, \\ \boldsymbol{P}_{\mathcal{N}\left(\boldsymbol{A}^{*}\right)}=\boldsymbol{I}-\boldsymbol{A} \boldsymbol{A}^{\dagger}=\boldsymbol{U}_{2} \boldsymbol{U}_{2}^{*}=\sum_{j=r+1}^{m} \boldsymbol{u}_{j} \boldsymbol{u}_{j}^{*} \in \mathbb{C}^{m \times m} . \end{gathered} | |  |
| 610 | | 223 | \boldsymbol{v}=\boldsymbol{U} \boldsymbol{U}^{*} \boldsymbol{v}=\left[\boldsymbol{U}_{1}, \boldsymbol{U}_{2}\left[\begin{array}{l} \boldsymbol{U}_{1}^{*} \\ \boldsymbol{U}_{2}^{*} \end{array}\right] \boldsymbol{v}=\boldsymbol{s}+\boldsymbol{t},\right. | |  |
| 611 | | 224 | \boldsymbol{A}:=\frac{1}{\sqrt{2}}\left[\begin{array}{l} 1 \\ 1 \end{array}\right][2] \frac{1}{\sqrt{2}}\left[\begin{array}{ll} 1 & 1 \end{array}\right], \boldsymbol{A}^{\dagger}=\frac{1}{\sqrt{2}}\left[\begin{array}{l} 1 \\ 1 \end{array}\right]\left[\frac{1}{2}\right] \frac{1}{\sqrt{2}}\left[\begin{array}{ll} 1 & 1 \end{array}\right]=\frac{1}{4} \boldsymbol{A}, \boldsymbol{A}^{\dagger} \boldsymbol{b}=\left[\begin{array}{l} 1 / 2 \\ 1 / 2 \end{array}\right] . | |  |
| 612 | | 224 | \boldsymbol{x}=\boldsymbol{A}^{\dagger} \boldsymbol{b}=\boldsymbol{A}^{\dagger} \boldsymbol{b}_{1}, \quad \boldsymbol{A}^{\dagger}=\left(\boldsymbol{A}^{*} \boldsymbol{A}\right)^{-1} \boldsymbol{A}^{*}, | |  |
| 613 | 9.13 | 225 | \frac{1}{K(\boldsymbol{A})} \frac{\left\|\boldsymbol{e}_{1}\right\|}{\left\|\boldsymbol{b}_{1}\right\|} \leq \frac{\|\boldsymbol{y}-\boldsymbol{x}\|}{\|\boldsymbol{x}\|} \leq K(\boldsymbol{A}) \frac{\left\|\boldsymbol{e}_{1}\right\|}{\left\|\boldsymbol{b}_{1}\right\|}, \quad K(\boldsymbol{A})=\|\boldsymbol{A}\|\left\|\boldsymbol{A}^{\dagger}\right\| . | |  |
| 614 | | 225 | \boldsymbol{A}=\left[\begin{array}{ll} 1 & 1 \\ 0 & 1 \\ 0 & 0 \end{array}\right], \quad \boldsymbol{b}=\left[\begin{array}{c} 10^{-4} \\ 0 \\ 1 \end{array}\right], \quad \boldsymbol{e}=\left[\begin{array}{c} 10^{-6} \\ 0 \\ 0 \end{array}\right] . | |  |
| 615 | | 226 | \boldsymbol{A}^{*} \boldsymbol{A}=\left[\begin{array}{ll} 1 & 1 \\ 1 & 2 \end{array}\right],\left(\boldsymbol{A}^{*} \boldsymbol{A}\right)^{-1}=\left[\begin{array}{cc} 2 & -1 \\ -1 & 1 \end{array}\right], \boldsymbol{A}^{\dagger}=\left(\boldsymbol{A}^{*} \boldsymbol{A}\right)^{-1} \boldsymbol{A}^{*}=\left[\begin{array}{ccc} 1 & -1 & 0 \\ 0 & 1 & 0 \end{array}\right] . | |  |
| 616 | | 226 | \boldsymbol{b}_{1}=\boldsymbol{A} \boldsymbol{A}^{\dagger} \boldsymbol{b}=\left[10^{-4}, 0,0\right]^{*}, \quad \text { and } \quad \boldsymbol{e}_{1}=\boldsymbol{A} \boldsymbol{A}^{\dagger} \boldsymbol{e}=\left[10^{-6}, 0,0\right]^{*} . | |  |
| 617 | 9.14 | 226 | \frac{1}{4} 10^{-2} \leq \frac{\|\boldsymbol{y}-\boldsymbol{x}\|_{\infty}}{\|\boldsymbol{x}\|_{\infty}} \leq 4 \cdot 10^{-2} . | |  |
| 618 | | 226 | \frac{\|\boldsymbol{x}-\boldsymbol{y}\|_{\infty}}{\|\boldsymbol{x}\|_{\infty}}=\frac{10^{-6}}{10^{-4}}=10^{-2} | |  |
| 619 | 9.15 | 226 | \rho=\frac{\|\boldsymbol{x}-\boldsymbol{y}\|_{2}}{\|\boldsymbol{x}\|_{2}} \leq \frac{1}{\alpha} K(1+\beta K) \frac{\|\boldsymbol{E}\|_{2}}{\|\boldsymbol{A}\|_{2}}, \quad \beta=\frac{\left\|\boldsymbol{b}_{2}\right\|_{2}}{\left\|\boldsymbol{b}_{1}\right\|_{2}}, \quad K=\|\boldsymbol{A}\|_{2}\left\|\boldsymbol{A}^{\dagger}\right\|_{2} . | |  |
| 620 | 9.16 | 227 | \sigma_{k}=\min _{\operatorname{dim}(\mathcal{S})=n-k+1} \max _{\substack{\boldsymbol{x} \in \mathcal{S} \\ \boldsymbol{x} \neq \mathbf{0}}} \frac{\|\boldsymbol{A} \boldsymbol{x}\|_{2}}{\|\boldsymbol{x}\|_{2}}=\max _{\operatorname{dim}(\mathcal{S})=k} \min _{\substack{\boldsymbol{x} \in \mathcal{S} \\ \boldsymbol{x} \neq \mathbf{0}}} \frac{\|\boldsymbol{A} \boldsymbol{x}\|_{2}}{\|\boldsymbol{x}\|_{2}} . | |  |
| 621 | | 227 | \frac{\|\boldsymbol{A} \boldsymbol{x}\|_{2}^{2}}{\|\boldsymbol{x}\|_{2}^{2}}=\frac{(\boldsymbol{A} \boldsymbol{x})^{*}(\boldsymbol{A} \boldsymbol{x})}{\boldsymbol{x}^{*} \boldsymbol{x}}=\frac{\boldsymbol{x}^{*} \boldsymbol{A}^{*} \boldsymbol{A} \boldsymbol{x}}{\boldsymbol{x}^{*} \boldsymbol{x}}=R_{\boldsymbol{A}^{*} \boldsymbol{A}}(\boldsymbol{x}), | |  |
| 622 | 9.17 | 227 | \sigma_{1}=\max _{\substack{\boldsymbol{x} \in \mathbb{C}^{n} \\ \boldsymbol{x} \neq \mathbf{0}}} \frac{\|\boldsymbol{A} \boldsymbol{x}\|_{2}}{\|\boldsymbol{x}\|_{2}}, \quad \sigma_{n}=\min _{\substack{\boldsymbol{x} \in \mathbb{C}^{n} \\ \boldsymbol{x} \neq \mathbf{0}}} \frac{\|\boldsymbol{A} \boldsymbol{x}\|_{2}}{\|\boldsymbol{x}\|_{2}} . | |  |
| 623 | 9.18 | 228 | \left|\alpha_{j}-\beta_{j}\right| \leq\|\boldsymbol{A}-\boldsymbol{B}\|_{2}, \text { for } j=1,2, \ldots, n . | |  |
| 624 | | 228 | \begin{aligned} \alpha_{j} & \leq \max _{\substack{\boldsymbol{x} \in \mathcal{S} \\ \boldsymbol{x} \neq \mathbf{0}}} \frac{\|(\boldsymbol{B}+(\boldsymbol{A}-\boldsymbol{B})) \boldsymbol{x}\|_{2}}{\|\boldsymbol{x}\|_{2}} \leq \max _{\substack{\boldsymbol{x} \in \mathcal{S} \\ \boldsymbol{x} \neq \mathbf{0}}} \frac{\|\boldsymbol{B} \boldsymbol{x}\|_{2}}{\|\boldsymbol{x}\|_{2}}+\max _{\substack{\boldsymbol{x} \in \mathcal{S} \\ \boldsymbol{x} \neq \mathbf{0}}} \frac{\|(\boldsymbol{A}-\boldsymbol{B}) \boldsymbol{x}\|_{2}}{\|\boldsymbol{x}\|_{2}} \\ & \leq \beta_{j}+\|\boldsymbol{A}-\boldsymbol{B}\|_{2} . \end{aligned} | |  |
| 625 | | 228 | \left\|(\boldsymbol{A}+\boldsymbol{E})^{\dagger}\right\|_{2}=\frac{1}{\beta_{r}} \leq \frac{1}{\alpha_{r}-\epsilon_{1}}=\frac{1 / \alpha_{r}}{1-\epsilon_{1} / \alpha_{r}}=\frac{\left\|\boldsymbol{A}^{\dagger}\right\|_{2}}{1-\left\|\boldsymbol{A}^{\dagger}\right\|_{2}\|\boldsymbol{E}\|_{2}} . | |  |
| 626 | 9.19 | 228 | \sum_{j=1}^{n}\left|\mu_{j}-\lambda_{j}\right|^{2} \leq\|\boldsymbol{A}-\boldsymbol{B}\|_{F}^{2}:=\sum_{i=1}^{n} \sum_{j=1}^{n}\left|a_{i j}-b_{i j}\right|^{2}, | |  |
| 627 | 9.20 | 229 | \sum_{j=1}^{n}\left|\beta_{j}-\alpha_{j}\right|^{2} \leq\|\boldsymbol{A}-\boldsymbol{B}\|_{F}^{2} . | |  |
| 628 | | 229 | \boldsymbol{C}:=\left[\begin{array}{cc} \mathbf{0} & \boldsymbol{A} \\ \boldsymbol{A}^{*} & \mathbf{0} \end{array}\right] \text { and } \boldsymbol{D}:=\left[\begin{array}{cc} \mathbf{0} & \boldsymbol{B} \\ \boldsymbol{B}^{*} & \mathbf{0} \end{array}\right] \in \mathbb{C}^{(m+n) \times(m+n)} . | |  |
| 629 | 9.21 | 229 | \sum_{j=1}^{m+n}\left|\lambda_{j}-\mu_{j}\right|^{2} \leq\|\boldsymbol{C}-\boldsymbol{D}\|_{F}^{2} . | |  |
| 630 | | 229 | \begin{aligned} {\left[\begin{array}{cc} \boldsymbol{0} & \boldsymbol{A} \\ \boldsymbol{A}^{*} & \boldsymbol{0} \end{array}\right]\left[\begin{array}{l} \boldsymbol{u}_{i} \\ \boldsymbol{v}_{i} \end{array}\right] } & =\left[\begin{array}{c} \boldsymbol{A} \boldsymbol{v}_{i} \\ \boldsymbol{A}^{*} \boldsymbol{u}_{i} \end{array}\right]=\left[\begin{array}{l} \alpha_{i} \boldsymbol{u}_{i} \\ \alpha_{i} \boldsymbol{v}_{i} \end{array}\right]=\alpha_{i}\left[\begin{array}{l} \boldsymbol{u}_{i} \\ \boldsymbol{v}_{i} \end{array}\right], \quad i=1, \ldots, r, \\ {\left[\begin{array}{cc} \mathbf{0} & \boldsymbol{A} \\ \boldsymbol{A}^{*} & \mathbf{0} \end{array}\right]\left[\begin{array}{c} \boldsymbol{u}_{i} \\ -\boldsymbol{v}_{i} \end{array}\right] } & =\left[\begin{array}{c} -\boldsymbol{A} \boldsymbol{v}_{i} \\ \boldsymbol{A}^{*} \boldsymbol{u}_{i} \end{array}\right]=\left[\begin{array}{c} -\alpha_{i} \boldsymbol{u}_{i} \\ \alpha_{i} \boldsymbol{v}_{i} \end{array}\right]=-\alpha_{i}\left[\begin{array}{c} \boldsymbol{u}_{i} \\ -\boldsymbol{v}_{i} \end{array}\right], \quad i=1, \ldots, r, \\ {\left[\begin{array}{cc} \mathbf{0} & \boldsymbol{A} \\ \boldsymbol{A}^{*} & \mathbf{0} \end{array}\right]\left[\begin{array}{c} \boldsymbol{u}_{i} \\ \mathbf{0} \end{array}\right] } & =\left[\begin{array}{c} \mathbf{0} \\ \boldsymbol{A}^{*} \boldsymbol{u}_{i} \end{array}\right]=\left[\begin{array}{l} \mathbf{0} \\ \mathbf{0} \end{array}\right]=0\left[\begin{array}{c} \boldsymbol{u}_{i} \\ \mathbf{0} \end{array}\right], \quad i=r+1, \ldots, m, \\ {\left[\begin{array}{cc} \mathbf{0} & \boldsymbol{A} \\ \boldsymbol{A}^{*} & \mathbf{0} \end{array}\right]\left[\begin{array}{c} \mathbf{0} \\ \boldsymbol{v}_{i} \end{array}\right] } & =\left[\begin{array}{c} \boldsymbol{A} \boldsymbol{v}_{i} \\ \mathbf{0} \end{array}\right]=\left[\begin{array}{l} \mathbf{0} \\ \mathbf{0} \end{array}\right]=0\left[\begin{array}{c} \mathbf{0} \\ \boldsymbol{v}_{i} \end{array}\right], \quad i=r+1, \ldots, n \end{aligned} | |  |
| 631 | | 229 | t:=\max (r, s) . | |  |
| 632 | | 229 | \begin{gathered} \lambda_{1} \geq \cdots \geq \lambda_{m+n}=\alpha_{1} \geq \cdots \geq \alpha_{t} \geq 0=\cdots=0 \geq-\alpha_{t} \geq \cdots \geq-\alpha_{1}, \\ \mu_{1} \geq \cdots \geq \mu_{m+n}=\beta_{1} \geq \cdots \geq \beta_{t} \geq 0=\cdots=0 \geq-\beta_{t} \geq \cdots \geq-\beta_{1} . \end{gathered} | |  |
| 633 | | 230 | \sum_{j=1}^{m+n}\left|\lambda_{j}-\mu_{j}\right|^{2}=\sum_{i=1}^{t}\left|\alpha_{i}-\beta_{i}\right|^{2}+\sum_{i=1}^{t}\left|-\alpha_{i}+\beta_{i}\right|^{2}=2 \sum_{i=1}^{t}\left|\alpha_{i}-\beta_{i}\right|^{2} | |  |
| 634 | | 230 | \|\boldsymbol{C}-\boldsymbol{D}\|_{F}^{2}=\left\|\left[\begin{array}{cc} \mathbf{0} & \boldsymbol{A}-\boldsymbol{B} \\ \boldsymbol{A}^{*}-\boldsymbol{B}^{*} & \mathbf{0} \end{array}\right]\right\|_{F}^{2}=\|\boldsymbol{B}-\boldsymbol{A}\|_{F}^{2}+\left\|(\boldsymbol{B}-\boldsymbol{A})^{*}\right\|_{F}^{2}=2\|\boldsymbol{B}-\boldsymbol{A}\|_{F}^{2} . | |  |
| 635 | 9.22 | 230 | \left(t_{i}-c_{1}\right)^{2}+\left(y_{i}-c_{2}\right)^{2}=r^{2}, i=1, \ldots, m, | |  |
| 636 | 9.23 | 230 | t_{i} x_{1}+y_{i} x_{2}+x_{3}=t_{i}^{2}+y_{i}^{2}, i=1, \ldots, m, | |  |
| 637 | | 231 | \sum_{i=1}^{3}\left\|p\left(x_{i}\right)-y_{i}\right\|^{2}, | |  |
| 638 | | 231 | F(\boldsymbol{x}):=\|\boldsymbol{A} \boldsymbol{x}-\boldsymbol{b}\|_{2}^{2}+\|\boldsymbol{x}\|_{2}^{2} . | |  |
| 639 | | 231 | F(\boldsymbol{x})=\boldsymbol{x}^{T} \boldsymbol{B} \boldsymbol{x}-2 \boldsymbol{c}^{T} \boldsymbol{x}+\boldsymbol{b}^{T} \boldsymbol{b}, \quad \text { where } \quad \boldsymbol{c}=\boldsymbol{A}^{T} \boldsymbol{b} . | |  |
| 640 | 9.24 | 231 | \|\boldsymbol{r}(\boldsymbol{x})\|_{D}^{2}:=\sum_{i=1}^{m} r_{i}(x)^{2} d_{i}, \quad \boldsymbol{x} \in \mathbb{R}^{n} | |  |
| 641 | | 232 | r=r(x)=b-A x . | |  |
| 642 | | 232 | \boldsymbol{A}^{T} \boldsymbol{D} \boldsymbol{A} \boldsymbol{x}=\boldsymbol{A}^{T} \boldsymbol{D} \boldsymbol{b} . | |  |
| 643 | | 232 | K_{2}\left(\boldsymbol{A}^{T} \boldsymbol{D} \boldsymbol{A}\right) \leq K_{2}\left(\boldsymbol{A}^{T} \boldsymbol{A}\right) K_{2}(\boldsymbol{D}), | |  |
| 644 | | 232 | \begin{array}{cl} \boldsymbol{A B A}=\boldsymbol{A} & (1) \quad \boldsymbol{A C A}=\boldsymbol{A}, \\ \boldsymbol{B A B}=\boldsymbol{B} & (2) \quad \boldsymbol{C A C}=\boldsymbol{C}, \\ (\boldsymbol{A B})^{*}=\boldsymbol{A B} & (3) \quad(\boldsymbol{A C})^{*}=\boldsymbol{A C}, \\ (\boldsymbol{B A})^{*}=\boldsymbol{B A} & (4) \quad(\boldsymbol{C A})^{*}=\boldsymbol{C A} . \end{array} | |  |
| 645 | | 232 | \begin{aligned} B & =(B A) B=\left(A^{*}\right) B^{*} B=\left(A^{*} C^{*}\right) A^{*} B^{*} B=C A\left(A^{*} B^{*}\right) B \\ & =C A(B A B)=(C) A B=C(A C) A B=C C^{*} A^{*}(A B) \\ & =C C^{*}\left(A^{*} B^{*} A^{*}\right)=C\left(C^{*} A^{*}\right)=C A C=C . \end{aligned} | |  |
| 646 | 9.25 | 233 | \begin{aligned} & \boldsymbol{P}_{\mathcal{R}\left(\boldsymbol{A}^{*}\right)}=\boldsymbol{A}^{\dagger} \boldsymbol{A}=\boldsymbol{V}_{1} \boldsymbol{V}_{1}^{*}=\sum_{j=1}^{r} \boldsymbol{v}_{j} \boldsymbol{v}_{j}^{*} \in \mathbb{C}^{n \times n} \\ & \boldsymbol{P}_{\mathcal{N}(\boldsymbol{A})}=\boldsymbol{I}-\boldsymbol{A}^{\dagger} \boldsymbol{A}=\boldsymbol{V}_{2} \boldsymbol{V}_{2}^{*}=\sum_{j=r+1}^{n} \boldsymbol{v}_{j} \boldsymbol{v}_{j}^{*} \in \mathbb{C}^{n \times n} . \end{aligned} | |  |
| 647 | | 233 | \boldsymbol{A}^{\dagger}=\frac{1}{\alpha} \boldsymbol{A}^{*}, \quad \alpha=\|\boldsymbol{u}\|_{2}^{2}\|\boldsymbol{v}\|_{2}^{2} . | |  |
| 648 | | 233 | \lambda_{i}^{\dagger}=\left\{\begin{array}{cc} 1 / \lambda_{i}, & \lambda_{i} \neq 0 \\ 0 & \lambda_{i}=0 . \end{array}\right. | |  |
| 649 | | 234 | U^{*} A V=\left[\begin{array}{cc} \Sigma_{1} & 0 \\ \mathbf{0} & \mathbf{0} \end{array}\right] | |  |
| 650 | | 234 | \left[\begin{array}{cc} \Sigma_{1} & 0 \\ 0 & 0 \end{array}\right] y=c . | |  |
| 651 | | 234 | \text { (1) } \quad \boldsymbol{A} \boldsymbol{x}=\boldsymbol{b}, \quad \text { (2) } \quad \boldsymbol{A}^{*} \boldsymbol{y}=\mathbf{0}, \boldsymbol{y}^{*} \boldsymbol{b} \neq 0 \text {. } | |  |
| 652 | | 234 | A=\left[\begin{array}{l} B \\ C \end{array}\right] | |  |
| 653 | | 235 | \boldsymbol{A}=\left[\begin{array}{ll} 1 & 2 \\ 1 & 1 \\ 1 & 1 \end{array}\right], \quad \boldsymbol{b}=\left[\begin{array}{l} b_{1} \\ b_{2} \\ b_{3} \end{array}\right] . | |  |
| 654 | | 235 | \boldsymbol{A}=\left[\begin{array}{cc} 1 & 1 \\ 1 & 1 \\ 1 & 1+\epsilon \end{array}\right], \quad \boldsymbol{b}=\left[\begin{array}{l} 2 \\ 3 \\ 2 \end{array}\right], \quad \in \in \mathbb{R} . | |  |
| 655 | | 235 | \left|\sigma_{i}(\epsilon)-\sigma_{i}(0)\right| \leq|\epsilon|, \quad i=1, \ldots, n, | |  |
| 656 | 10.1 | 238 | \begin{gathered} -\Delta u:=-\frac{\partial^{2} u}{\partial x^{2}}-\frac{\partial^{2} u}{\partial y^{2}}=f \text { on } \Omega, \\ u:=0 \text { on } \partial \Omega . \end{gathered} | |  |
| 657 | | 238 | \bar{\Omega}_{h}:=\{(j h, k h): j, k=0,1, \ldots, m+1\}, \quad \text { where } \quad h=1 /(m+1) . | |  |
| 658 | 10.2 | 239 | \frac{\partial^{2} u(j h, k h)}{\partial x^{2}} \approx \frac{v_{j-1, k}-2 v_{j, k}+v_{j+1, k}}{h^{2}}, \frac{\partial^{2} u(j h, k h)}{\partial y^{2}} \approx \frac{v_{j, k-1}-2 v_{j, k}+v_{j, k+1}}{h^{2}} . | |  |
| 659 | 10.3 | 239 | \begin{aligned} -\Delta_{h} v_{j, k} & =f_{j, k}, \quad(j h, k h) \in \Omega_{h}, \\ v_{j, k} & =0, \quad(j h, k h) \in \partial \Omega_{h}, \end{aligned} | |  |
| 660 | 10.4 | 239 | -\Delta_{h} v_{j, k}:=\frac{-v_{j-1, k}+2 v_{j, k}-v_{j+1, k}}{h^{2}}+\frac{-v_{j, k-1}+2 v_{j, k}-v_{j, k+1}}{h^{2}} . | |  |
| 661 | 10.5 | 239 | \boldsymbol{T} \boldsymbol{V}+\boldsymbol{V} \boldsymbol{T}=h^{2} \boldsymbol{F} \quad \text { with } \quad h=1 /(m+1), | |  |
| 662 | 10.6 | 239 | \boldsymbol{V}:=\left[\begin{array}{ccc} v_{1,1} & \cdots & v_{1, m} \\ \vdots & & \vdots \\ v_{m, 1} & \cdots & v_{m, m} \end{array}\right] \in \mathbb{R}^{m \times m}, \quad \boldsymbol{F}:=\left[\begin{array}{ccc} f_{1,1} & \cdots & f_{1, m} \\ \vdots & & \vdots \\ f_{m, 1} & \cdots & f_{m, m} \end{array}\right] \in \mathbb{R}^{m \times m} . | |  |
| 663 | | 239 | \sum_{i=1}^{m} \boldsymbol{T}_{j, i} v_{i, k}+\sum_{i=1}^{m} v_{j, i} \boldsymbol{T}_{i, k}=-v_{j-1, k}+2 v_{j, k}-v_{j+1, k}-v_{j, k-1}+2 v_{j, k}-v_{j, k+1}, | |  |
| 664 | | 239 | \operatorname{vec}(\boldsymbol{B}):=\left[b_{11}, \ldots, b_{m 1}, b_{12}, \ldots, b_{m 2}, \ldots, b_{1 n}, \ldots, b_{m n}\right]^{T} \in \mathbb{R}^{m n} | |  |
| 665 | | 240 | 4 x_{i}-x_{i-1}-x_{i+1}-x_{i-m}-x_{i+m}=b_{i}, | |  |
| 666 | 10.7 | 240 | \boldsymbol{A} \boldsymbol{x}=\boldsymbol{b}, \quad \boldsymbol{A} \in \mathbb{R}^{n \times n}, \quad b \in \mathbb{R}^{n}, \quad n=m^{2}, | |  |
| 667 | 10.8 | 240 | \begin{aligned} a_{i i}=4, & i=1, \ldots, n, \\ a_{i+1, i}=a_{i, i+1}=-1, & i=1, \ldots, n-1, \quad i \neq m, 2 m, \ldots,(m-1) m, \\ a_{i+m, i}=a_{i, i+m}=-1, & i=1, \ldots, n-m, \\ a_{i j}=0, & \text { otherwise } . \end{aligned} | |  |
| 668 | | 241 | \boldsymbol{A}=\left[\begin{array}{rrrrrrrr} 4 & -1 & 0 & -1 & 0 & 0 & 0 & 0 \\ -1 & 4 & -1 & 0 & -1 & 0 & 0 & 0 \\ 0 & -1 & 4 & 0 & 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 4 & -1 & 0 & -1 & 0 \\ 0 & -1 & 0 & -1 & 4 & -1 & 0 & -1 \\ 0 & 0 & -1 & 0 & -1 & 4 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 & 4 & -1 \\ 0 & 0 & 0 & 0 & -1 & 0 & -1 & 4 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & -1 \end{array}\right] . | |  |
| 669 | 10.9 | 241 | \boldsymbol{T}_{1}:=\operatorname{tridiag}(a, d, a) . | |  |
| 670 | 10.10 | 241 | \begin{aligned} a_{i i} & =2 d, i=1, \ldots, n, \\ a_{i, i+1}=a_{i+1, i} & =a, \quad i=1, \ldots, n-1, \quad i \neq m, 2 m, \ldots,(m-1) m, \\ a_{i, i+m}=a_{i+m, i} & =a, \quad i=1, \ldots, n-m, \\ a_{i j} & =0, \text { otherwise } \end{aligned} | |  |
| 671 | 10.11 | 242 | \boldsymbol{T}_{2}=\left[\begin{array}{rrr|rrr|rrr} 2 d & a & 0 & a & 0 & 0 & 0 & 0 & 0 \\ a & 2 d & a & 0 & a & 0 & 0 & 0 & 0 \\ 0 & a & 2 d & 0 & 0 & a & 0 & 0 & 0 \\ \hline a & 0 & 0 & 2 d & a & 0 & a & 0 & 0 \\ 0 & a & 0 & a & 2 d & a & 0 & a & 0 \\ 0 & 0 & a & 0 & a & 2 d & 0 & 0 & a \\ \hline 0 & 0 & 0 & a & 0 & 0 & 2 d & a & 0 \\ 0 & 0 & 0 & 0 & a & 0 & a & 2 d & a \\ 0 & 0 & 0 & 0 & 0 & a & 0 & a & 2 d \end{array}\right] . | |  |
| 672 | | 242 | \boldsymbol{C}=\left[\begin{array}{cccc} \boldsymbol{A} b_{1,1} & \boldsymbol{A} b_{1,2} & \cdots & \boldsymbol{A} b_{1, s} \\ \boldsymbol{A} b_{2,1} & \boldsymbol{A} b_{2,2} & \cdots & \boldsymbol{A} b_{2, s} \\ \vdots & \vdots & \ddots & \vdots \\ \boldsymbol{A} b_{r, 1} & \boldsymbol{A} b_{r, 2} & \cdots & \boldsymbol{A} b_{r, s} \end{array}\right] . | |  |
| 673 | | 242 | \boldsymbol{T}_{1}=\left[\begin{array}{ccc} d & a & 0 \\ a & d & a \\ 0 & a & d \end{array}\right], \quad \boldsymbol{I}=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] | |  |
| 674 | | 243 | \boldsymbol{T}_{1} \otimes \boldsymbol{I}+\boldsymbol{I} \otimes \boldsymbol{T}_{1}=\left[\begin{array}{ccc} \boldsymbol{T}_{1} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \boldsymbol{T}_{1} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \boldsymbol{T}_{1} \end{array}\right]+\left[\begin{array}{ccc} d \boldsymbol{I} & a \boldsymbol{I} & \mathbf{0} \\ a \boldsymbol{I} & d \boldsymbol{I} & a \boldsymbol{I} \\ \mathbf{0} & a \boldsymbol{I} & d \boldsymbol{I} \end{array}\right]=\boldsymbol{T}_{2} | |  |
| 675 | 10.12 | 243 | \boldsymbol{T}_{2}=\boldsymbol{T}_{1} \otimes \boldsymbol{I}+\boldsymbol{I} \otimes \boldsymbol{T}_{1}, \quad \boldsymbol{T}_{1}, \boldsymbol{I} \in \mathbb{R}^{m \times m}, \quad \boldsymbol{T}_{2} \in \mathbb{R}^{\left(m^{2}\right) \times\left(m^{2}\right)} . | |  |
| 676 | 10.13 | 243 | \begin{aligned} (\lambda \boldsymbol{A}) \otimes(\mu \boldsymbol{B}) & =\lambda \mu(\boldsymbol{A} \otimes \boldsymbol{B}) \\ \left(\boldsymbol{A}_{1}+\boldsymbol{A}_{2}\right) \otimes \boldsymbol{B} & =\boldsymbol{A}_{1} \otimes \boldsymbol{B}+\boldsymbol{A}_{2} \otimes \boldsymbol{B}, \\ \boldsymbol{A} \otimes\left(\boldsymbol{B}_{1}+\boldsymbol{B}_{2}\right) & =\boldsymbol{A} \otimes \boldsymbol{B}_{1}+\boldsymbol{A} \otimes \boldsymbol{B}_{2} \\ (\boldsymbol{A} \otimes \boldsymbol{B}) \otimes \boldsymbol{C} & =\boldsymbol{A} \otimes(\boldsymbol{B} \otimes \boldsymbol{C}) . \end{aligned} | |  |
| 677 | 10.14 | 243 | (\boldsymbol{A} \otimes \boldsymbol{B})(\boldsymbol{C} \otimes \boldsymbol{D})=(\boldsymbol{A} \boldsymbol{C}) \otimes(\boldsymbol{B} \boldsymbol{D}) . | |  |
| 678 | | 243 | (\boldsymbol{A} \otimes \boldsymbol{B})(\boldsymbol{C} \otimes \boldsymbol{D})=\left[\begin{array}{ccc} \boldsymbol{A} b_{1,1} & \cdots & \boldsymbol{A} b_{1, t} \\ \vdots & & \vdots \\ \boldsymbol{A} b_{r, 1} & \cdots & \boldsymbol{A} b_{r, t} \end{array}\right]\left[\begin{array}{ccc} \boldsymbol{C} d_{1,1} & \cdots & \boldsymbol{C} d_{1, s} \\ \vdots & & \vdots \\ \boldsymbol{C} d_{t, 1} & \cdots & \boldsymbol{C} d_{t, s} \end{array}\right] . | |  |
| 679 | | 244 | ((\boldsymbol{A} \otimes \boldsymbol{B})(\boldsymbol{C} \otimes \boldsymbol{D}))_{i, j}=\boldsymbol{A} \boldsymbol{C} \sum_{k=1}^{t} b_{i, k} d_{k, j}=(\boldsymbol{A} \boldsymbol{C})(\boldsymbol{B} \boldsymbol{D})_{i, j}=((\boldsymbol{A} \boldsymbol{C}) \otimes(\boldsymbol{B} \boldsymbol{D}))_{i, j} . | |  |
| 680 | | 245 | \begin{aligned} (\boldsymbol{A} \otimes \boldsymbol{B}) \operatorname{vec}(\boldsymbol{V}) & =\operatorname{vec}(\boldsymbol{F}) \\ & \Leftrightarrow\left[\begin{array}{ccc} \boldsymbol{A} b_{11} & \cdots & \boldsymbol{A} b_{1 s} \\ \vdots & & \vdots \\ \boldsymbol{A} b_{s 1} & \cdots & \boldsymbol{A} b_{s s} \end{array}\right]\left[\begin{array}{c} \boldsymbol{v}_{1} \\ \vdots \\ \boldsymbol{v}_{s} \end{array}\right]=\left[\begin{array}{c} \boldsymbol{f}_{1} \\ \vdots \\ \boldsymbol{f}_{s} \end{array}\right] \\ & \Leftrightarrow \boldsymbol{A}\left[\sum_{j} b_{1 j} \boldsymbol{v}_{j}, \ldots, \sum_{j} b_{s j} \boldsymbol{v}_{j}\right]=\left[\boldsymbol{f}_{1}, \ldots, \boldsymbol{f}_{s}\right] \\ & \Leftrightarrow \boldsymbol{A}\left[\boldsymbol{V} \boldsymbol{b}_{1}, \ldots, \boldsymbol{V} \boldsymbol{b}_{s}\right]=\boldsymbol{F} \quad \Leftrightarrow \quad \boldsymbol{A} \boldsymbol{V} \boldsymbol{B}^{T}=\boldsymbol{F} . \end{aligned} | |  |
| 681 | | 245 | \begin{aligned} \left(\boldsymbol{A} \otimes \boldsymbol{I}_{s}+\boldsymbol{I}_{r} \otimes \boldsymbol{B}\right) \operatorname{vec}(\boldsymbol{V}) & =\operatorname{vec}(\boldsymbol{F}) \\ \Leftrightarrow \quad\left(\boldsymbol{A} \boldsymbol{V} \boldsymbol{I}_{s}^{T}+\boldsymbol{I}_{r} \boldsymbol{V} \boldsymbol{B}^{T}\right) & =\boldsymbol{F} \quad \Leftrightarrow \quad \boldsymbol{A} \boldsymbol{V}+\boldsymbol{V} \boldsymbol{B}^{T}=\boldsymbol{F} . \end{aligned} | |  |
| 682 | 10.15 | 245 | \lambda_{j}=d+2 a \cos (j \pi h), \quad h:=\frac{1}{m+1}, | |  |
| 683 | 10.16 | 246 | \boldsymbol{s}_{j}=[\sin (j \pi h), \sin (2 j \pi h), \ldots, \sin (m j \pi h)]^{T} . | |  |
| 684 | 10.17 | 246 | \boldsymbol{s}_{j}^{T} \boldsymbol{s}_{k}=\frac{m+1}{2} \delta_{j, k}=\frac{1}{2 h} \delta_{j, k}, \quad j, k=1, \ldots, m . | |  |
| 685 | 10.18 | 246 | \boldsymbol{T}_{2}\left(\boldsymbol{s}_{j} \otimes \boldsymbol{s}_{k}\right)=\left(\lambda_{j}+\lambda_{k}\right)\left(\boldsymbol{s}_{j} \otimes \boldsymbol{s}_{k}\right) \quad j, k=1, \ldots, m, | |  |
| 686 | 10.19 | 246 | \left(\boldsymbol{s}_{j} \otimes \boldsymbol{s}_{k}\right)^{T}\left(\boldsymbol{s}_{p} \otimes \boldsymbol{s}_{q}\right)=\frac{1}{4 h^{2}} \delta_{j, p} \delta_{k, q}, \quad j, k, p, q=1, \ldots, m, | |  |
| 687 | 4h | 246 | \left(\boldsymbol{s}_{j} \otimes \boldsymbol{s}_{k}\right)^{T}\left(\boldsymbol{s}_{p} \otimes \boldsymbol{s}_{q}\right)=\left(\boldsymbol{s}_{j}^{T} \otimes \boldsymbol{s}_{k}^{T}\right)\left(\boldsymbol{s}_{p} \otimes \boldsymbol{s}_{q}\right)=\left(\boldsymbol{s}_{j}^{T} \boldsymbol{s}_{p}\right) \otimes\left(\boldsymbol{s}_{k}^{T} \boldsymbol{s}_{q}\right)=\frac{1}{4 h^{2}} \delta_{j, p} \delta_{k, q} | |  |
| 688 | 10.20 | 246 | \|\boldsymbol{A}\|_{2}\left\|\boldsymbol{A}^{-1}\right\|_{2}=\frac{\cos ^{2} w}{\sin ^{2} w}, \quad w:=\frac{\pi}{2(m+1)} | |  |
| 689 | | 246 | \lambda_{j, k}=4-2 \cos (2 j w)-2 \cos (2 k w)=4 \sin ^{2}(j w)+4 \sin ^{2}(k w), \quad j, k=1, \ldots, m . | |  |
| 690 | | 246 | \lambda_{\max }=8 \cos ^{2} w, \quad \lambda_{\min }=8 \sin ^{2} w . | |  |
| 691 | | 247 | \boldsymbol{A}=\left[\begin{array}{rrrr} 2 d & a & a & 0 \\ a & 2 d & 0 & a \\ a & 0 & 2 d & a \\ 0 & a & a & 2 d \end{array}\right] . | |  |
| 692 | 10.22 | 248 | \boldsymbol{T} \boldsymbol{V}+\boldsymbol{V} \boldsymbol{T}-\frac{1}{6} \boldsymbol{T} \boldsymbol{V} \boldsymbol{T}=h^{2} \mu \boldsymbol{F} | |  |
| 693 | 10.23 | 248 | \begin{array}{ll} \Delta^{2} u(s, t):=\Delta(\Delta u(s, t))=f(s, t) & (s, t) \in \Omega, \\ u(s, t)=0, & \Delta u(s, t)=0 \end{array} | |  |
| 694 | 10.24 | 248 | \begin{array}{rlrl} -\Delta v(s, t) & =f(s, t) & & (s, t) \in \Omega \\ -\Delta u(s, t) & =v(s, t) & & (s, t) \in \Omega \\ u(s, t) & =v(s, t)=0 & (s, t) \in \partial \Omega \end{array} | |  |
| 695 | 10.21 | 248 | \boldsymbol{T} \boldsymbol{V}+\boldsymbol{V} \boldsymbol{T}=h^{2} \boldsymbol{F}, \quad \boldsymbol{T} \boldsymbol{U}+\boldsymbol{U} \boldsymbol{T}=h^{2} \boldsymbol{V} . | |  |
| 696 | | 249 | (\boldsymbol{T} \otimes \boldsymbol{I}+\boldsymbol{I} \otimes \boldsymbol{T}) \operatorname{vec}(\boldsymbol{V})=h^{2} \operatorname{vec}(\boldsymbol{F}), \quad(\boldsymbol{T} \otimes \boldsymbol{I}+\boldsymbol{I} \otimes \boldsymbol{T}) \operatorname{vec}(\boldsymbol{U})=h^{2} \operatorname{vec}(\boldsymbol{V}) . | |  |
| 697 | 10.25 | 249 | \boldsymbol{T}^{2} \boldsymbol{U}+2 \boldsymbol{T} \boldsymbol{U} \boldsymbol{T}+\boldsymbol{U} \boldsymbol{T}^{2}=h^{4} \boldsymbol{F} \quad \text { and } \quad \boldsymbol{A} \boldsymbol{x}=\boldsymbol{b}, | |  |
| 698 | | 250 | \begin{aligned} \boldsymbol{A} & =\left[\begin{array}{rrr|rrr|rrr} 4 & -1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\ -1 & 4 & -1 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 4 & 0 & 0 & -1 & 0 & 0 & 0 \\ \hline-1 & 0 & 0 & 4 & -1 & 0 & -1 & 0 & 0 \\ 0 & -1 & 0 & -1 & 4 & -1 & 0 & -1 & 0 \\ 0 & 0 & -1 & 0 & -1 & 4 & 0 & 0 & -1 \\ \hline 0 & 0 & 0 & -1 & 0 & 0 & 4 & -1 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 & -1 & 4 & -1 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0-1 & 4 \end{array}\right] \\ & =\left[\begin{array}{ccc} \boldsymbol{T}+2 \boldsymbol{I} & -\boldsymbol{I} & \mathbf{0} \\ -\boldsymbol{I} & \boldsymbol{T}+2 \boldsymbol{I} & -\boldsymbol{I} \\ \mathbf{0} & -\boldsymbol{I} & \boldsymbol{T}+2 \boldsymbol{I} \end{array}\right], \end{aligned} | |  |
| 699 | 11.1 | 251 | \left[\begin{array}{cccc} \boldsymbol{D}_{1} & \boldsymbol{C}_{1} & & \\ \boldsymbol{A}_{1} & \boldsymbol{D}_{2} & \boldsymbol{C}_{2} & \\ & \ddots & \ddots & \ddots \\ & & \boldsymbol{A}_{m-2} & \boldsymbol{D}_{m-1} \\ & & \boldsymbol{A}_{m-1} & \boldsymbol{D}_{m} \end{array}\right]=\left[\begin{array}{cccc} \boldsymbol{I} & & & \\ \boldsymbol{L}_{1} & \boldsymbol{I} & & \\ & \ddots & \ddots & \\ & & \boldsymbol{L}_{m-1} & \boldsymbol{I} \end{array}\right]\left[\begin{array}{cccc} \boldsymbol{U}_{1} & \boldsymbol{C}_{1} & & \\ & \ddots & \ddots & \\ & & \boldsymbol{U}_{m-1} & \boldsymbol{C}_{m-1} \\ & & & \boldsymbol{U}_{m} \end{array}\right] . | |  |
| 700 | 11.2 | 252 | \boldsymbol{U}_{1}=\boldsymbol{D}_{1}, \quad \boldsymbol{L}_{k}=\boldsymbol{A}_{k} \boldsymbol{U}_{k}^{-1}, \quad \boldsymbol{U}_{k+1}=\boldsymbol{D}_{k+1}-\boldsymbol{L}_{k} \boldsymbol{C}_{k}, \quad k=1,2, \ldots, m-1 . | |  |
| 701 | 11.3 | 252 | \begin{aligned} \boldsymbol{y}_{1} & =\boldsymbol{b}_{1}, \quad \boldsymbol{y}_{k}=\boldsymbol{b}_{k}-\boldsymbol{L}_{k-1} \boldsymbol{y}_{k-1}, \quad k=2,3, \ldots, m, \\ \boldsymbol{x}_{m} & =\boldsymbol{U}_{m}^{-1} \boldsymbol{y}_{m}, \quad \boldsymbol{x}_{k}=\boldsymbol{U}_{k}^{-1}\left(\boldsymbol{y}_{k}-\boldsymbol{C}_{k} \boldsymbol{x}_{k+1}\right), \quad k=m-1, \ldots, 2,1 . \end{aligned} | |  |
| 702 | | 252 | \boldsymbol{T} \boldsymbol{V}+\boldsymbol{V} \boldsymbol{T}=h^{2} \boldsymbol{F} \quad \text { with } \quad h=1 /(m+1), | |  |
| 703 | | 253 | \begin{aligned} \boldsymbol{T s}_{j} & =\lambda_{j} \boldsymbol{s}_{j}, \quad j=1, \ldots, m, \\ \boldsymbol{s}_{j} & =[\sin (j \pi h), \sin (2 j \pi h), \ldots, \sin (m j \pi h)]^{T}, \\ \lambda_{j} & =2-2 \cos (j \pi h)=4 \sin ^{2}(j \pi h / 2), \quad h=1 /(m+1), \\ \boldsymbol{s}_{j}^{T} \boldsymbol{s}_{k} & =\delta_{j, k} /(2 h) \text { for all } j, k . \end{aligned} | |  |
| 704 | 11.4 | 253 | \boldsymbol{S}:=\left[\boldsymbol{s}_{1}, \ldots, \boldsymbol{s}_{m}\right]=[\sin (j k \pi h)]_{j, k=1}^{m} \in \mathbb{R}^{m \times m}, \quad \boldsymbol{D}=\operatorname{diag}\left(\lambda_{1}, \ldots, \lambda_{m}\right) . | |  |
| 705 | 2h | 253 | \boldsymbol{T} \boldsymbol{S}=\left[\boldsymbol{T} \boldsymbol{s}_{1}, \ldots, \boldsymbol{T} \boldsymbol{s}_{m}\right]=\left[\lambda_{1} \boldsymbol{s}_{1}, \ldots, \lambda_{m} \boldsymbol{s}_{m}\right]=\boldsymbol{S} \boldsymbol{D}, \quad \boldsymbol{S}^{2}=\boldsymbol{S}^{T} \boldsymbol{S}=\frac{1}{2 h} \boldsymbol{I} . | |  |
| 706 | | 253 | \begin{array}{rl} \boldsymbol{T} \boldsymbol{V}+\boldsymbol{V} \boldsymbol{T} & =h^{2} \boldsymbol{F} \\ \boldsymbol{V}^{\boldsymbol{V}=\boldsymbol{S} \boldsymbol{X} \boldsymbol{S}} \boldsymbol{T} \boldsymbol{S} \boldsymbol{X} \boldsymbol{S}+\boldsymbol{S} \boldsymbol{X} \boldsymbol{S} \boldsymbol{T} & =h^{2} \boldsymbol{F} \\ \stackrel{\boldsymbol{S}() \boldsymbol{S}}{\Longleftrightarrow} \boldsymbol{S} \boldsymbol{T} \boldsymbol{S} \boldsymbol{X} \boldsymbol{S}^{2}+\boldsymbol{S}^{2} \boldsymbol{X} \boldsymbol{S} \boldsymbol{T} \boldsymbol{S} & =h^{2} \boldsymbol{S} \boldsymbol{F} \boldsymbol{S}=h^{2} \boldsymbol{G} \\ \stackrel{\boldsymbol{T}}{\Leftrightarrow} \boldsymbol{S}^{\mathbf{2}} \boldsymbol{D} & \boldsymbol{S} \boldsymbol{X} \boldsymbol{S}^{2}+\boldsymbol{S}^{2} \boldsymbol{X} \boldsymbol{S}^{2} \boldsymbol{D} \\ \boldsymbol{S}^{2} & =h^{2} \boldsymbol{G} \\ \Longleftrightarrow & \boldsymbol{I} /(2 h) \end{array} | |  |
| 707 | | 253 | (\boldsymbol{D} \boldsymbol{X}+\boldsymbol{X} \boldsymbol{D})_{j, k}=\sum_{\ell=1}^{m} d_{j, \ell} x_{\ell, k}+\sum_{\ell=1}^{m} x_{j, \ell} d_{\ell, k}=\lambda_{j} x_{j, k}+\lambda_{k} x_{j, k} | |  |
| 708 | | 253 | x_{j, k}=4 h^{4} g_{j, k} /\left(\lambda_{j}+\lambda_{k}\right)=h^{4} g_{j, k} /\left(\sigma_{j}+\sigma_{k}\right), \quad \sigma_{j}:=\lambda_{j} / 4=\sin ^{2}(j \pi h / 2) . | |  |
| 709 | | 254 | \begin{aligned} & \boldsymbol{X}=h^{4} \boldsymbol{G} / \boldsymbol{M}, \text { where } \boldsymbol{M}:=\left[\begin{array}{c} \sigma_{1} \\ \vdots \\ \sigma_{m} \end{array}\right][1, \ldots, 1]+\left[\begin{array}{c} 1 \\ \vdots \\ i \end{array}\right]\left[\sigma_{1} \ldots . \sigma_{m}\right], \\ & \boldsymbol{S}=\sin \left(\pi h\left[\begin{array}{c} 1 \\ 2 \\ \vdots \\ m \end{array}\right][12 \ldots m]\right), \quad \boldsymbol{\sigma}=\sin \left(\frac{\pi h}{2}\left[\begin{array}{c} 1 \\ 2 \\ \vdots \\ m \end{array}\right]\right) \wedge 2 . \end{aligned} | |  |
| 710 | | 255 | w_{j}=\sum_{k=1}^{m} \sin \left(\frac{j k \pi}{m+1}\right) v_{k}, \quad j=1, \ldots, m | |  |
| 711 | 11.5 | 255 | \omega_{N}=\exp ^{-2 \pi i / N}=\cos (2 \pi / N)-i \sin (2 \pi / N), | |  |
| 712 | | 256 | z_{j+1}=\sum_{k=0}^{N-1} \omega_{N}^{j k} y_{k+1}, \quad j=0, \ldots, N-1, | |  |
| 713 | | 256 | \omega_{4}=\exp ^{-2 \pi i / 4}=\cos (\pi / 2)-i \sin (\pi / 2)=-i | |  |
| 714 | 11.6 | 256 | \boldsymbol{F}_{4}=\left[\begin{array}{cccc} 1 & 1 & 1 & 1 \\ 1 & \omega_{4} & \omega_{4}^{2} & \omega_{4}^{3} \\ 1 & \omega_{4}^{2} & \omega_{4}^{4} & \omega_{4}^{6} \\ 1 & \omega_{4}^{3} & \omega_{4}^{6} & \omega_{4}^{9} \end{array}\right]=\left[\begin{array}{rrrr} 1 & 1 & 1 & 1 \\ 1 & -i & -1 & i \\ 1 & -1 & 1 & -1 \\ 1 & i & -1 & -i \end{array}\right] . | |  |
| 715 | | 256 | \sin w=\frac{e^{i w}-e^{-i w}}{2 i} . | |  |
| 716 | | 256 | \boldsymbol{z}^{T}=\left[0, \boldsymbol{x}^{T}, 0,-\boldsymbol{x}_{B}^{T}\right] \in \mathbb{R}^{2 m+2}, \quad \boldsymbol{x}_{B}^{T}:=\left[x_{m}, \ldots, x_{2}, x_{1}\right] . | |  |
| 717 | | 256 | (\boldsymbol{S} \boldsymbol{x})_{k}=\frac{i}{2}\left(\boldsymbol{F}_{2 m+2} \boldsymbol{z}\right)_{k+1}, \quad k=1, \ldots, m . | |  |
| 718 | | 256 | \omega^{j k}=e^{-\pi i j k /(m+1)}, \quad \omega^{(2 m+2-j) k}=e^{-2 \pi i} e^{\pi i j k /(m+1)}=e^{\pi i j k /(m+1)} . | |  |
| 719 | | 257 | \begin{aligned} \left(\boldsymbol{F}_{2 m+2} \boldsymbol{z}\right)_{k+1} & =\sum_{j=0}^{2 m-1} \omega^{j k} z_{j+1}=\sum_{j=1}^{m} x_{j} \omega^{j k}-\sum_{j=1}^{m} x_{j} \omega^{(2 m+2-j) k} \\ & =\sum_{j=1}^{m} x_{j}\left(e^{-\pi i j k /(m+1)}-e^{\pi i j k /(m+1)}\right) \\ & =-2 i \sum_{j=1}^{m} x_{j} \sin \left(\frac{j k \pi}{m+1}\right)=-2 i\left(\boldsymbol{S}_{m} \boldsymbol{x}\right)_{k} . \end{aligned} | |  |
| 720 | | 257 | \boldsymbol{P}_{N}=\left[\boldsymbol{e}_{1}, \boldsymbol{e}_{3}, \ldots, \boldsymbol{e}_{N-1}, \boldsymbol{e}_{2}, \boldsymbol{e}_{4}, \ldots, \boldsymbol{e}_{N}\right], | |  |
| 721 | | 257 | \boldsymbol{A} \boldsymbol{P}_{N}=\left[\boldsymbol{a}_{1}, \boldsymbol{a}_{3}, \ldots, \boldsymbol{a}_{N-1}, \boldsymbol{a}_{2}, \boldsymbol{a}_{4}, \ldots, \boldsymbol{a}_{N}\right], | |  |
| 722 | | 257 | \boldsymbol{P}_{4}=\left[\begin{array}{lll} \boldsymbol{e}_{1} & \boldsymbol{e}_{3} & \boldsymbol{e}_{2} \\ \boldsymbol{e}_{4} \end{array}\right]=\left[\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \quad \boldsymbol{F}_{4} \boldsymbol{P}_{4}=\left[\begin{array}{rr|rr} 1 & 1 & 1 & 1 \\ 1 & -1 & -i & i \\ \hline 1 & 1 & -1 & -1 \\ 1 & -1 & i & -i \end{array}\right], | |  |
| 723 | | 258 | \boldsymbol{D}_{2}=\operatorname{diag}\left(1, \omega_{4}\right)=\left[\begin{array}{rr} 1 & 0 \\ 0 & -i \end{array}\right] . | |  |
| 724 | | 258 | \boldsymbol{F}_{2}=\left[\begin{array}{rr} 1 & 1 \\ 1 & -1 \end{array}\right], \quad \boldsymbol{D}_{2} \boldsymbol{F}_{2}=\left[\begin{array}{rr} 1 & 1 \\ -i & i \end{array}\right], | |  |
| 725 | | 258 | \boldsymbol{F}_{4} \boldsymbol{P}_{4}=\left[\begin{array}{r|r} \boldsymbol{F}_{2} & \boldsymbol{D}_{2} \boldsymbol{F}_{2} \\ \hline \boldsymbol{F}_{2} & -\boldsymbol{D}_{2} \boldsymbol{F}_{2} \end{array}\right] . | |  |
| 726 | 11.7 | 258 | \boldsymbol{F}_{2 m} \boldsymbol{P}_{2 m}=\left[\begin{array}{c|c} \boldsymbol{F}_{m} & \boldsymbol{D}_{m} \boldsymbol{F}_{m} \\ \hline \boldsymbol{F}_{m} & -\boldsymbol{D}_{m} \boldsymbol{F}_{m} \end{array}\right], | |  |
| 727 | 11.8 | 258 | \boldsymbol{D}_{m}=\operatorname{diag}\left(1, \omega_{N}, \omega_{N}^{2}, \ldots, \omega_{N}^{m-1}\right) . | |  |
| 728 | | 258 | \omega_{m}^{m}=1, \omega_{2 m}^{2 k}=\omega_{m}^{k}, \omega_{2 m}^{m}=-1,\left(\boldsymbol{F}_{m}\right)_{p, q}=\omega_{m}^{j k},\left(\boldsymbol{D}_{m} \boldsymbol{F}_{m}\right)_{p, q}=\omega_{2 m}^{j} \omega_{m}^{j k}, | |  |
| 729 | | 258 | \begin{array}{lll} \left(\boldsymbol{F}_{2 m} \boldsymbol{P}_{2 m}\right)_{p, q} & =\omega_{2 m}^{j(2 k)} & =\omega_{m}^{j k}, \\ \left(\boldsymbol{F}_{2 m} \boldsymbol{P}_{2 m}\right)_{p+m, q} & =\omega_{2 m}^{(j+m)(2 k)} & =\omega_{m}^{j+m) k} \\ \left(\boldsymbol{F}_{2 m} \boldsymbol{P}_{2 m}\right)_{p, q+m} & =\omega_{2 m}^{j(2 k+1)} & =\omega_{2 m}^{j} \omega_{m}^{j k}, \\ \left(\boldsymbol{F}_{2 m} \boldsymbol{P}_{2 m}\right)_{p+m, q+m} & \left.=\omega_{2 m}^{j+m}\right)(2 k+1) & \left.=\omega_{2 m}^{j+m} \omega_{m}^{j+m}\right) k \\ & \\ (j+m & & \end{array} | |  |
| 730 | | 258 | \boldsymbol{w}_{1}^{T}=\left[y_{1}, y_{3}, \ldots, y_{2 m-1}\right], \quad \boldsymbol{w}_{2}^{T}=\left[y_{2}, y_{4}, \ldots, y_{2 m}\right] . | |  |
| 731 | | 259 | \begin{aligned} \boldsymbol{F}_{2 m} \boldsymbol{y} & =\boldsymbol{F}_{2 m} \boldsymbol{P}_{2 m} \boldsymbol{P}_{2 m}^{T} \boldsymbol{y}=\boldsymbol{F}_{2 m} \boldsymbol{P}_{2 m} \boldsymbol{w} \\ & =\left[\begin{array}{r|r} \boldsymbol{F}_{m} & \boldsymbol{D}_{m} \boldsymbol{F}_{m} \\ \hline \boldsymbol{F}_{m} & -\boldsymbol{D}_{m} \boldsymbol{F}_{m} \end{array}\right]\left[\begin{array}{l} \boldsymbol{w}_{1} \\ \boldsymbol{w}_{2} \end{array}\right]=\left[\begin{array}{l} \boldsymbol{q}_{1}+\boldsymbol{q}_{2} \\ \boldsymbol{q}_{1}-\boldsymbol{q}_{2} \end{array}\right], \end{aligned} | |  |
| 732 | | 259 | \boldsymbol{q}_{1}=\boldsymbol{F}_{m} \boldsymbol{w}_{1}, \quad \text { and } \quad \boldsymbol{q}_{2}=\boldsymbol{D}_{m}\left(\boldsymbol{F}_{m} \boldsymbol{w}_{2}\right) . | |  |
| 733 | | 260 | \frac{8 N^{2}}{5 N \log _{2} N} \approx 84000 . | |  |
| 734 | | 260 | 8 \gamma m(m+1) \log _{2}(2 m+2) \approx 8 \gamma m^{2} \log _{2} m=4 \gamma n \log _{2} n, | |  |
| 735 | | 261 | v_{i j}=\frac{1}{(m+1)^{4}} \sum_{p=1}^{m} \sum_{r=1}^{m} \sum_{k=1}^{m} \sum_{l=1}^{m} \frac{\sin \left(\frac{i p \pi}{m+1}\right) \sin \left(\frac{j r \pi}{m+1}\right) \sin \left(\frac{k p \pi}{m+1}\right) \sin \left(\frac{l r \pi}{m+1}\right)}{\left[\sin \left(\frac{p \pi}{2(m+1)}\right)\right]^{2}+\left[\sin \left(\frac{r \pi}{2(m+1)}\right)\right]^{2}} f_{k, l} . | |  |
| 736 | | 261 | \boldsymbol{T} \boldsymbol{X}+\boldsymbol{X} \boldsymbol{D}=\boldsymbol{C}, \text { where } \boldsymbol{X}=\boldsymbol{V} \boldsymbol{S}, \text { and } \boldsymbol{C}=h^{2} \boldsymbol{F} \boldsymbol{S} . | |  |
| 737 | 11.9 | 261 | \left(\boldsymbol{T}+\lambda_{j} \boldsymbol{I}\right) \boldsymbol{x}_{j}=\boldsymbol{c}_{j} \quad j=1, \ldots, m, | |  |
| 738 | | 261 | \boldsymbol{T} \boldsymbol{V}+\boldsymbol{V} \boldsymbol{T}-\frac{1}{6} \boldsymbol{T} \boldsymbol{V} \boldsymbol{T}=h^{2} \mu \boldsymbol{F}, | |  |
| 739 | | 261 | \boldsymbol{D} \boldsymbol{X}+\boldsymbol{X} \boldsymbol{D}-\frac{1}{6} \boldsymbol{D} \boldsymbol{X} \boldsymbol{D}=4 h^{4} \boldsymbol{G}, \text { where } \boldsymbol{G}=\boldsymbol{S} \mu \boldsymbol{F} \boldsymbol{S}, | |  |
| 740 | | 262 | x_{j, k}=\frac{h^{4} g_{j, k}}{\sigma_{j}+\sigma_{k}-\frac{2}{3} \sigma_{j} \sigma_{k}}, \text { where } \sigma_{j}=\sin ^{2}((j \pi h) / 2) \text { for } j, k=1,2, \ldots, m . | |  |
| 741 | | 262 | \boldsymbol{T}^{2} \boldsymbol{U}+2 \boldsymbol{T} \boldsymbol{U} \boldsymbol{T}+\boldsymbol{U} \boldsymbol{T}^{2}=h^{4} \boldsymbol{F} . | |  |
| 742 | | 262 | \boldsymbol{D}^{2} \boldsymbol{X}+2 \boldsymbol{D} \boldsymbol{X} \boldsymbol{D}+\boldsymbol{X} \boldsymbol{D}^{2}=4 h^{6} \boldsymbol{G}, \text { where } \boldsymbol{G}=\boldsymbol{S} \boldsymbol{F} \boldsymbol{S}, | |  |
| 743 | | 262 | x_{j, k}=\frac{h^{6} g_{j, k}}{4\left(\sigma_{j}+\sigma_{k}\right)^{2}}, \text { where } \sigma_{j}=\sin ^{2}((j \pi h) / 2) \text { for } j, k=1,2, \ldots, m . | |  |
| 744 | 12.1 | 266 | \boldsymbol{x}(i)=\left(-\sum_{j=1}^{i-1} a_{i j} \boldsymbol{x}(j)-\sum_{j=i+1}^{n} a_{i j} \boldsymbol{x}(j)+b_{i}\right) / a_{i i}, \quad i=1,2, \ldots, n . | |  |
| 745 | 12.2 | 266 | \boldsymbol{x}_{k+1}(i)=\left(-\sum_{j=1}^{i-1} a_{i j} \boldsymbol{x}_{k}(j)-\sum_{j=i+1}^{n} a_{i j} \boldsymbol{x}_{k}(j)+b_{i}\right) / a_{i i}, \text { for } i=1,2, \ldots, n . | |  |
| 746 | 12.3 | 266 | \boldsymbol{x}_{k+1}(i)=\left(-\sum_{j=1}^{i-1} a_{i j} \boldsymbol{x}_{k+1}(j)-\sum_{j=i+1}^{n} a_{i j} \boldsymbol{x}_{k}(j)+b_{i}\right) / a_{i i}, \text { for } i=1,2, \ldots, n . | |  |
| 747 | 12.4 | 266 | \boldsymbol{x}_{k+1}(i)=\omega\left(-\sum_{j=1}^{i-1} a_{i j} \boldsymbol{x}_{k+1}(j)-\sum_{j=i+1}^{n} a_{i j} \boldsymbol{x}_{k}(j)+b_{i}\right) / a_{i i}+(1-\omega) \boldsymbol{x}_{k}(i) . | |  |
| 748 | 12.5 | 267 | \boldsymbol{x}_{k+1}(i)=\omega\left(-\sum_{j=1}^{i-1} a_{i j} \boldsymbol{x}_{k+1 / 2}(j)-\sum_{j=i+1}^{n} a_{i j} \boldsymbol{x}_{k+1}(j)+b_{i}\right) / a_{i i}+(1-\omega) \boldsymbol{x}_{k+1 / 2}(i) | |  |
| 749 | | 267 | 4 \boldsymbol{v}(i, j)-\boldsymbol{v}(i-1, j)-\boldsymbol{v}(i+1, j)-\boldsymbol{v}(i, j-1)-\boldsymbol{v}(i, j+1)=h^{2} f_{i, j}, \quad i, j=1, \ldots, m, | |  |
| 750 | 12.6 | 267 | \boldsymbol{v}(i, j)=\left(\boldsymbol{v}(i-1, j)+\boldsymbol{v}(i+1, j)+\boldsymbol{v}(i, j-1)+v(i, j+1)+e_{i, j}\right) / 4, | |  |
| 751 | | 267 | \begin{array}{r} J: \boldsymbol{v}_{k+1}(i, j)=\left(\boldsymbol{v}_{k}(i-1, j)+\boldsymbol{v}_{k}(i, j-1)+\boldsymbol{v}_{k}(i+1, j)+\boldsymbol{v}_{k}(i, j+1)\right. \\ +\boldsymbol{e}(i, j)) / 4 \\ G S: \boldsymbol{v}_{k+1}(i, j)=\left(\boldsymbol{v}_{k+1}(i-1, j)+\boldsymbol{v}_{k+1}(i, j-1)+\boldsymbol{v}_{k}(i+1, j)+\boldsymbol{v}_{k}(i, j+1)\right. \\ +\boldsymbol{e}(i, j)) / 4 \end{array} | |  |
| 752 | 12.7 | 268 | \begin{array}{r} S O R: \boldsymbol{v}_{k+1}(i, j)=\omega\left(\boldsymbol{v}_{k+1}(i-1, j)+\boldsymbol{v}_{k+1}(i, j-1)+\boldsymbol{v}_{k}(i+1, j)\right. \\ \left.+\boldsymbol{v}_{k}(i, j+1)+\boldsymbol{e}(i, j)\right) / 4+(1-\omega) \boldsymbol{v}_{k}(i, j) . \end{array} | |  |
| 753 | 12.8 | 270 | \boldsymbol{M} \boldsymbol{x}=(\boldsymbol{M}-\boldsymbol{A}) \boldsymbol{x}+\boldsymbol{b} . | |  |
| 754 | 12.9 | 270 | \boldsymbol{M} \boldsymbol{x}_{k+1}=(\boldsymbol{M}-\boldsymbol{A}) \boldsymbol{x}_{k}+\boldsymbol{b} | |  |
| 755 | 12.10 | 270 | \boldsymbol{x}_{k+1}:=\boldsymbol{G} \boldsymbol{x}_{k}+\boldsymbol{c}, \quad \boldsymbol{G}=\boldsymbol{I}-\boldsymbol{M}^{-1} \boldsymbol{A}, \quad, \boldsymbol{c}=\boldsymbol{M}^{-1} b . | |  |
| 756 | | 270 | \boldsymbol{x}=\lim _{k \rightarrow \infty} \boldsymbol{x}_{k+1}=\lim _{k \rightarrow \infty}\left(\boldsymbol{G} \boldsymbol{x}_{k}+\boldsymbol{c}\right)=\boldsymbol{G} \lim _{k \rightarrow \infty} \boldsymbol{x}_{k}+\boldsymbol{c}=\boldsymbol{G} \boldsymbol{x}+\boldsymbol{c} . | |  |
| 757 | 12.11 | 270 | \boldsymbol{A}_{\boldsymbol{L}}:=\left[\begin{array}{ccc} 0 & & \\ -a_{2,1} & 0 & \\ \vdots & \ddots & \ddots \\ -a_{n, 1} & \cdots & -a_{n, n-1} \end{array}\right], \quad \boldsymbol{A}_{\boldsymbol{R}}:=\left[\begin{array}{ccc} 0 & -a_{1,2} & \cdots \\ \ddots & \ddots & \vdots \\ & 0 & -a_{n-1, n} \\ & & 0 \end{array}\right] . | |  |
| 758 | 12.12 | 271 | \boldsymbol{M}_{J}=\boldsymbol{D}, \quad \boldsymbol{M}_{1}=\boldsymbol{D}-\boldsymbol{A}_{L}, \quad \boldsymbol{M}_{\omega}=\omega^{-1} \boldsymbol{D}-\boldsymbol{A}_{L} . | |  |
| 759 | 12.13 | 271 | \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} . | |  |
| 760 | | 271 | \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} 1 \\ 1 \end{array}\right] | |  |
| 761 | | 271 | \boldsymbol{A}_{L}=\left[\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right], \quad \boldsymbol{D}=\left[\begin{array}{ll} 2 & 0 \\ 0 & 2 \end{array}\right], \quad \boldsymbol{A}_{R}=\left[\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right], | |  |
| 762 | | 271 | \boldsymbol{M}_{J}=\boldsymbol{D}=\left[\begin{array}{ll} 2 & 0 \\ 0 & 2 \end{array}\right], \quad \boldsymbol{M}_{\omega}=\omega^{-1} \boldsymbol{D}-\boldsymbol{A}_{L}=\left[\begin{array}{cc} 2 \omega^{-1} & 0 \\ -1 & 2 \omega^{-1} \end{array}\right] . | |  |
| 763 | 12.14 | 271 | \boldsymbol{G}_{\omega}=\left[\begin{array}{ll} l & 0 \\ 0 & 1 \end{array}\right]-\left[\begin{array}{cc} \omega / 2 & 0 \\ \omega^{2} / 4 & \omega / 2 \end{array}\right]\left[\begin{array}{cc} 2 & -1 \\ -1 & 2 \end{array}\right]=\left[\begin{array}{cc} 1-\omega & \omega / 2 \\ \omega(1-\omega) / 2 & 1-\omega+\omega^{2} / 4 \end{array}\right] . | |  |
| 764 | 12.15 | 271 | \boldsymbol{G}_{J}=\boldsymbol{I}-\boldsymbol{D}^{-1} \boldsymbol{A}=\left[\begin{array}{cc} 0 & 1 / 2 \\ 1 / 2 & 0 \end{array}\right], \quad \boldsymbol{G}_{1}=\left[\begin{array}{ll} 0 & 1 / 2 \\ 0 & 1 / 4 \end{array}\right] . | |  |
| 765 | | 272 | x_{k+1}(1)=\frac{1}{2} x_{k}(2)+\frac{1}{2}, \quad x_{k+1}(2)=\frac{1}{2} x_{k+1}(1)+\frac{1}{2} . | |  |
| 766 | | 272 | x_{k+1}(2)=\frac{1}{2}\left(\frac{1}{2} x_{k}(2)+\frac{1}{2}\right)+\frac{1}{2}=\frac{1}{4} x_{k}(2)+\frac{3}{4} . | |  |
| 767 | | 272 | \boldsymbol{x}_{k+1}=\left[\begin{array}{l} \boldsymbol{x}_{k+1}(1) \\ \boldsymbol{x}_{k+1}(2) \end{array}\right]=\left[\begin{array}{ll} 0 & 1 / 2 \\ 0 & 1 / 4 \end{array}\right]\left[\begin{array}{l} \boldsymbol{x}_{k}(1) \\ \boldsymbol{x}_{k}(2) \end{array}\right]+\left[\begin{array}{l} 1 / 2 \\ 3 / 4 \end{array}\right]=\boldsymbol{G}_{1} \boldsymbol{x}_{k}+\boldsymbol{c} . | |  |
| 768 | 12.16 | 272 | \boldsymbol{x}_{k}-\boldsymbol{x}=\boldsymbol{G}^{k}\left(\boldsymbol{x}_{0}-\boldsymbol{x}\right), \quad k=0,1,2, \ldots | |  |
| 769 | | 273 | \left\|\boldsymbol{x}_{k}-\boldsymbol{x}\right\|=\left\|\boldsymbol{G}^{k}\left(\boldsymbol{x}_{0}-\boldsymbol{x}\right)\right\| \leq\left\|\boldsymbol{G}^{k}\right\|\left\|\boldsymbol{x}_{0}-\boldsymbol{x}\right\| \leq\|\boldsymbol{G}\|^{k}\left\|\boldsymbol{x}_{0}-\boldsymbol{x}\right\| \rightarrow \mathbf{0}, \quad k \rightarrow \infty . | |  |
| 770 | 12.17 | 273 | \rho(\boldsymbol{A}):=\max _{\lambda \in \sigma(\boldsymbol{A})}|\lambda| . | |  |
| 771 | 12.18 | 273 | \boldsymbol{x}_{k+1}=\boldsymbol{x}_{k}+\alpha\left(\boldsymbol{b}-\boldsymbol{A} \boldsymbol{x}_{k}\right) . | |  |
| 772 | 12.19 | 273 | \begin{gathered} \rho(\boldsymbol{I}-\alpha \boldsymbol{A})>\rho\left(\boldsymbol{I}-\alpha_{o} \boldsymbol{A}\right)=\frac{\kappa-1}{\kappa+1}, \quad \alpha \in \mathbb{R} \backslash\left\{\alpha_{o}\right\}, \\ \kappa:=\frac{\lambda_{1}}{\lambda_{n}}, \quad \alpha_{o}:=\frac{2}{\lambda_{1}+\lambda_{n}} . \end{gathered} | |  |
| 773 | | 274 | \rho_{\alpha}:=\rho(\boldsymbol{I}-\alpha \boldsymbol{A}):=\max _{j}\left|1-\alpha \lambda_{j}\right|= \begin{cases}1-\alpha \lambda_{1}, & \text { if } \alpha \leq 0 \\ 1-\alpha \lambda_{n}, & \text { if } 0<\alpha \leq \alpha_{o} \\ \alpha \lambda_{1}-1, & \text { if } \alpha>\alpha_{o},\end{cases} | |  |
| 774 | | 274 | \rho_{\alpha_{o}}=\alpha_{o} \lambda_{1}-1=\frac{\lambda_{1}-\lambda_{n}}{\lambda_{1}+\lambda_{n}}=\frac{\kappa-1}{\kappa+1}<1 . | |  |
| 775 | 12.20 | 274 | \left\|\boldsymbol{x}_{k}-\boldsymbol{x}\right\|_{2} \leq\left(\frac{\kappa-1}{\kappa+1}\right)^{k}\left\|\boldsymbol{x}_{0}-\boldsymbol{x}\right\|_{2}, \quad k=0,1,2, \ldots | |  |
| 776 | 12.21 | 275 | \boldsymbol{G}_{\omega}=(\boldsymbol{I}-\omega \boldsymbol{L})^{-1}(\omega \boldsymbol{R}+(1-\omega) \boldsymbol{I}) . | |  |
| 777 | 12.22 | 275 | \omega^{-1}(2-\omega)|1-\lambda|^{2} \boldsymbol{x}^{*} \boldsymbol{D} \boldsymbol{x}=\left(1-|\lambda|^{2}\right) \boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{x}, | |  |
| 778 | | 275 | \boldsymbol{G}_{\omega} \boldsymbol{x}=\left(\boldsymbol{I}-\boldsymbol{M}_{\omega}^{-1} \boldsymbol{A}\right) \boldsymbol{x}=\boldsymbol{x}-\left(\omega^{-1} \boldsymbol{D}-\boldsymbol{A}_{\boldsymbol{L}}\right)^{-1} \boldsymbol{A} \boldsymbol{x} | |  |
| 779 | 12.23 | 276 | \boldsymbol{A} \boldsymbol{x}=\left(\omega^{-1} \boldsymbol{D}-\boldsymbol{A}_{\boldsymbol{L}}\right) \boldsymbol{y}, \quad \boldsymbol{y}:=(1-\lambda) \boldsymbol{x} . | |  |
| 780 | 12.24 | 276 | \boldsymbol{E} \boldsymbol{y}=\lambda \boldsymbol{A} \boldsymbol{x}, \quad \boldsymbol{E}:=\omega^{-1} \boldsymbol{D}+\boldsymbol{A}_{\boldsymbol{R}}-\boldsymbol{D}=\omega^{-1} \boldsymbol{D}-\boldsymbol{A}_{\boldsymbol{L}}-\boldsymbol{A} . | |  |
| 781 | | 276 | \boldsymbol{E} \boldsymbol{y}=\left(\omega^{-1} \boldsymbol{D}-\boldsymbol{A}_{\boldsymbol{L}}-\boldsymbol{A}\right) \boldsymbol{y}=\boldsymbol{A} \boldsymbol{x}-\boldsymbol{A} \boldsymbol{y}=\boldsymbol{A} \boldsymbol{x}-(1-\lambda) \boldsymbol{A} \boldsymbol{x}=\lambda \boldsymbol{A} \boldsymbol{x} . | |  |
| 782 | | 276 | \begin{aligned} (\boldsymbol{A} \boldsymbol{x})^{*} \boldsymbol{y}+\boldsymbol{y}^{*}(\lambda \boldsymbol{A} \boldsymbol{x}) & =\boldsymbol{y}^{*}\left(\omega^{-1} \boldsymbol{D}-\boldsymbol{A}_{\boldsymbol{R}}\right) \boldsymbol{y}+\boldsymbol{y}^{*}\left(\omega^{-1} \boldsymbol{D}+\boldsymbol{A}_{\boldsymbol{R}}-\boldsymbol{D}\right) \boldsymbol{y} \\ & =\boldsymbol{y}^{*}\left(2 \omega^{-1}-1\right) \boldsymbol{D} \boldsymbol{y}=\omega^{-1}(2-\omega)|1-\lambda|^{2} \boldsymbol{x}^{*} \boldsymbol{D} \boldsymbol{x} . \end{aligned} | |  |
| 783 | | 276 | (\boldsymbol{A} \boldsymbol{x})^{*} \boldsymbol{y}+\boldsymbol{y}^{*}(\lambda \boldsymbol{A} \boldsymbol{x})=(1-\lambda) \boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{x}+\lambda(1-\bar{\lambda}) \boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{x}=\left(1-|\lambda|^{2}\right) \boldsymbol{x}^{*} \boldsymbol{A} \boldsymbol{x}, | |  |
| 784 | | 276 | \lambda_{j, k}=4-2 \cos (j \pi h)-2 \cos (k \pi h), \quad j, k=1, \ldots, m, h=1 /(m+1) . | |  |
| 785 | 12.25 | 276 | \mu_{j, k}=1-\frac{1}{4} \lambda_{j, k}=\frac{1}{2} \cos (j \pi h)+\frac{1}{2} \cos (k \pi h), \quad j, k=1, \ldots, m . | |  |
| 786 | 12.26 | 277 | \left\|\boldsymbol{x}_{k}-\boldsymbol{x}\right\|_{2} \leq\left\|\boldsymbol{G}_{J}\right\|_{2}^{k}\left\|\boldsymbol{x}_{0}-\boldsymbol{x}\right\|_{2}=\cos ^{k}(\pi h)\left\|\boldsymbol{x}_{0}-\boldsymbol{x}\right\|_{2}, \quad k=0,1,2, \ldots | |  |
| 787 | 12.27 | 277 | \rho\left(\boldsymbol{G}_{\omega}\right)= \begin{cases}\frac{1}{4}\left(\omega \beta+\sqrt{(\omega \beta)^{2}-4(\omega-1)}\right)^{2}, & \text { for } 0<\omega \leq \omega^{*}, \\ \omega-1, & \text { for } \omega^{*}<\omega<2,\end{cases} | |  |
| 788 | 12.28 | 277 | \omega^{*}:=\frac{2}{1+\sqrt{1-\beta^{2}}}>1 . | |  |
| 789 | 12.29 | 277 | \rho\left(\boldsymbol{G}_{\omega}\right)>\rho\left(\boldsymbol{G}_{\omega^{*}}\right) \text { for } \omega \in(0,2) \backslash\left\{\omega^{*}\right\} . | |  |
| 790 | 12.30 | 277 | \omega^{*}=\frac{2}{1+\sin (\pi h)}, \quad \rho\left(\boldsymbol{G}_{\omega^{*}}\right)=\omega^{*}-1=\frac{1-\sin (\pi h)}{1+\sin (\pi h)}, \quad h=\frac{1}{m+1} . | |  |
| 791 | | 278 | \left\|\boldsymbol{x}_{k}-\boldsymbol{x}\right\| \leq\left\|\boldsymbol{G}^{k}\right\|\left\|\boldsymbol{x}_{0}-\boldsymbol{x}\right\| \approx \rho(\boldsymbol{G})^{k}\left\|\boldsymbol{x}_{0}-\boldsymbol{x}\right\| . | |  |
| 792 | 12.31 | 278 | \tilde{k}:=\frac{s \log (10)}{\eta} | |  |
| 793 | | 279 | k=-\frac{s \log (10)}{\log (1-\eta)} \approx \frac{s \log (10)}{\eta}=\tilde{k} . | |  |
| 794 | | 279 | \tilde{k}_{n}=\frac{2 \log (10) s}{\pi^{2}} n+O\left(n^{-1}\right)=O(n) . | |  |
| 795 | | 279 | \tilde{k}_{n}=\frac{\log (10) s}{\pi^{2}} n+O\left(n^{-1}\right)=O(n) . | |  |
| 796 | | 279 | \tilde{k}_{n}=\frac{\log (10) s}{2 \pi} \sqrt{n}+O\left(n^{-1 / 2}\right)=O(\sqrt{n}) . | |  |
| 797 | 12.32 | 280 | \left\|\boldsymbol{x}_{k}-\boldsymbol{x}_{k-1}\right\| \geq \frac{1-\|\boldsymbol{G}\|}{\|\boldsymbol{G}\|}\left\|\boldsymbol{x}_{k}-\boldsymbol{x}\right\|, \quad k \geq 1 . | |  |
| 798 | | 280 | \begin{aligned} \left\|\boldsymbol{x}_{k}-\boldsymbol{x}\right\| & =\left\|\boldsymbol{G}\left(\boldsymbol{x}_{k-1}-\boldsymbol{x}\right)\right\| \leq\|\boldsymbol{G}\|\left\|\boldsymbol{x}_{k-1}-\boldsymbol{x}\right\| \\ & =\|\boldsymbol{G}\|\left\|\boldsymbol{x}_{k-1}-\boldsymbol{x}_{k}+\boldsymbol{x}_{k}-\boldsymbol{x}\right\| \leq\|\boldsymbol{G}\|\left(\left\|\boldsymbol{x}_{k-1}-\boldsymbol{x}_{k}\right\|+\left\|\boldsymbol{x}_{k}-\boldsymbol{x}\right\|\right) \end{aligned} | |  |
| 799 | 12.33 | 280 | \boldsymbol{x}_{k+1}=\boldsymbol{x}_{k}+\boldsymbol{M}^{-1} \boldsymbol{r}_{k}, \quad \boldsymbol{r}_{k}=\boldsymbol{b}-\boldsymbol{A} \boldsymbol{x}_{k} . | |  |
| 800 | | 281 | \lim _{k \rightarrow \infty} \boldsymbol{A}^{k}=\mathbf{0} \Longleftrightarrow \rho(\boldsymbol{A})<1, | |  |
| 801 | | 281 | \boldsymbol{D}_{t} \boldsymbol{R} \boldsymbol{D}_{t}^{-1}=\left[\begin{array}{ccc} \lambda_{1} & t^{-1} r_{12} & t^{-2} r_{13} \\ 0 & \lambda_{2} & t^{-1} r_{23} \\ 0 & 0 & \lambda_{3} \end{array}\right] . | |  |
| 802 | | 281 | \begin{aligned} \|\boldsymbol{A}\| & =\left\|\boldsymbol{D}_{t} \boldsymbol{U}^{*} \boldsymbol{A} \boldsymbol{U} \boldsymbol{D}_{t}^{-1}\right\|_{1}=\left\|\boldsymbol{D}_{t} \boldsymbol{R} \boldsymbol{D}_{t}^{-1}\right\|_{1}=\max _{1 \leq j \leq n} \sum_{i=1}^{n}\left|\left(\boldsymbol{D}_{t} \boldsymbol{R} \boldsymbol{D}_{t}^{-1}\right)_{i j}\right| \\ & \leq \max _{1 \leq j \leq n}\left(\left|\lambda_{j}\right|+\epsilon\right)=\rho(\boldsymbol{A})+\epsilon . \end{aligned} | |  |
| 803 | 12.34 | 282 | \lim _{k \rightarrow \infty}\left\|\boldsymbol{A}^{k}\right\|^{1 / k}=\rho(\boldsymbol{A}) . | |  |
| 804 | | 282 | \left\|\boldsymbol{A}^{k}\right\|=\left\|(\rho(\boldsymbol{A})+\epsilon)^{k} \boldsymbol{B}^{k}\right\|=(\rho(\boldsymbol{A})+\epsilon)^{k}\left\|\boldsymbol{B}^{k}\right\|<(\rho(\boldsymbol{A})+\epsilon)^{k} . | |  |
| 805 | | 282 | \sum_{k=0}^{\infty} \boldsymbol{B}^{k} | |  |
| 806 | 12.35 | 282 | \left\|(\boldsymbol{I}-\boldsymbol{B})^{-1}\right\| \leq \frac{1}{1-\|\boldsymbol{B}\|} . | |  |
| 807 | | 283 | \left\|\boldsymbol{S}_{l}-\boldsymbol{S}_{m}\right\|=\left\|\sum_{k=m+1}^{l} \boldsymbol{B}^{k}\right\| \leq \sum_{k=m+1}^{l}\|\boldsymbol{B}\|^{k} \leq\|\boldsymbol{B}\|^{m+1} \sum_{k=0}^{\infty}\|\boldsymbol{B}\|^{k}=\frac{\|\boldsymbol{B}\|^{m+1}}{1-\|\boldsymbol{B}\|} . | |  |
| 808 | 12.36 | 283 | \left(\sum_{k=0}^{m} \boldsymbol{B}^{k}\right)(\boldsymbol{I}-\boldsymbol{B})=\boldsymbol{I}+\boldsymbol{B}+\cdots+\boldsymbol{B}^{m}-\left(\boldsymbol{B}+\cdots+\boldsymbol{B}^{m+1}\right)=\boldsymbol{I}-\boldsymbol{B}^{m+1} . | |  |
| 809 | 12.37 | 283 | \frac{1}{4}\left(v_{i-1, j}+v_{i, j-1}+v_{i+1, j}+v_{i, j+1}\right)=\mu v_{i, j}, \quad i, j=1, \ldots, m, | |  |
| 810 | 12.38 | 283 | (\omega \boldsymbol{R}+\lambda \omega \boldsymbol{L}) \boldsymbol{w}=(\lambda+\omega-1) \boldsymbol{w} . | |  |
| 811 | 12.39 | 284 | \frac{\omega}{4}\left(\lambda w_{i-1, j}+\lambda w_{i, j-1}+w_{i+1, j}+w_{i, j+1}\right)=(\lambda+\omega-1) w_{i, j} | |  |
| 812 | 12.40 | 284 | \mu:=\frac{\lambda+\omega-1}{\omega \lambda^{1 / 2}} | |  |
| 813 | 12.41 | 284 | \mu \omega \lambda^{1 / 2}=\lambda+\omega-1 | |  |
| 814 | | 284 | \frac{\omega}{4}\left(v_{i-1, j}+v_{i, j-1}+v_{i+1, j}+v_{i, j+1}\right)=\lambda^{-1 / 2}(\lambda+\omega-1) v_{i, j} . | |  |
| 815 | | 284 | \boldsymbol{G}_{J} \boldsymbol{v}=(\boldsymbol{L}+\boldsymbol{R}) \boldsymbol{v}=\frac{\lambda+\omega-1}{\omega \lambda^{1 / 2}} \boldsymbol{v}=\mu \boldsymbol{v} . | |  |
| 816 | | 284 | \frac{1}{4}\left(\lambda^{1 / 2} w_{i-1, j}+\lambda^{1 / 2} w_{i, j-1}+\lambda^{-1 / 2} w_{i+1, j}+\lambda^{-1 / 2} w_{i, j+1}\right)=\mu w_{i, j} . | |  |
| 817 | | 284 | \frac{\omega}{4}\left(\lambda w_{i-1, j}+\lambda w_{i, j-1}+w_{i+1, j}+w_{i, j+1}\right)=\omega \mu \lambda^{1 / 2} w_{i, j} | |  |
| 818 | 12.42 | 285 | \lambda(\mu):=\frac{1}{4}\left(\omega \mu \pm \sqrt{(\omega \mu)^{2}-4(\omega-1)}\right)^{2} . | |  |
| 819 | | 285 | d(\omega):=(\omega \mu)^{2}-4(\omega-1) . | |  |
| 820 | | 285 | d^{\prime}(\omega)=2\left(\omega \mu^{2}-2\right)<2(\omega-2)<0 . | |  |
| 821 | 12.43 | 285 | \omega=\tilde{\omega}(\mu):=2 \frac{1-\sqrt{1-\mu^{2}}}{\mu^{2}}=\frac{2}{1+\sqrt{1-\mu^{2}}} . | |  |
| 822 | | 285 | |\lambda(\mu)|=\frac{1}{4}\left((\omega \mu)^{2}+4(\omega-1)-(\omega \mu)^{2}\right)=\omega-1, \quad \tilde{\omega}(\mu)<\omega<2 . | |  |
| 823 | 12.44 | 285 | \lambda(\mu)=\frac{1}{4}\left(\omega \mu+\sqrt{(\omega \mu)^{2}-4(\omega-1)}\right)^{2}, \quad 0<\omega \leq \tilde{\omega}(\mu) . | |  |
| 824 | | 285 | \frac{d}{d \omega}\left(\omega \beta+\sqrt{(\omega \beta)^{2}-4(\omega-1)}\right)=\frac{\beta \sqrt{(\omega \beta)^{2}-4(\omega-1)}+\omega \beta^{2}-2}{\sqrt{(\omega \beta)^{2}-4(\omega-1)}} . | |  |
| 825 | | 286 | \rho\left(\boldsymbol{G}_{J}\right)=\sqrt{\left|a_{21} a_{12} / a_{11} a_{22}\right|} . | |  |
| 826 | | 286 | \rho\left(\boldsymbol{G}_{1}\right)=\left|a_{21} a_{12} / a_{11} a_{22}\right| . | |  |
| 827 | | 287 | \boldsymbol{x}:=\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]=\left[\begin{array}{ll} 0 & a \\ a & 0 \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]+\left[\begin{array}{l} 1-a \\ 1-a \end{array}\right]=: \boldsymbol{G} \boldsymbol{x}+\boldsymbol{c} . | |  |
| 828 | | 287 | \boldsymbol{x}_{k}(1)=\boldsymbol{x}_{k}(2)=1-a^{k}, \quad k \geq 0, | |  |
| 829 | 12.45 | 287 | \boldsymbol{M} \boldsymbol{x}_{k+1}=\boldsymbol{N} \boldsymbol{x}_{k}+\boldsymbol{b}, | |  |
| 830 | 12.46 | 287 | \boldsymbol{A}=\boldsymbol{M}-\boldsymbol{N}, \quad \boldsymbol{M}=(\boldsymbol{I}-\boldsymbol{L})\left(\boldsymbol{I}-\boldsymbol{L}^{T}\right), \quad \boldsymbol{N}=\boldsymbol{L} \boldsymbol{L}^{T} . | |  |
| 831 | 12.47 | 287 | \lambda=\frac{\boldsymbol{x}^{T} \boldsymbol{N} \boldsymbol{x}}{\boldsymbol{x}^{T} \boldsymbol{A x}+\boldsymbol{x}^{T} \boldsymbol{N} \boldsymbol{x}} . | |  |
| 832 | 12.48 | 288 | x(i)=b(i)-\sum_{j \neq i} a(i, j) x(j) | |  |
| 833 | | 288 | A:=\left[\begin{array}{lll} 3 & 0 & 1 \\ 0 & 7 & 2 \\ 1 & 2 & 4 \end{array}\right] | |  |
| 834 | | 289 | \boldsymbol{A}:=\left[\begin{array}{cccccc} \lambda & 0 & \cdots & 0 & 0 \\ 0 & \lambda & a & \cdots & 0 & 0 \\ 0 & 0 & \lambda & \cdots & 0 & 0 \\ \vdots & & & & \vdots \\ 0 & 0 & 0 & \cdots & \lambda & a \\ 0 & 0 & 0 & \cdots & 0 & \lambda \end{array}\right] \in \mathbb{R}^{n \times n} . | |  |
| 835 | 13.1 | 290 | \langle\boldsymbol{x}, \boldsymbol{y}\rangle_{\boldsymbol{A}}:=\boldsymbol{x}^{T} \boldsymbol{A} \boldsymbol{y}, \quad\|\boldsymbol{y}\|_{\boldsymbol{A}}:=\sqrt{\boldsymbol{y}^{T} \boldsymbol{A} \boldsymbol{y}}, \quad \boldsymbol{x}, \boldsymbol{y} \in \mathbb{R}^{n} . | |  |
| 836 | 13.2 | 291 | \left|\boldsymbol{x}^{T} \boldsymbol{A y}\right|^{2} \leq\left(\boldsymbol{x}^{T} \boldsymbol{A x}\right)\left(\boldsymbol{y}^{T} \boldsymbol{A y}\right), \quad \boldsymbol{x}, \boldsymbol{y} \in \mathbb{R}^{n} . | |  |
| 837 | 13.3 | 291 | Q(\boldsymbol{y}):=\frac{1}{2} \boldsymbol{y}^{T} \boldsymbol{A} \boldsymbol{y}-\boldsymbol{b}^{T} \boldsymbol{y}+c . | |  |
| 838 | 13.4 | 291 | Q(x, y):=\frac{1}{2}\left[\begin{array}{ll} x & y \end{array}\right]\left[\begin{array}{cc} 2 & -1 \\ -1 & 2 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=x^{2}-x y+y^{2} | |  |
| 839 | 13.5 | 291 | Q(\boldsymbol{y}+\varepsilon \boldsymbol{h})=Q(\boldsymbol{y})-\varepsilon \boldsymbol{h}^{T} r(\boldsymbol{y})+\frac{1}{2} \varepsilon^{2} \boldsymbol{h}^{T} \boldsymbol{A} \boldsymbol{h}, \text { where } r(\boldsymbol{y}):=\boldsymbol{b}-\boldsymbol{A} \boldsymbol{y} . | |  |
| 840 | 13.6 | 292 | \begin{aligned} \frac{\partial}{\partial y_{i}} Q(\boldsymbol{y}) & :=\lim _{\varepsilon \rightarrow 0} \frac{1}{\varepsilon}\left(Q\left(\boldsymbol{y}+\varepsilon \boldsymbol{e}_{i}\right)-Q(\boldsymbol{y})\right) \\ & \left.=\lim _{\varepsilon \rightarrow 0} \frac{1}{\varepsilon}\left(-\varepsilon \boldsymbol{e}_{i}^{T} \boldsymbol{r}(\boldsymbol{y})\right)+\frac{1}{2} \varepsilon^{2} \boldsymbol{e}_{i}^{T} \boldsymbol{A} \boldsymbol{e}_{i}\right)=-\boldsymbol{e}_{i}^{T} \boldsymbol{r}(\boldsymbol{y}), \quad i=1, \ldots, n, \end{aligned} | |  |
| 841 | | 293 | Q\left(\boldsymbol{x}_{k+1}\right)=Q\left(\boldsymbol{x}_{k}+\alpha_{k} \boldsymbol{p}_{k}\right)=\min _{\alpha \in \mathbb{R}} Q\left(\boldsymbol{x}_{k}+\alpha \boldsymbol{p}_{k}\right) . | |  |
| 842 | 13.7 | 293 | \alpha_{k}:=\frac{\boldsymbol{p}_{k}^{T} \boldsymbol{r}_{k}}{\boldsymbol{p}_{k}^{T} \boldsymbol{A} \boldsymbol{p}_{k}} . | |  |
| 843 | 13.8 | 293 | \boldsymbol{x}_{k+1}=\boldsymbol{x}_{k}+\left(\frac{\boldsymbol{r}_{k}^{T} \boldsymbol{r}_{k}}{\boldsymbol{r}_{k}^{T} \boldsymbol{A} \boldsymbol{r}_{k}}\right) \boldsymbol{r}_{k} . | |  |
| 844 | 13.9 | 293 | \begin{aligned} \boldsymbol{p}_{k} & =\boldsymbol{r}_{k}, \boldsymbol{t}_{k}=\boldsymbol{A} \boldsymbol{p}_{k}, \\ \alpha_{k} & =\left(\boldsymbol{p}_{k}^{T} \boldsymbol{r}_{k}\right) /\left(\boldsymbol{p}_{k}^{T} \boldsymbol{t}_{k}\right), \\ \boldsymbol{x}_{k+1} & =\boldsymbol{x}_{k}+\alpha_{k} \boldsymbol{p}_{k}, \\ \boldsymbol{r}_{k+1} & =\boldsymbol{r}_{k}-\alpha_{k} \boldsymbol{t}_{k} . \end{aligned} | |  |
| 845 | 13.10 | 293 | \boldsymbol{r}_{k+1}=\boldsymbol{b}-\boldsymbol{A} \boldsymbol{x}_{k+1}=\boldsymbol{b}-\boldsymbol{A}\left(\boldsymbol{x}_{k}+\alpha_{k} \boldsymbol{p}_{k}\right)=\boldsymbol{r}_{k}-\alpha_{k} \boldsymbol{A} \boldsymbol{p}_{k} . | |  |
| 846 | | 293 | \begin{aligned} & \boldsymbol{t}_{0}=3\left[\begin{array}{c} 1 \\ -1 / 2 \end{array}\right], \quad \alpha_{0}=\frac{1}{2}, \quad \boldsymbol{x}_{1}=-4^{-1}\left[\begin{array}{l} 1 \\ 2 \end{array}\right], \quad \boldsymbol{r}_{1}=3 * 4^{-1}\left[\begin{array}{l} 0 \\ 1 \end{array}\right] \\ & \boldsymbol{t}_{1}=3 * 4^{-1}\left[\begin{array}{c} -1 \\ 2 \end{array}\right], \quad \alpha_{1}=\frac{1}{2}, \quad \boldsymbol{x}_{2}=-4^{-1}\left[\begin{array}{c} 1 \\ 1 / 2 \end{array}\right], \quad \boldsymbol{r}_{2}=3 * 4^{-1}\left[\begin{array}{c} 1 / 2 \\ 0 \end{array}\right], \end{aligned} | |  |
| 847 | 13.11 | 294 | \begin{aligned} & \boldsymbol{t}_{2 k-2}=3 * 4^{1-k}\left[\begin{array}{c} 1 \\ -1 / 2 \end{array}\right], \quad \boldsymbol{x}_{2 k-1}=-4^{-k}\left[\begin{array}{l} 1 \\ 2 \end{array}\right], \quad \boldsymbol{r}_{2 k-1}=3 * 4^{-k}\left[\begin{array}{l} 0 \\ 1 \end{array}\right] \\ & \boldsymbol{t}_{2 k-1}=3 * 4^{-k}\left[\begin{array}{c} -1 \\ 2 \end{array}\right], \quad \boldsymbol{x}_{2 k}=-4^{-k}\left[\begin{array}{c} 1 \\ 1 / 2 \end{array}\right], \quad \boldsymbol{r}_{2 k}=3 * 4^{-k}\left[\begin{array}{c} 1 / 2 \\ 0 \end{array}\right] . \end{aligned} | |  |
| 848 | 13.12 | 294 | \begin{aligned} \boldsymbol{p}_{j} & :=\boldsymbol{r}_{j}-\sum_{i=0}^{j-1}\left(\frac{\boldsymbol{r}_{j}^{T} \boldsymbol{A} \boldsymbol{p}_{i}}{\boldsymbol{p}_{i}^{T} \boldsymbol{A} \boldsymbol{p}_{i}}\right) \boldsymbol{p}_{i}, \\ \boldsymbol{x}_{j+1} & :=\boldsymbol{x}_{j}+\alpha_{j} \boldsymbol{p}_{j} \quad \alpha_{j}:=\frac{\boldsymbol{r}_{j}^{T} \boldsymbol{r}_{j}}{\boldsymbol{p}_{j}^{T} \boldsymbol{A} \boldsymbol{p}_{j}}, \\ \boldsymbol{r}_{j+1} & =\boldsymbol{r}_{j}-\alpha_{j} \boldsymbol{A} \boldsymbol{p}_{j} . \end{aligned} | |  |
| 849 | | 295 | \begin{aligned} \boldsymbol{r}_{k+1}^{T} \boldsymbol{r}_{j} & \stackrel{(13.13)}{=}\left(\boldsymbol{r}_{k}-\alpha_{k} \boldsymbol{A} \boldsymbol{p}_{k}\right)^{T} \boldsymbol{r}_{j} \\ & \stackrel{(13.11)}{=} \boldsymbol{r}_{k}^{T} \boldsymbol{r}_{j}-\alpha_{k} \boldsymbol{p}_{k}^{T} \boldsymbol{A}\left(\boldsymbol{p}_{j}+\sum_{i=0}^{j-1}\left(\frac{\boldsymbol{r}_{j}^{T} \boldsymbol{A} \boldsymbol{p}_{i}}{\boldsymbol{p}_{i}^{T} \boldsymbol{A} \boldsymbol{p}_{i}}\right) \boldsymbol{p}_{i}\right) \\ & \stackrel{\boldsymbol{p}_{k}^{T} \boldsymbol{A} \boldsymbol{p}_{i}=0}{=} \boldsymbol{r}_{k}^{T} \boldsymbol{r}_{j}-\alpha_{k} \boldsymbol{p}_{k}^{T} \boldsymbol{A} \boldsymbol{p}_{j}=0, \quad j=0,1, \ldots, k . \end{aligned} | |  |
| 850 | 13.16 | 295 | \boldsymbol{r}_{j}^{T} \boldsymbol{A} \boldsymbol{p}_{i} \stackrel{(13.13)}{=} \boldsymbol{r}_{j}^{T}\left(\frac{\boldsymbol{r}_{i}-\boldsymbol{r}_{i+1}}{\alpha_{i}}\right)=0, \quad i=0,1, \ldots, j-2 . | |  |
| 851 | 13.14 | 295 | \beta_{k}:=-\frac{\boldsymbol{r}_{k+1}^{T} \boldsymbol{A} \boldsymbol{p}_{k}}{\boldsymbol{p}_{k}^{T} \boldsymbol{A} \boldsymbol{p}_{k}} \stackrel{\text { (13.13) }}{=} \frac{\boldsymbol{r}_{k+1}^{T}\left(\boldsymbol{r}_{k+1}-\boldsymbol{r}_{k}\right)}{\alpha_{k} \boldsymbol{p}_{k}^{T} \boldsymbol{A} \boldsymbol{p}_{k}} \stackrel{\text { (13.12) }}{=} \frac{\boldsymbol{r}_{k+1}^{T} \boldsymbol{r}_{k+1}}{\boldsymbol{r}_{k}^{T} \boldsymbol{r}_{k}} . | |  |
| 852 | 13.15 | 295 | \begin{aligned} & \text { For } k=0,1,2, \ldots \\ & \qquad \boldsymbol{x}_{k+1}:=\boldsymbol{x}_{k}+\alpha_{k} \boldsymbol{p}_{k}, \quad \alpha_{k}:=\frac{\boldsymbol{r}_{k}^{T} \boldsymbol{r}_{k}}{\boldsymbol{p}_{k}^{T} \boldsymbol{A} \boldsymbol{p}_{k}}, \\ & \boldsymbol{r}_{k+1}:=\boldsymbol{r}_{k}-\alpha_{k} \boldsymbol{A} \boldsymbol{p}_{k}, \end{aligned} | |  |
| 853 | 13.17 | 296 | \boldsymbol{p}_{k+1}:=\boldsymbol{r}_{k+1}+\beta_{k} \boldsymbol{p}_{k}, \quad \beta_{k}:=\frac{\boldsymbol{r}_{k+1}^{T} \boldsymbol{r}_{k+1}}{\boldsymbol{r}_{k}^{T} \boldsymbol{r}_{k}} . | |  |
| 854 | 13.18 | 296 | \begin{aligned} \boldsymbol{t}_{k} & =\boldsymbol{A} \boldsymbol{p}_{k}, \\ \alpha_{k} & =\left(\boldsymbol{r}_{k}^{T} \boldsymbol{r}_{k}\right) /\left(\boldsymbol{p}_{k}^{T} \boldsymbol{t}_{k}\right), \\ \boldsymbol{x}_{k+1} & =\boldsymbol{x}_{k}+\alpha_{k} \boldsymbol{p}_{k}, \\ \boldsymbol{r}_{k+1} & =\boldsymbol{r}_{k}-\alpha_{k} \boldsymbol{t}_{k}, \\ \beta_{k} & =\left(\boldsymbol{r}_{k+1}^{T} \boldsymbol{r}_{k+1}\right) /\left(\boldsymbol{r}_{k}^{T} \boldsymbol{r}_{k}\right), \\ \boldsymbol{p}_{k+1} & :=\boldsymbol{r}_{k+1}+\beta_{k} \boldsymbol{p}_{k} . \end{aligned} | |  |
| 855 | | 296 | \begin{array}{ll} \boldsymbol{t}_{0}=\left[\begin{array}{c} 3 \\ -3 / 2 \end{array}\right], \quad \alpha_{0}=1 / 2, \quad \boldsymbol{x}_{1}=\left[\begin{array}{l} -1 / 4 \\ -1 / 2 \end{array}\right], \quad \boldsymbol{r}_{1}=\left[\begin{array}{c} 0 \\ 3 / 4 \end{array}\right], \quad \beta_{0}=1 / 4, \\ \boldsymbol{p}_{1}=\left[\begin{array}{l} 3 / 8 \\ 3 / 4 \end{array}\right], \quad \boldsymbol{t}_{1}=\left[\begin{array}{c} 0 \\ 9 / 8 \end{array}\right], \quad \alpha_{1}=2 / 3, \quad \boldsymbol{x}_{2}=\mathbf{0}, \quad \boldsymbol{r}_{2}=\mathbf{0} . \end{array} | |  |
| 856 | | 298 | \boldsymbol{A}=\operatorname{tridiag}_{m}(a, d, a) \otimes \boldsymbol{I}_{m}+\boldsymbol{I}_{m} \otimes \operatorname{tridiag}_{m}(a, d, a) \in \mathbb{R}^{\left(m^{2}\right) \times\left(m^{2}\right)} | |  |
| 857 | 13.19 | 299 | \frac{\left\|\boldsymbol{x}-\boldsymbol{x}_{k}\right\|_{\boldsymbol{A}}}{\left\|\boldsymbol{x}-\boldsymbol{x}_{0}\right\|_{\boldsymbol{A}}} \leq\left(\frac{\kappa-1}{\kappa+1}\right)^{k}<e^{-\frac{2}{\kappa} k}, \quad, k>0, | |  |
| 858 | 13.20 | 299 | \frac{\left\|\boldsymbol{x}-\boldsymbol{x}_{k}\right\|_{\boldsymbol{A}}}{\left\|\boldsymbol{x}-\boldsymbol{x}_{0}\right\|_{\boldsymbol{A}}} \leq 2\left(\frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1}\right)^{k}<2 e^{-\frac{2}{\sqrt{\kappa}} k}, \quad k \geq 0 . | |  |
| 859 | | 300 | \boldsymbol{T}_{2}:=\operatorname{tridiag}_{m}(a, d, a) \otimes \boldsymbol{I}_{m}+\boldsymbol{I}_{m} \otimes \operatorname{tridiag}_{m}(a, d, a) \in \mathbb{R}^{\left(m^{2}\right) \times\left(m^{2}\right)} . | |  |
| 860 | 13.21 | 300 | \lambda_{j, k}=2 d+2 a \cos (j \pi h)+2 a \cos (k \pi h), \quad j, k=1, \ldots, m . | |  |
| 861 | | 300 | \kappa_{A}=\frac{5+4 \cos (\pi h)}{5-4 \cos (\pi h)} \leq 9, | |  |
| 862 | | 300 | \kappa_{P}=\frac{\lambda_{\max }}{\lambda_{\min }}=\frac{\cos ^{2}(\pi h / 2)}{\sin ^{2}(\pi h / 2)} \text { and } \sqrt{\kappa_{P}}=\frac{\cos (\pi h / 2)}{\sin (\pi h / 2)} \approx \frac{2}{\pi h} \approx \frac{2}{\pi} \sqrt{n} \text {. } | |  |
| 863 | | 300 | \mathbb{W}_{k}=\operatorname{span}\left(\boldsymbol{r}_{0}, \boldsymbol{A} \boldsymbol{r}_{0}, \boldsymbol{A}^{2} \boldsymbol{r}_{0}, \ldots, \boldsymbol{A}^{k-1} \boldsymbol{r}_{0}\right), \quad k=1,2,3, \cdots . | |  |
| 864 | | 301 | \mathbb{W}_{0} \subset \mathbb{W}_{1} \subset \mathbb{W}_{2} \subset \cdots \subset \mathbb{W}_{n} \subset \mathbb{R}^{n} | |  |
| 865 | 13.22 | 301 | \boldsymbol{x}_{k}-\boldsymbol{x}_{0} \in \mathbb{W}_{k}, \quad \boldsymbol{r}_{k}, \boldsymbol{p}_{k} \in \mathbb{W}_{k+1}, \quad k=0,1, \ldots, | |  |
| 866 | 13.23 | 301 | \boldsymbol{r}_{k}^{T} \boldsymbol{w}=\boldsymbol{p}_{k}^{T} \boldsymbol{A} \boldsymbol{w}=0, \quad \boldsymbol{w} \in \mathbb{W}_{k} . | |  |
| 867 | 13.24 | 301 | \left\|\boldsymbol{x}-\boldsymbol{x}_{k}\right\|_{\boldsymbol{A}}=\min _{\boldsymbol{w} \in \mathbb{W}_{k}}\left\|\boldsymbol{x}-\boldsymbol{x}_{0}-\boldsymbol{w}\right\|_{\boldsymbol{A}} . | |  |
| 868 | | 301 | \left\|\boldsymbol{x}-\boldsymbol{x}_{0}-\boldsymbol{w}\right\|_{\boldsymbol{A}}=\left\|\boldsymbol{x}-\boldsymbol{x}_{k}+\boldsymbol{u}\right\|_{\boldsymbol{A}} \geq\left\|\boldsymbol{x}-\boldsymbol{x}_{k}\right\|_{\boldsymbol{A}}, | |  |
| 869 | 13.25 | 302 | \left\|\boldsymbol{x}-\boldsymbol{x}_{0}-\boldsymbol{w}\right\|_{\boldsymbol{A}}^{2}=\sum_{j=1}^{n} \frac{\sigma_{j}^{2}}{\lambda_{j}} Q\left(\lambda_{j}\right)^{2}, \quad Q(t):=1-t P(t) . | |  |
| 870 | 13.26 | 302 | \left\|\boldsymbol{x}-\boldsymbol{x}_{0}-P(\boldsymbol{A}) \boldsymbol{r}_{0}\right\|_{\boldsymbol{A}}^{2}=\boldsymbol{c}^{T} \boldsymbol{A}^{-1} \boldsymbol{c} \text { where } \boldsymbol{c}=(\boldsymbol{I}-\boldsymbol{A} P(\boldsymbol{A})) \boldsymbol{r}_{0}=Q(\boldsymbol{A}) \boldsymbol{r}_{0} . | |  |
| 871 | 13.27 | 302 | \boldsymbol{c}=\sum_{j=1}^{n} \sigma_{j} Q\left(\lambda_{j}\right) \boldsymbol{u}_{j}, \quad \boldsymbol{A}^{-1} \boldsymbol{c}=\sum_{i=1}^{n} \sigma_{i} \frac{Q\left(\lambda_{i}\right)}{\lambda_{i}} \boldsymbol{u}_{i} . | |  |
| 872 | 13.28 | 303 | \frac{\left\|\boldsymbol{x}-\boldsymbol{x}_{k}\right\|_{\boldsymbol{A}}}{\left\|\boldsymbol{x}-\boldsymbol{x}_{0}\right\|_{\boldsymbol{A}}} \leq \min _{\substack{Q \in \Pi_{k} \\ Q(0)=1}} \max _{a \leq x \leq b}|Q(x)|, | |  |
| 873 | | 303 | \left\|\boldsymbol{x}-\boldsymbol{x}_{k}\right\|_{\boldsymbol{A}}^{2} \leq\left\|\boldsymbol{x}-\boldsymbol{x}_{0}-\boldsymbol{w}\right\|_{\boldsymbol{A}}^{2} \leq \max _{a \leq x \leq b}|Q(x)|^{2} \sum_{j=1}^{n} \frac{\sigma_{j}^{2}}{\lambda_{j}}=\max _{a \leq x \leq b}|Q(x)|^{2}\left\|\boldsymbol{x}-\boldsymbol{x}_{0}\right\|_{\boldsymbol{A}}^{2}, | |  |
| 874 | 13.29 | 303 | \frac{\left\|\boldsymbol{x}-\boldsymbol{x}_{k}\right\|_{\boldsymbol{A}}}{\left\|\boldsymbol{x}-\boldsymbol{x}_{0}\right\|_{\boldsymbol{A}}} \leq \min _{\substack{Q \in \Pi_{k} \\ Q(0)=1}} \max _{\lambda_{\min } \leq x \leq \lambda_{\max }}|Q(x)|=\frac{2}{a^{-k}+a^{k}}, \quad a:=\frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1}, | |  |
| 875 | 13.30 | 303 | \frac{x-1}{x+1}<e^{-2 / x} \quad \text { for } \quad x>1 | |  |
| 876 | | 303 | e^{2 / x}=e^{2 y}=\sum_{k=0}^{\infty} \frac{(2 y)^{k}}{k!}<-1+2 \sum_{k=0}^{\infty} y^{k}=\frac{1+y}{1-y}=\frac{x+1}{x-1} | |  |
| 877 | | 304 | \|f\|_{\infty}=\max _{a \leq x \leq b}|f(x)| . | |  |
| 878 | | 304 | \left\|Q^{*}\right\|_{\infty}=\min _{Q \in \mathcal{S}_{k}}\|Q\|_{\infty} . | |  |
| 879 | | 304 | T_{n+1}(t)=2 t T_{n}(t)-T_{n-1}(t), \quad n \geq 1, \quad t \in \mathbb{R}, | |  |
| 880 | | 304 | P_{n+1}(t)+P_{n-1}(t)=\cos (n+1) \phi+\cos (n-1) \phi=2 \cos \phi \cos n \phi=2 t P_{n}(t), | |  |
| 881 | 13.31 | 304 | x_{n+1}-2 t x_{n}+x_{n-1}=0 \text { for } n \geq 1, \text { with } x_{0}=1, x_{1}=t . | |  |
| 882 | | 305 | z_{1}=t+\sqrt{t^{2}-1}, \quad z_{2}=t-\sqrt{t^{2}-1}=\left(t+\sqrt{t^{2}-1}\right)^{-1} . | |  |
| 883 | 13.32 | 305 | Q^{*}(x)=\frac{T_{k}(u(x))}{T_{k}(u(c))}, \quad u(x)=\frac{b+a-2 x}{b-a} . | |  |
| 884 | | 305 | \sigma_{j}=f\left(\mu_{j-1}\right) f\left(\mu_{j}\right) . | |  |
| 885 | | 305 | Q\left(\mu_{j}\right) \leq\|Q\|_{\infty} \leq\left\|Q^{*}\right\|_{\infty}=Q^{*}\left(\mu_{j}\right) | |  |
| 886 | | 305 | -Q\left(\mu_{j-1}\right) \leq\|Q\|_{\infty} \leq\left\|Q^{*}\right\|_{\infty}=-Q^{*}\left(\mu_{j-1}\right) . | |  |
| 887 | 13.33 | 306 | Q^{*}(x)=T_{k}\left(\frac{b+a-2 x}{b-a}\right) / T_{k}\left(\frac{b+a}{b-a}\right) . | |  |
| 888 | 13.34 | 306 | \max _{a \leq x \leq b}\left|T_{k}\left(\frac{b+a-2 x}{b-a}\right)\right|=\max _{-1 \leq t \leq 1}\left|T_{k}(t)\right|=1 . | |  |
| 889 | | 306 | t+\sqrt{t^{2}-1}=\frac{\sqrt{\kappa}+1}{\sqrt{\kappa}-1}, \quad \kappa=b / a . | |  |
| 890 | 13.35 | 306 | T_{k}\left(\frac{b+a}{b-a}\right)=T_{k}\left(\frac{\kappa+1}{\kappa-1}\right)=\frac{1}{2}\left[\left(\frac{\sqrt{\kappa}+1}{\sqrt{\kappa}-1}\right)^{k}+\left(\frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1}\right)^{k}\right] | |  |
| 891 | 13.36 | 307 | 1 \leq \frac{\left(\boldsymbol{y}^{T} \boldsymbol{A y}\right)\left(\boldsymbol{y}^{T} \boldsymbol{A}^{-1} \boldsymbol{y}\right)}{\left(\boldsymbol{y}^{T} \boldsymbol{y}\right)^{2}} \leq \frac{(M+m)^{2}}{4 M m} \quad \boldsymbol{y} \neq \mathbf{0}, \boldsymbol{y} \in \mathbb{R}^{n}, | |  |
| 892 | 13.37 | 307 | a:=\frac{\boldsymbol{y}^{T} \boldsymbol{A y}}{\boldsymbol{y}^{T} \boldsymbol{y}}=\sum_{i=1}^{n} t_{i} \lambda_{i}, \quad b:=\frac{\boldsymbol{y}^{T} \boldsymbol{A}^{-1} \boldsymbol{y}}{\boldsymbol{y}^{T} \boldsymbol{y}}=\sum_{i=1}^{n} \frac{t_{i}}{\lambda_{i}}, | |  |
| 893 | 13.38 | 307 | t_{i}=\frac{c_{i}^{2}}{\sum_{j=1}^{n} c_{j}^{2}} \geq 0, \quad i=1, \ldots, n \text { and } \sum_{i=1}^{n} t_{i}=1 . | |  |
| 894 | | 307 | \sqrt{a b}=\sqrt{(a c)(b / c)} \leq(a c+b / c) / 2=\frac{1}{2} \sum_{i=1}^{n} t_{i}\left(\lambda_{i} c+1 /\left(\lambda_{i} c\right)\right)=\frac{1}{2} \sum_{i=1}^{n} t_{i} f\left(\lambda_{i} c\right), | |  |
| 895 | | 307 | \sqrt{a b} \leq \frac{1}{2} \max _{m c \leq x \leq M c} f(x) . | |  |
| 896 | 13.39 | 307 | \sqrt{a b} \leq \frac{1}{2} \max \{f(m c), f(M c)\} . | |  |
| 897 | | 308 | \frac{\left(\boldsymbol{y}^{T} \boldsymbol{A y}\right)\left(\boldsymbol{y}^{T} \boldsymbol{A}^{-1} \boldsymbol{y}\right)}{\left(\boldsymbol{y}^{T} \boldsymbol{y}\right)^{2}}=a b \leq \frac{(M+m)^{2}}{4 M m}, | |  |
| 898 | | 308 | 1=\left(\sum_{i=1}^{n} t_{i}\right)^{2}=\left(\sum_{i=1}^{n}\left(t_{i} \lambda_{i}\right)^{1 / 2}\left(t_{i} / \lambda_{i}\right)^{1 / 2}\right)^{2} \leq\left(\sum_{i=1}^{n} t_{i} \lambda_{i}\right)\left(\sum_{i=1}^{n} t_{i} / \lambda_{i}\right)=a b . | |  |
| 899 | 13.40 | 308 | \frac{\left\|\boldsymbol{\epsilon}_{k+1}\right\|_{\boldsymbol{A}}^{2}}{\left\|\boldsymbol{\epsilon}_{k}\right\|_{\boldsymbol{A}}^{2}} \leq\left(\frac{\kappa-1}{\kappa+1}\right)^{2}, \quad k=0,1,2, \ldots, | |  |
| 900 | | 308 | \boldsymbol{\epsilon}_{k+1}=\boldsymbol{\epsilon}_{k}-\alpha_{k} \boldsymbol{r}_{k}, \quad \alpha_{k}:=\frac{\boldsymbol{r}_{k}^{T} \boldsymbol{r}_{k}}{\boldsymbol{r}_{k}^{T} \boldsymbol{A} \boldsymbol{r}_{k}} . | |  |
| 901 | | 308 | \begin{aligned} \left\|\boldsymbol{\epsilon}_{k}\right\|_{\boldsymbol{A}}^{2} & =\boldsymbol{\epsilon}_{k}^{T} \boldsymbol{A} \boldsymbol{\epsilon}_{k}=\boldsymbol{r}_{k}^{T} \boldsymbol{A}^{-1} \boldsymbol{r}_{k}, \\ \left\|\boldsymbol{\epsilon}_{k+1}\right\|_{\boldsymbol{A}}^{2} & =\left(\boldsymbol{\epsilon}_{k}-\alpha_{k} \boldsymbol{r}_{k}\right)^{T} \boldsymbol{A}\left(\boldsymbol{\epsilon}_{k}-\alpha_{k} \boldsymbol{r}_{k}\right) \\ & =\boldsymbol{\epsilon}_{k}^{T} \boldsymbol{A} \boldsymbol{\epsilon}_{k}-2 \alpha_{k} \boldsymbol{r}_{k}^{T} \boldsymbol{A} \boldsymbol{\epsilon}_{k}+\alpha_{k}^{2} \boldsymbol{r}_{k}^{T} \boldsymbol{A} \boldsymbol{r}_{k}=\left\|\boldsymbol{\epsilon}_{k}\right\|_{\boldsymbol{A}}^{2}-\frac{\left(\boldsymbol{r}_{k}^{T} \boldsymbol{r}_{k}\right)^{2}}{\boldsymbol{r}_{k}^{T} \boldsymbol{A} \boldsymbol{r}_{k}} \end{aligned} | |  |
| 902 | | 308 | \frac{\left\|\boldsymbol{\epsilon}_{k+1}\right\|_{\boldsymbol{A}}^{2}}{\left\|\boldsymbol{\epsilon}_{k}\right\|_{\boldsymbol{A}}^{2}}=1-\frac{\left(\boldsymbol{r}_{k}^{T} \boldsymbol{r}_{k}\right)^{2}}{\left(\boldsymbol{r}_{k}^{T} \boldsymbol{A} \boldsymbol{r}_{k}\right)\left(\boldsymbol{r}_{k}^{T} \boldsymbol{A}^{-1} \boldsymbol{r}_{k}\right)} \leq 1-\frac{4 \lambda_{\min } \lambda_{\max }}{\left(\lambda_{\min }+\lambda_{\max }\right)^{2}}=\left(\frac{\kappa-1}{\kappa+1}\right)^{2} | |  |
| 903 | | 309 | \left\|\boldsymbol{\epsilon}_{k}\right\|_{2}^{2}-\left\|\boldsymbol{\epsilon}_{k+1}\right\|_{2}^{2}=\frac{\left\|\boldsymbol{p}_{k}\right\|_{2}^{2}}{\left\|\boldsymbol{p}_{k}\right\|_{\boldsymbol{A}}^{2}}\left(\left\|\boldsymbol{\epsilon}_{k}\right\|_{\boldsymbol{A}}^{2}+\left\|\boldsymbol{\epsilon}_{k+1}\right\|_{\boldsymbol{A}}^{2}\right) | |  |
| 904 | | 309 | \boldsymbol{\epsilon}_{j}=\boldsymbol{x}_{m+1}-\boldsymbol{x}_{j}=\boldsymbol{x}_{m}-\boldsymbol{x}_{j}+\alpha_{m} \boldsymbol{p}_{m}=\boldsymbol{x}_{m-1}-\boldsymbol{x}_{j}+\alpha_{m-1} \boldsymbol{p}_{m-1}+\alpha_{m} \boldsymbol{p}_{m}=\ldots | |  |
| 905 | 13.41 | 309 | \boldsymbol{\epsilon}_{j}=\sum_{i=j}^{m} \alpha_{i} \boldsymbol{p}_{i}, \quad \alpha_{i}=\frac{\boldsymbol{r}_{i}^{T} \boldsymbol{r}_{i}}{\boldsymbol{p}_{i}^{T} \boldsymbol{A} \boldsymbol{p}_{i}} . | |  |
| 906 | 13.42 | 309 | \left\|\boldsymbol{\epsilon}_{j}\right\|_{\boldsymbol{A}}^{2}=\boldsymbol{\epsilon}_{j} \boldsymbol{A} \boldsymbol{\epsilon}_{j}=\sum_{i=j}^{m} \alpha_{i}^{2} \boldsymbol{p}_{i}^{T} \boldsymbol{A} \boldsymbol{p}_{i}=\sum_{i=j}^{m} \frac{\left(\boldsymbol{r}_{i}^{T} \boldsymbol{r}_{i}\right)^{2}}{\boldsymbol{p}_{i}^{T} \boldsymbol{A} \boldsymbol{p}_{i}} . | |  |
| 907 | | 309 | \boldsymbol{p}_{i}^{T} \boldsymbol{p}_{k}=\left(\boldsymbol{r}_{i}+\beta_{i-1} \boldsymbol{p}_{i-1}\right)^{T} \boldsymbol{p}_{k}=\beta_{i-1} \boldsymbol{p}_{i-1}^{T} \boldsymbol{p}_{k}=\cdots=\beta_{i-1} \cdots \beta_{k}\left(\boldsymbol{p}_{k}^{T} \boldsymbol{p}_{k}\right), | |  |
| 908 | 13.43 | 309 | \boldsymbol{p}_{i}^{T} \boldsymbol{p}_{k}=\frac{\boldsymbol{r}_{i}^{T} \boldsymbol{r}_{i}}{\boldsymbol{r}_{k}^{T} \boldsymbol{r}_{k}} \boldsymbol{p}_{k}^{T} \boldsymbol{p}_{k}, \quad i \geq k | |  |
| 909 | | 309 | \left\|\boldsymbol{\epsilon}_{k}\right\|_{2}^{2}=\left\|\boldsymbol{\epsilon}_{k+1}+\boldsymbol{x}_{k+1}-\boldsymbol{x}_{k}\right\|_{2}^{2}=\left\|\boldsymbol{\epsilon}_{k+1}+\alpha_{k} \boldsymbol{p}_{k}\right\|_{2}^{2}, | |  |
| 910 | 13.46 | 310 | \begin{aligned} & \left\|\boldsymbol{\epsilon}_{k}\right\|_{2}^{2}-\left\|\boldsymbol{\epsilon}_{k+1}\right\|_{2}^{2}=\alpha_{k}\left(2 \boldsymbol{p}_{k}^{T} \boldsymbol{\epsilon}_{k+1}+\alpha_{k} \boldsymbol{p}_{k}^{T} \boldsymbol{p}_{k}\right) \\ & \quad \stackrel{\text { (13.41) }}{=} \alpha_{k}\left(2 \sum_{i=k+1}^{m} \alpha_{i} \boldsymbol{p}_{i}^{T} \boldsymbol{p}_{k}+\alpha_{k} \boldsymbol{p}_{k}^{T} \boldsymbol{p}_{k}\right)=\left(\sum_{i=k}^{m}+\sum_{i=k+1}^{m}\right) \alpha_{k} \alpha_{i} \boldsymbol{p}_{i}^{T} \boldsymbol{p}_{k} \\ & \quad \stackrel{\text { (13.43) }}{=}\left(\sum_{i=k}^{m}+\sum_{i=k+1}^{m}\right) \frac{\boldsymbol{r}_{k}^{T} \boldsymbol{r}_{k}}{\boldsymbol{p}_{k}^{T} \boldsymbol{A} \boldsymbol{p}_{k}} \frac{\boldsymbol{r}_{i}^{T} \boldsymbol{r}_{i}}{\boldsymbol{p}_{i}^{T} \boldsymbol{A} \boldsymbol{p}_{i}} \frac{\boldsymbol{r}_{i}^{T} \boldsymbol{r}_{i}}{\boldsymbol{r}_{k}^{T} \boldsymbol{r}_{k}} \boldsymbol{p}_{k}^{T} \boldsymbol{p}_{k} \\ & \quad \stackrel{\text { (13.42) }}{=} \frac{\left\|\boldsymbol{p}_{k}\right\|_{2}^{2}}{\left\|\boldsymbol{p}_{k}\right\|_{\boldsymbol{A}}^{2}}\left(\left\|\boldsymbol{\epsilon}_{k}\right\|_{\boldsymbol{A}}^{2}+\left\|\boldsymbol{\epsilon}_{k+1}\right\|_{\boldsymbol{A}}^{2}\right) . \end{aligned} | |  |
| 911 | 13.44 | 310 | \boldsymbol{B} \boldsymbol{A} \boldsymbol{x}=\boldsymbol{B} \boldsymbol{b} \Leftrightarrow \boldsymbol{C}^{T}\left(\boldsymbol{C} \boldsymbol{A} \boldsymbol{C}^{T}\right) \boldsymbol{C}^{-T} \boldsymbol{x}=\boldsymbol{C}^{T} \boldsymbol{C} \boldsymbol{b} \Leftrightarrow\left(\boldsymbol{C} \boldsymbol{A} \boldsymbol{C}^{T}\right) \boldsymbol{y}=\boldsymbol{C} \boldsymbol{b}, | |  |
| 912 | 13.45 | 310 | \begin{aligned} A x & =b \\ B A x & =B b \\ \left(C A C^{T}\right) y & =C b, \& x=C^{T} y . \end{aligned} | |  |
| 913 | 13.49 | 311 | \boldsymbol{C}^{T}\left(\boldsymbol{C} \boldsymbol{A} \boldsymbol{C}^{T}\right) \boldsymbol{C}^{-T}=\boldsymbol{B} \boldsymbol{A} . | |  |
| 914 | 13.48 | 311 | \begin{aligned} & \boldsymbol{y}_{k+1}=\boldsymbol{y}_{k}+\alpha_{k} \boldsymbol{q}_{k}, \quad \alpha_{k}=z_{k}^{T} z_{k} / \boldsymbol{q}_{k}^{T}\left(\boldsymbol{C} \boldsymbol{A} \boldsymbol{C}^{T}\right) \boldsymbol{q}_{k}, \\ & \boldsymbol{z}_{k+1}=z_{k}-\alpha_{k}\left(\boldsymbol{C} \boldsymbol{A} \boldsymbol{C}^{T}\right) \boldsymbol{q}_{k}, \\ & \boldsymbol{q}_{k+1}=z_{k+1}+\beta_{k} \boldsymbol{q}_{k}, \quad \beta_{k}=z_{k+1}^{T} z_{k+1} / z_{k}^{T} z_{k} . \end{aligned} | |  |
| 915 | 13.47 | 311 | \boldsymbol{x}_{k}:=\boldsymbol{C}^{T} \boldsymbol{y}_{k}, \quad \boldsymbol{p}_{k}:=\boldsymbol{C}^{T} \boldsymbol{q}_{k}, \quad \boldsymbol{s}_{k}:=\boldsymbol{C}^{T} z_{k}, \quad \boldsymbol{r}_{k}:=\boldsymbol{C}^{-1} z_{k} | |  |
| 916 | 13.50 | 311 | \begin{aligned} & \boldsymbol{x}_{k+1}=\boldsymbol{x}_{k}+\alpha_{k} \boldsymbol{p}_{k}, \quad \alpha_{k}=\frac{\boldsymbol{s}_{k}^{T} \boldsymbol{r}_{k}}{\boldsymbol{p}_{k}^{T} \boldsymbol{A} \boldsymbol{p}_{k}}, \\ & \boldsymbol{r}_{k+1}=\boldsymbol{r}_{k}-\alpha_{k} \boldsymbol{A} \boldsymbol{p}_{k}, \\ & \boldsymbol{s}_{k+1}=\boldsymbol{s}_{k}-\alpha_{k} \boldsymbol{B} \boldsymbol{A} \boldsymbol{p}_{k}, \\ & \boldsymbol{p}_{k+1}=\boldsymbol{s}_{k+1}+\beta_{k} \boldsymbol{p}_{k}, \quad \beta_{k}=\frac{\boldsymbol{s}_{k+1}^{T} \boldsymbol{r}_{k+1}}{\boldsymbol{s}_{k}^{T} \boldsymbol{r}_{k}} . \end{aligned} | |  |
| 917 | | 312 | \frac{\left\|\boldsymbol{x}-\boldsymbol{x}_{k}\right\|_{\boldsymbol{A}}}{\left\|\boldsymbol{x}-\boldsymbol{x}_{0}\right\|_{\boldsymbol{A}}} \leq 2\left(\frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1}\right)^{k} \quad \text { for } \quad k \geq 0 | |  |
| 918 | | 312 | \frac{\left\|\boldsymbol{y}-\boldsymbol{y}_{k}\right\|_{\boldsymbol{C A C}^{T}}}{\left\|\boldsymbol{y}-\boldsymbol{y}_{0}\right\|_{\boldsymbol{C A C}^{T}}} \leq 2\left(\frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1}\right)^{k}, \quad \text { for } \quad k \geq 0, | |  |
| 919 | | 312 | \begin{aligned} \left\|\boldsymbol{y}-\boldsymbol{y}_{k}\right\|_{\boldsymbol{C} \boldsymbol{A} \boldsymbol{C}^{T}}^{2} & =\left(\boldsymbol{y}-\boldsymbol{y}_{k}\right)^{T}\left(\boldsymbol{C} \boldsymbol{A} \boldsymbol{C}^{T}\right)\left(\boldsymbol{y}-\boldsymbol{y}_{k}\right) \\ & =\left(\boldsymbol{C}^{T}\left(\boldsymbol{y}-\boldsymbol{y}_{k}\right)\right)^{T} \boldsymbol{A}\left(\boldsymbol{C}^{T}\left(\boldsymbol{y}-\boldsymbol{y}_{k}\right)\right)=\left\|\boldsymbol{x}-\boldsymbol{x}_{k}\right\|_{\boldsymbol{A}}^{2}, \end{aligned} | |  |
| 920 | 13.52 | 313 | \begin{aligned} -\frac{\partial}{\partial x}\left(c(x, y) \frac{\partial u}{\partial x}\right)-\frac{\partial}{\partial y}\left(c(x, y) \frac{\partial u}{\partial y}\right) & =f(x, y), & & (x, y) \in \Omega=(0,1)^{2}, \\ u(x, y) & =0, & & (x, y) \in \partial \Omega . \end{aligned} | |  |
| 921 | | 313 | \begin{aligned} I_{m} & :=\{(j, k): 1 \leq j, k \leq m\}, \\ \bar{I}_{m} & :=\{(j, k): 0 \leq j, k \leq m+1\}, \\ \partial I_{m} & :=\bar{I}_{m} \backslash I_{m} . \end{aligned} | |  |
| 922 | | 313 | \left\{\left(x_{j}, y_{k}\right)=(j h, k h):(j, k) \in \bar{I}_{m}\right\} | |  |
| 923 | 13.54 | 314 | \begin{aligned} \frac{d}{d t}\left(f(t) \frac{d}{d t} g(t)\right) & \approx\left(f\left(t+\frac{h}{2}\right) \frac{d}{d t} g(t+h / 2)-f\left(t-\frac{h}{2}\right) \frac{d}{d t} g\left(t-\frac{h}{2}\right)\right) / h \\ & \approx\left(f\left(t+\frac{h}{2}\right)(g(t+h)-g(t))-f\left(t-\frac{h}{2}\right)(g(t)-g(t-h))\right) / h^{2} \end{aligned} | |  |
| 924 | 13.53 | 314 | \begin{array}{rlr} \left(L_{h} v\right)_{j, k} & :=\frac{(d v)_{j, k}}{h^{2}}=f_{j, k}, & (j, k) \in I_{m}, \\ v_{j, k} & =0, & (j, k) \in \partial I_{m}, \end{array} | |  |
| 925 | 13.55 | 314 | \begin{aligned} & \left(d_{1} v\right)_{j, k}:=c_{j-\frac{1}{2}, k}\left(v_{j, k}-v_{j-1, k}\right)-c_{j+\frac{1}{2}, k}\left(v_{j+1, k}-v_{j, k}\right) \approx-h^{2} \frac{\partial}{\partial x}\left(c \frac{\partial u}{\partial x}\right)_{j, k}, \\ & \left(d_{2} v\right)_{j, k}:=c_{j, k-\frac{1}{2}}\left(v_{j, k}-v_{j, k-1}\right)-c_{j, k+\frac{1}{2}}\left(v_{j, k+1}-v_{j, k}\right) \approx-h^{2} \frac{\partial}{\partial y}\left(c \frac{\partial u}{\partial y}\right)_{j, k}, \end{aligned} | |  |
| 926 | 13.56 | 314 | L_{h} v=\boldsymbol{F}, \quad L_{h} v:=\frac{1}{h^{2}}\left[\begin{array}{ccc} (d v)_{1,1} & \ldots & (d v)_{1, m} \\ \vdots & & \vdots \\ (d v)_{m, 1} & \ldots & (d v)_{m, m} \end{array}\right], \quad \boldsymbol{F}:=\left[\begin{array}{ccc} f_{1,1} & \ldots & f_{1, m} \\ \vdots & & \vdots \\ f_{m, 1} & \ldots & f_{m, m} \end{array}\right] . | |  |
| 927 | | 314 | \boldsymbol{V}:=\left[\begin{array}{ccc} v_{1,1} & \ldots & v_{1, m} \\ \vdots & & \vdots \\ v_{m, 1} & \ldots & v_{m, m} \end{array}\right] | |  |
| 928 | 13.57 | 314 | \boldsymbol{A} \boldsymbol{x}=\boldsymbol{A} \operatorname{vec}(\boldsymbol{V}):=h^{2} \operatorname{vec}\left(L_{h} \boldsymbol{v}\right) . | |  |
| 929 | 13.58 | 315 | \langle\boldsymbol{V}, \boldsymbol{W}\rangle:=h^{2} \sum_{j, k=1}^{m} v_{j, k} w_{j, k}, | |  |
| 930 | 13.59 | 315 | \begin{aligned} \left\langle L_{h} v, \boldsymbol{W}\right\rangle & =\sum_{j=1}^{m} \sum_{k=0}^{m} c_{j, k+\frac{1}{2}}\left(v_{j, k+1}-v_{j, k}\right)\left(w_{j, k+1}-w_{j, k}\right) \\ & +\sum_{j=0}^{m} \sum_{k=1}^{m} c_{j+\frac{1}{2}, k}\left(v_{j+1, k}-v_{j, k}\right)\left(w_{j+1, k}-w_{j, k}\right) . \end{aligned} | |  |
| 931 | 13.60 | 315 | \sum_{i=1}^{m}\left(a_{i-1}\left(b_{i}-b_{i-1}\right)-a_{i}\left(b_{i+1}-b_{i}\right)\right) c_{i}=\sum_{i=0}^{m} a_{i}\left(b_{i+1}-b_{i}\right)\left(c_{i+1}-c_{i}\right) . | |  |
| 932 | | 315 | \sum_{i=0}^{m} a_{i}\left(b_{i+1}-b_{i}\right) c_{i+1}-\sum_{i=0}^{m} a_{i}\left(b_{i+1}-b_{i}\right) c_{i}, | |  |
| 933 | | 316 | c(x, y)=e^{-x+y} \text { and } f(x, y)=1 . | |  |
| 934 | | 317 | \kappa=\frac{\lambda_{\max }}{\lambda_{\min }} \leq \frac{c_{1}}{c_{0}} . | |  |
| 935 | | 317 | \lambda=\frac{x^{T} \boldsymbol{A x}}{x^{T} \boldsymbol{A}_{p} x} . | |  |
| 936 | | 317 | \boldsymbol{x}^{T} \boldsymbol{A}_{p} x=\sum_{i=1}^{m} \sum_{j=0}^{m}\left(v_{i, j+1}-v_{i, j}\right)^{2}+\sum_{j=1}^{m} \sum_{i=0}^{m}\left(v_{i+1, j}-v_{i, j}\right)^{2} . | |  |
| 937 | | 317 | c_{0}\left(x^{T} \boldsymbol{A}_{p} x\right) \leq x^{T} \boldsymbol{A} \boldsymbol{x} \leq c_{1}\left(x^{T} \boldsymbol{A}_{p} x\right) | |  |
| 938 | | 318 | Q(\boldsymbol{y})=\frac{1}{2} \sum_{j=1}^{n} \lambda_{j} v_{j}^{2}-\sum_{j=1}^{n} c_{j} v_{j} . | |  |
| 939 | | 318 | \left[\begin{array}{rr} 2 & -1 \\ -1 & 2 \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]=\left[\begin{array}{l} 0 \\ 3 \end{array}\right] | |  |
| 940 | | 319 | \boldsymbol{w}^{T} \boldsymbol{A}^{T} \boldsymbol{A} \hat{\boldsymbol{x}}=\boldsymbol{w}^{T} \boldsymbol{A}^{T} \boldsymbol{b}, \quad \text { for all } \boldsymbol{w} \in \mathbb{W}, | |  |
| 941 | | 319 | \|\boldsymbol{b}-\boldsymbol{A} \hat{\boldsymbol{x}}\|_{2} \leq\|\boldsymbol{b}-\boldsymbol{A} \boldsymbol{w}\|_{2}, \quad \text { for all } \boldsymbol{w} \in \mathbb{W} . | |  |
| 942 | | 319 | \mathbb{W}_{k}:=\operatorname{span}\left(\boldsymbol{b}, \boldsymbol{A} \boldsymbol{b}, \ldots, \boldsymbol{A}^{k-1} \boldsymbol{b}\right) . | |  |
| 943 | | 319 | \langle\boldsymbol{v}, \boldsymbol{w}\rangle_{\boldsymbol{A}}:=\boldsymbol{v}^{T} \boldsymbol{A}^{T} \boldsymbol{A} \boldsymbol{w}, \quad \boldsymbol{v}, \boldsymbol{w} \in \mathbb{R}^{n} . | |  |
| 944 | | 320 | \left\langle\boldsymbol{p}_{k}, \boldsymbol{w}\right\rangle_{\boldsymbol{A}}=0, \quad \text { for all } \boldsymbol{w} \in \mathbb{W}_{k} . | |  |
| 945 | | 320 | \boldsymbol{A} \boldsymbol{p}_{k-1} \in \operatorname{span}\left(\boldsymbol{p}_{k-2}, \boldsymbol{p}_{k-1}, \boldsymbol{p}_{k}\right) . | |  |
| 946 | | 320 | \boldsymbol{A}=\left[\begin{array}{rrr} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{array}\right], \quad \text { and } \quad \boldsymbol{b}=\left[\begin{array}{l} 4 \\ 0 \\ 0 \end{array}\right] . | |  |
| 947 | | 320 | \begin{aligned} & {\left[\boldsymbol{x}_{0}, \boldsymbol{x}_{1}, \boldsymbol{x}_{2}, \boldsymbol{x}_{3}\right]=\left[\begin{array}{rrrr} 0 & 2 & 8 / 3 & 3 \\ 0 & 0 & 4 / 3 & 2 \\ 0 & 0 & 0 & 1 \end{array}\right],} \\ & {\left[\boldsymbol{r}_{0}, \boldsymbol{r}_{1}, \boldsymbol{r}_{2}, \boldsymbol{r}_{3}\right]=\left[\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 4 / 3 & 0 \end{array}\right],} \end{aligned} | |  |
| 948 | 13.61 | 321 | \begin{aligned} {\left[\boldsymbol{A} \boldsymbol{p}_{0}, \boldsymbol{A} \boldsymbol{p}_{1}, \boldsymbol{A} \boldsymbol{p}_{2}\right] } & =\left[\begin{array}{rcr} 8 & 0 & 0 \\ -4 & 3 & 0 \\ 0 & -2 & 16 / 9 \end{array}\right], \\ {\left[\boldsymbol{p}_{0}, \boldsymbol{p}_{1}, \boldsymbol{p}_{2}, \boldsymbol{p}_{3}\right] } & =\left[\begin{array}{rrrr} 4 & 1 & 4 / 9 & 0 \\ 0 & 2 & 8 / 9 & 0 \\ 0 & 0 & 12 / 9 & 0 \end{array}\right], \end{aligned} | |  |
| 949 | 13.62 | 321 | \begin{aligned} \langle\boldsymbol{x}, \boldsymbol{y}\rangle & =\langle\boldsymbol{y}, \boldsymbol{x}\rangle, \quad \boldsymbol{x}, \boldsymbol{y} \in \mathbb{R}^{n}, \\ \langle\boldsymbol{C} \boldsymbol{x}, \boldsymbol{y}\rangle & =\left\langle\boldsymbol{y}, \boldsymbol{C}^{T} \boldsymbol{x}\right\rangle \stackrel{(13.61)}{=}\left\langle\boldsymbol{C}^{T} \boldsymbol{x}, \boldsymbol{y}\right\rangle, \quad \boldsymbol{x}, \boldsymbol{y} \in \mathbb{R}^{n}, \boldsymbol{C} \in \mathbb{R}^{n \times n} . \end{aligned} | |  |
| 950 | 13.64 | 321 | \begin{aligned} & \langle\boldsymbol{B} \boldsymbol{x}, \boldsymbol{x}\rangle=0, \quad \boldsymbol{x} \in \mathbb{R}^{n}, \\ & \langle\boldsymbol{A} \boldsymbol{x}, \boldsymbol{x}\rangle=\langle\boldsymbol{x}, \boldsymbol{x}\rangle=\|\boldsymbol{x}\|_{2}^{2} . \end{aligned} | |  |
| 951 | 13.63 | 321 | \left\|\boldsymbol{A}^{-1}\right\|_{2}=\max _{\boldsymbol{x} \neq \mathbf{0}} \frac{\|\boldsymbol{x}\|_{2}}{\|\boldsymbol{A} \boldsymbol{x}\|_{2}}, | |  |
| 952 | 13.65 | 321 | \langle\boldsymbol{A} \boldsymbol{x}, \boldsymbol{w}\rangle=\langle\boldsymbol{b}, \boldsymbol{w}\rangle \text { for all } \boldsymbol{w} \in \mathcal{W}, | |  |
| 953 | 13.66 | 322 | \sum_{j=1}^{k} x_{j}\left\langle\boldsymbol{A} \boldsymbol{w}_{j}, \boldsymbol{w}_{i}\right\rangle=\left\langle\boldsymbol{b}, \boldsymbol{w}_{i}\right\rangle, \quad i=1, \ldots, k | |  |
| 954 | 13.67 | 322 | \left\|\boldsymbol{x}^{*}-\boldsymbol{x}\right\|_{2} \leq\|\boldsymbol{A}\|_{2} \min _{\boldsymbol{w} \in \mathcal{W}}\left\|\boldsymbol{x}^{*}-\boldsymbol{w}\right\|_{2} . | |  |
| 955 | | 322 | \mathcal{W}_{k}:=\operatorname{span}\left(\boldsymbol{b}, \boldsymbol{B} \boldsymbol{b}, \ldots, \boldsymbol{B}^{k-1} \boldsymbol{b}\right) . | |  |
| 956 | | 322 | \left\langle\boldsymbol{A} \boldsymbol{x}_{k}, \boldsymbol{w}\right\rangle=\langle\boldsymbol{b}, \boldsymbol{w}\rangle, \text { for all } \boldsymbol{w} \in \mathcal{W}_{k} . | |  |
| 957 | | 322 | \boldsymbol{r}_{k}:=\boldsymbol{b}-\boldsymbol{A} \boldsymbol{x}_{k}, \text { and } \rho_{k}:=\left\|\boldsymbol{r}_{k}\right\|_{2}^{2} . | |  |
| 958 | | 322 | \rho_{k} \neq 0, \quad k=0, \ldots, m . | |  |
| 959 | 13.68 | 322 | \omega_{k}:= \begin{cases}1, & \text { if } k=0 \\ \left(1+\omega_{k-1}^{-1} \rho_{k} / \rho_{k-1}\right)^{-1}, & \text { otherwise. }\end{cases} | |  |
| 960 | 13.69 | 323 | \begin{aligned} & \boldsymbol{x}_{k+1}=\left(1-\omega_{k}\right) \boldsymbol{x}_{k-1}+\omega_{k}\left(\boldsymbol{x}_{k}+\boldsymbol{r}_{k}\right), \\ & \boldsymbol{r}_{k+1}=\left(1-\omega_{k}\right) \boldsymbol{r}_{k-1}+\omega_{k} \boldsymbol{B} \boldsymbol{r}_{k}, \end{aligned} | |  |
| 961 | 13.71 | 323 | \boldsymbol{B} \boldsymbol{r}_{k}=\alpha_{k} \boldsymbol{r}_{k+1}+\beta_{k} \boldsymbol{r}_{k-1}, | |  |
| 962 | 13.72 | 323 | \left\langle\boldsymbol{r}_{k+1}, \boldsymbol{A}^{-1} \boldsymbol{r}_{k+1}\right\rangle=\left\langle\boldsymbol{r}_{k+1}, \boldsymbol{A}^{-1} \boldsymbol{r}_{j}\right\rangle, \quad j=0,1, \ldots, k . | |  |
| 963 | 13.70 | 323 | T_{n}(t)=\cosh (n \operatorname{arccosh} t) \text { for } t \geq 1, | |  |
| 964 | | 324 | \boldsymbol{A} \boldsymbol{x}=\left[\begin{array}{cccc} a_{1,1} & -c_{\frac{3}{2}, 1} & -c_{1, \frac{3}{2}} & 0 \\ -c_{\frac{3}{2}, 1} & a_{2,2} & 0 & -c_{2, \frac{3}{2}} \\ -c_{1, \frac{3}{2}} & 0 & a_{3,3} & -c_{\frac{3}{2}, 2} \\ 0 & -c_{2, \frac{3}{2}} & -c_{\frac{3}{2}, 2} & a_{4,4} \end{array}\right]\left[\begin{array}{l} v_{1,1} \\ v_{2,1} \\ v_{1,2} \\ v_{2,2} \end{array}\right]=\left[\begin{array}{l} (d v)_{1,1} \\ (d v)_{2,1} \\ (d v)_{1,2} \\ (d v)_{2,2} \end{array}\right], | |  |
| 965 | | 324 | \left[\begin{array}{l} a_{1,1} \\ a_{2,2} \\ a_{3,3} \\ a_{4,4} \end{array}\right]=\left[\begin{array}{l} c_{\frac{1}{2}, 1}+c_{1, \frac{1}{2}}+c_{1, \frac{3}{2}}+c_{\frac{3}{2}, 1} \\ c_{\frac{3}{2}, 1}+c_{2, \frac{1}{2}}+c_{2, \frac{3}{2}}+c_{\frac{5}{2}, 1} \\ c_{\frac{1}{2}, 2}+c_{1, \frac{3}{2}}+c_{1, \frac{5}{2}}+c_{\frac{3}{2}, 2} \\ c_{\frac{3}{2}, 2}+c_{2, \frac{3}{2}}+c_{2, \frac{5}{2}}+c_{\frac{5}{2}, 2} \end{array}\right] . | |  |
| 966 | | 326 | \boldsymbol{A}=\operatorname{diag}\left(\boldsymbol{A}_{1}, \boldsymbol{A}_{2}, \ldots, \boldsymbol{A}_{r}\right), \quad \boldsymbol{A}_{i} \in \mathbb{C}^{m_{i} \times m_{i}} . | |  |
| 967 | | 326 | \begin{aligned} \boldsymbol{A} \boldsymbol{D} & =\operatorname{diag}\left(\boldsymbol{A}_{1}, \ldots, \boldsymbol{A}_{r}\right) \operatorname{diag}\left(\boldsymbol{X}_{1}, \ldots, \boldsymbol{X}_{r}\right)=\operatorname{diag}\left(\boldsymbol{A}_{1} \boldsymbol{X}_{1}, \ldots, \boldsymbol{A}_{r} \boldsymbol{X}_{r}\right) \\ & =\operatorname{diag}\left(\boldsymbol{X}_{1} \boldsymbol{D}_{1}, \ldots, \boldsymbol{X}_{r} \boldsymbol{D}_{r}\right)=\boldsymbol{X} \boldsymbol{D} . \end{aligned} | |  |
| 968 | | 327 | \operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})=\prod_{i=1}^{r} \operatorname{det}\left(\boldsymbol{A}_{i i}-\lambda \boldsymbol{I}\right) | |  |
| 969 | | 327 | \begin{gathered} R_{i}=\left\{z \in \mathbb{C}:\left|z-a_{i i}\right| \leq r_{i}\right\}, \quad r_{i}:=\sum_{\substack{j=1 \\ j \neq i}}^{n}\left|a_{i j}\right|, \\ C_{j}=\left\{z \in \mathbb{C}:\left|z-a_{j j}\right| \leq c_{j}\right\}, \quad c_{j}:=\sum_{\substack{i=1 \\ i \neq j}}^{n}\left|a_{i j}\right| . \end{gathered} | |  |
| 970 | | 328 | \left|\lambda-a_{i i}\right|=\left|\sum_{j \neq i} a_{i j} x_{j} / x_{i}\right| \leq \sum_{j \neq i}\left|a_{i j}\right|\left|x_{j} / x_{i}\right| \leq r_{i} | |  |
| 971 | | 328 | R_{i}=\{z \in \mathbb{R}:|z-2| \leq 2\}, \quad i=2,3, \ldots, m-1 . | |  |
| 972 | | 328 | \lambda_{j}=4\left[\sin \frac{j \pi}{2(m+1)}\right]^{2}, \quad j=1,2, \ldots, m . | |  |
| 973 | | 329 | \boldsymbol{A}(t):=\boldsymbol{D}+t(\boldsymbol{A}-\boldsymbol{D}), \quad \boldsymbol{D}:=\operatorname{diag}\left(a_{11}, \ldots, a_{n n}\right), \quad t \in[0,1] . | |  |
| 974 | | 329 | R_{i}(t)=\left\{z \in \mathbb{C}:\left|z-a_{i i}\right| \leq t r_{i}\right\}, \quad r_{i}:=\sum_{\substack{j=1 \\ j \neq i}}^{n}\left|a_{i j}\right| . | |  |
| 975 | | 330 | \boldsymbol{A}_{1}:=\left[\begin{array}{cccccc} 0 & 1 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 0 & 1 \\ 0 & 0 & 0 & \cdots & 0 & 0 \end{array}\right], \quad \boldsymbol{E}:=\left[\begin{array}{cccccc} 0 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 0 & 0 \\ \epsilon & 0 & 0 & \cdots & 0 & 0 \end{array}\right]=\epsilon \boldsymbol{e}_{n} \boldsymbol{e}_{1}^{T} . | |  |
| 976 | 14.1 | 330 | |\mu-\lambda| \leq K\|E\|_{2}^{1 / n}, \quad K=\left(\|A\|_{2}+\|A+E\|_{2}\right)^{1-1 / n} . | |  |
| 977 | | 330 | \begin{gathered} \left\|(\mu \boldsymbol{I}-\boldsymbol{A}) \boldsymbol{u}_{1}\right\|_{2}=\left\|(\boldsymbol{A}+\boldsymbol{E}) \boldsymbol{u}_{1}-\boldsymbol{A} \boldsymbol{u}_{1}\right\|_{2}=\left\|\boldsymbol{E} \boldsymbol{u}_{1}\right\|_{2} \leq\|\boldsymbol{E}\|_{2}, \\ \prod_{j=2}^{n}\left\|(\mu \boldsymbol{I}-\boldsymbol{A}) \boldsymbol{u}_{j}\right\|_{2} \leq \prod_{j=2}^{n}\left(|\mu|+\left\|\boldsymbol{A} \boldsymbol{u}_{j}\right\|_{2}\right) \leq\left(\|(\boldsymbol{A}+\boldsymbol{E})\|_{2}+\|\boldsymbol{A}\|_{2}\right)^{n-1} . \end{gathered} | |  |
| 978 | | 330 | \begin{aligned} & \left|\mu-\lambda_{1}\right|^{n} \leq \prod_{j=1}^{n}\left|\mu-\lambda_{j}\right|=|\operatorname{det}(\mu \boldsymbol{I}-\boldsymbol{A})|=\left|\operatorname{det}\left((\mu \boldsymbol{I}-\boldsymbol{A})\left[\boldsymbol{u}_{1}, \ldots, \boldsymbol{u}_{n}\right]\right)\right| \\ & \leq\left\|(\mu \boldsymbol{I}-\boldsymbol{A}) \boldsymbol{u}_{1}\right\|_{2} \prod_{j=2}^{n}\left\|(\mu \boldsymbol{I}-\boldsymbol{A}) \boldsymbol{u}_{j}\right\|_{2} \leq\|\boldsymbol{E}\|_{2}\left(\|(\boldsymbol{A}+\boldsymbol{E})\|_{2}+\|\boldsymbol{A}\|_{2}\right)^{n-1} . \end{aligned} | |  |
| 979 | 14.2 | 331 | |\lambda-\mu| \leq K_{p}(\boldsymbol{X})\|\boldsymbol{r}\|_{p}, \quad 1 \leq p \leq \infty, | |  |
| 980 | 14.3 | 331 | |\lambda-\mu| \leq K_{p}(\boldsymbol{X})\|\boldsymbol{E}\|_{p}, \quad 1 \leq p \leq \infty, | |  |
| 981 | | 331 | \boldsymbol{X} \boldsymbol{D}_{1}^{-1} \boldsymbol{X}^{-1} \boldsymbol{r}=\left(\boldsymbol{X}(\boldsymbol{D}-\mu \boldsymbol{I}) \boldsymbol{X}^{-1}\right)^{-1} \boldsymbol{r}=(\boldsymbol{A}-\mu \boldsymbol{I})^{-1}(\boldsymbol{A}-\mu \boldsymbol{I}) \boldsymbol{x}=\boldsymbol{x} . | |  |
| 982 | | 331 | 1=\|\boldsymbol{x}\|_{p}=\left\|\boldsymbol{X} \boldsymbol{D}_{1}^{-1} \boldsymbol{X}^{-1} \boldsymbol{r}\right\|_{p} \leq\left\|\boldsymbol{D}_{1}^{-1}\right\|_{p} K_{p}(\boldsymbol{X})\|\boldsymbol{r}\|_{p}=\frac{K_{p}(\boldsymbol{X})\|\boldsymbol{r}\|_{p}}{\min _{j}\left|\lambda_{j}-\mu\right|} . | |  |
| 983 | | 332 | \|\boldsymbol{A} \boldsymbol{x}\|_{p}=\left\|\left[\lambda_{1} x_{1}, \ldots, \lambda_{n} x_{n}\right]^{T}\right\|_{p}=\left(\sum_{j=1}^{n}\left|\lambda_{j}\right|^{p}\left|x_{j}\right|^{p}\right)^{1 / p} \leq \rho(\boldsymbol{A})\|\boldsymbol{x}\|_{p} . | |  |
| 984 | 14.4 | 332 | \frac{|\lambda-\mu|}{|\lambda|} \leq K_{p}(\boldsymbol{X}) K_{p}(\boldsymbol{A}) \frac{\|\boldsymbol{r}\|_{p}}{\|\boldsymbol{A}\|_{p}}, \quad 1 \leq p \leq \infty, | |  |
| 985 | 14.5 | 332 | \frac{|\lambda-\mu|}{|\lambda|} \leq K_{p}(\boldsymbol{X})\left\|\boldsymbol{A}^{-1} \boldsymbol{E}\right\|_{p} \leq K_{p}(\boldsymbol{X}) K_{p}(\boldsymbol{A}) \frac{\|\boldsymbol{E}\|_{p}}{\|\boldsymbol{A}\|_{p}}, \quad 1 \leq p \leq \infty, | |  |
| 986 | | 332 | \frac{1}{\lambda} \leq\left\|\boldsymbol{A}^{-1}\right\|_{p}=\frac{K_{p}(\boldsymbol{A})}{\|\boldsymbol{A}\|_{p}} | |  |
| 987 | | 333 | (\boldsymbol{B}+\boldsymbol{F}-\boldsymbol{I}) \boldsymbol{x}=\left(\mu \boldsymbol{A}^{-1}-\boldsymbol{A}^{-1} \boldsymbol{E}-\boldsymbol{I}\right) \boldsymbol{x}=\boldsymbol{A}^{-1}(\mu \boldsymbol{I}-(\boldsymbol{E}+\boldsymbol{A})) \boldsymbol{x}=\mathbf{0} . | |  |
| 988 | | 333 | \boldsymbol{A}_{k+1}^{*}=\left(\boldsymbol{H}_{k} \boldsymbol{A}_{k} \boldsymbol{H}_{k}\right)^{*}=\boldsymbol{H}_{k} \boldsymbol{A}_{k}^{*} \boldsymbol{H}_{k}=\boldsymbol{A}_{k+1} . | |  |
| 989 | | 333 | \boldsymbol{A}_{k}=\left[\begin{array}{ll} \boldsymbol{B}_{k} & \boldsymbol{C}_{k} \\ \boldsymbol{D}_{k} & \boldsymbol{E}_{k} \end{array}\right] . | |  |
| 990 | | 334 | \boldsymbol{H}_{k}=\left[\begin{array}{cc} \boldsymbol{I}_{k} & \mathbf{0} \\ \mathbf{0} & \boldsymbol{V}_{k} \end{array}\right] \in \mathbb{C}^{n \times n} . | |  |
| 991 | | 334 | \begin{aligned} \boldsymbol{A}_{k+1} & =\boldsymbol{H}_{k} \boldsymbol{A}_{k} \boldsymbol{H}_{k}=\left[\begin{array}{cc} \boldsymbol{I}_{k} & \mathbf{0} \\ \mathbf{0} & \boldsymbol{V}_{k} \end{array}\right]\left[\begin{array}{ll} \boldsymbol{B}_{k} & \boldsymbol{C}_{k} \\ \boldsymbol{D}_{k} & \boldsymbol{E}_{k} \end{array}\right]\left[\begin{array}{cc} \boldsymbol{I}_{k} & \mathbf{0} \\ \mathbf{0} & \boldsymbol{V}_{k} \end{array}\right] \\ & =\left[\begin{array}{cc} \boldsymbol{B}_{k} & \boldsymbol{C}_{k} \boldsymbol{V}_{k} \\ \boldsymbol{V}_{k} \boldsymbol{D}_{k} & \boldsymbol{V}_{k} \boldsymbol{E}_{k} \boldsymbol{V}_{k} \end{array}\right] . \end{aligned} | |  |
| 992 | | 335 | \boldsymbol{Q}_{n-1}=\boldsymbol{I} \text { and } \boldsymbol{Q}_{k}=\boldsymbol{H}_{k} \boldsymbol{Q}_{k+1} \text { for } k=n-2, n-3, \ldots, 1 . | |  |
| 993 | | 335 | \boldsymbol{Q}_{k}=\left[\begin{array}{cc} \boldsymbol{I}_{k} & \mathbf{0} \\ \mathbf{0} & \boldsymbol{I}-\boldsymbol{v}_{k} \boldsymbol{v}_{k}^{T} \end{array}\right] *\left[\begin{array}{cc} \boldsymbol{I}_{k} & \mathbf{0} \\ \mathbf{0} & \boldsymbol{U}_{k} \end{array}\right]=\left[\begin{array}{cc} \boldsymbol{I}_{k} & \mathbf{0} \\ \mathbf{0} & \boldsymbol{U}_{k}-\boldsymbol{v}_{k}\left(\boldsymbol{v}_{k}^{T} \boldsymbol{U}_{k}\right) \end{array}\right] . | |  |
| 994 | 14.6 | 336 | \boldsymbol{A}=\left[\begin{array}{ccccc} d_{1} & c_{1} & & & \\ c_{1} & d_{2} & c_{2} & & \\ & \ddots & \ddots & \ddots & \\ & & c_{n-2} & d_{n-1} & c_{n-1} \\ & & & c_{n-1} & d_{n} \end{array}\right] | |  |
| 995 | | 336 | \boldsymbol{A}_{1}=\left[\begin{array}{ccccc} d_{1} & c_{1} & & & \\ c_{1} & d_{2} & c_{2} & & \\ & \ddots & \ddots & \ddots & \\ & & c_{i-2} & d_{i-1} & c_{i-1} \\ & & & c_{i-1} & d_{i} \end{array}\right] \text { and } \boldsymbol{A}_{2}=\left[\begin{array}{ccccc} d_{i+1} & c_{i+1} & & & \\ c_{i+1} & d_{i+2} & c_{i+2} & & \\ & \ddots & \ddots & \ddots & \\ & & c_{n-2} & d_{n-1} & c_{n-1} \\ & & & c_{n-1} & d_{n} \end{array}\right] . | |  |
| 996 | 14.7 | 336 | p_{k+1}(x)=\left(x-d_{k+1}\right) p_{k}(x)-c_{k}^{2} p_{k-1}(x) . | |  |
| 997 | | 336 | p_{2}(-M)>0, \quad p_{2}\left(d_{1}\right)<0, \quad p_{2}(+M)>0 . | |  |
| 998 | | 336 | p_{3}(x)=\left(x-d_{3}\right) p_{2}(x)-c_{2}^{2} p_{1}(x)=\left(x-d_{3}\right)\left(x-y_{1}\right)\left(x-y_{2}\right)-c_{2}^{2}\left(x-d_{1}\right) . | |  |
| 999 | | 337 | p_{3}(-M)<0, \quad p_{3}\left(y_{1}\right)>0, \quad p_{3}\left(y_{2}\right)<0, \quad p_{3}(+M)>0 . | |  |
| 1000 | | 337 | y_{0}<y_{1}<z_{1}<y_{2}<z_{2} \cdots<z_{k-1}<y_{k}<y_{k+1} . | |  |
| 1001 | | 337 | p_{k+1}\left(y_{j}\right)=(-1)^{k+1-j}\left|p_{k+1}\left(y_{j}\right)\right| \neq 0, \text { for } j=0,1, \ldots, k+1 . | |  |
| 1002 | | 337 | p_{k+1}\left(y_{j}\right)=-c_{k}^{2} p_{k-1}\left(y_{j}\right)=-c_{k}^{2}\left(y_{j}-z_{1}\right) \cdots\left(y_{j}-z_{k-1}\right) . | |  |
| 1003 | | 337 | \boldsymbol{y}^{*} \boldsymbol{A} \boldsymbol{y}=\sum_{i=1}^{n} \lambda_{i}\left|y_{i}\right|^{2}=\sum_{i=1}^{k} \lambda_{i}\left|x_{i}\right|^{2}>0 . | |  |
| 1004 | | 338 | \boldsymbol{y}^{*} \boldsymbol{A} \boldsymbol{y}=\boldsymbol{y}^{*} \boldsymbol{E}^{*} \boldsymbol{B} \boldsymbol{E} \boldsymbol{y}=z^{*} \boldsymbol{B} z=\sum_{i=m+1}^{n} \mu_{i}\left|z_{i}\right|^{2} \leq 0, | |  |
| 1005 | 14.8 | 338 | d_{1}(\alpha)=d_{1}-\alpha, d_{k}(\alpha)=d_{k}-\alpha-c_{k-1}^{2} / d_{k-1}(\alpha), k=2,3, \ldots, n . | |  |
| 1006 | | 338 | \rho(x):=\#\left\{k: d_{k}(x)>0 \text { for } k=1, \ldots, n\right\} | |  |
| 1007 | | 340 | \boldsymbol{A}=\left(\begin{array}{llll} 4 & 1 & 0 & 0 \\ 1 & 4 & 1 & 0 \\ 0 & 1 & 4 & 1 \\ 0 & 0 & 1 & 4 \end{array}\right) . | |  |
| 1008 | | 340 | \boldsymbol{A}(t):=\boldsymbol{D}+t(\boldsymbol{A}-\boldsymbol{D}), \quad \boldsymbol{D}:=\operatorname{diag}\left(a_{11}, \ldots, a_{n n}\right), \quad t \in \mathbb{R} . | |  |
| 1009 | | 340 | |\lambda-\mu| \leq C\left(t_{2}-t_{1}\right)^{1 / n}, \text { where } C \leq 2\left(\|\boldsymbol{D}\|_{2}+\|\boldsymbol{A}-\boldsymbol{D}\|_{2}\right) . | |  |
| 1010 | 14.10 | 340 | a_{k j}=\left\{\begin{array}{ll} 1, & j=k+1, \\ 0, & \text { otherwise }, \end{array} \quad e_{k j}= \begin{cases}\epsilon, & k=n, j=1, \\ 0, & \text { otherwise },\end{cases}\right. | |  |
| 1011 | | 341 | \boldsymbol{A}:=\left[\begin{array}{llll} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right], \quad \boldsymbol{E}:=\left[\begin{array}{llll} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \epsilon & 0 & 0 & 0 \end{array}\right], \quad \boldsymbol{B}:=\left[\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \epsilon & 0 & 0 & 0 \end{array}\right] . | |  |
| 1012 | | 342 | A_{n}=\left[\begin{array}{ccccc} 10 & 1 & 0 & \cdots & 0 \\ 1 & 10 & 1 & \ddots & \vdots \\ 0 & \ddots & \ddots & \ddots & 0 \\ \vdots & \ddots & 1 & 10 & 1 \\ 0 & \cdots & 0 & 1 & 10 \end{array}\right] \in \mathbb{R}^{n \times n} . | |  |
| 1013 | | 342 | \boldsymbol{D}^{-1 / 2}:=\operatorname{diag}\left(d_{1}^{-1 / 2}, \ldots, d_{n}^{-1 / 2}\right) . | |  |
| 1014 | 15.1 | 344 | \boldsymbol{z}_{k}:=\boldsymbol{A}^{k} \boldsymbol{z}_{0}=\boldsymbol{A} \boldsymbol{z}_{k-1}, \quad k=1,2, \ldots . | |  |
| 1015 | | 344 | \boldsymbol{A}=\left[\begin{array}{cc} 2 & -1 \\ -1 & 2 \end{array}\right], \quad \boldsymbol{z}_{0}:=\left[\begin{array}{l} 1 \\ 0 \end{array}\right] . | |  |
| 1016 | | 345 | z_{1}=\boldsymbol{A} z_{0}=\left[\begin{array}{c} 2 \\ -1 \end{array}\right], \quad z_{2}=\boldsymbol{A} z_{1}=\left[\begin{array}{c} 5 \\ -4 \end{array}\right], \cdots, z_{k}=\frac{1}{2}\left[\begin{array}{l} 1+3^{k} \\ 1-3^{k} \end{array}\right], \cdots . | |  |
| 1017 | | 345 | z_{0}=\frac{1}{2}\left[\begin{array}{c} 1 \\ -1 \end{array}\right]+\frac{1}{2}\left[\begin{array}{l} 1 \\ 1 \end{array}\right]=c_{1} \boldsymbol{v}_{1}+c_{2} \boldsymbol{v}_{2} . | |  |
| 1018 | | 345 | z_{k}=c_{1} \lambda_{1}^{k} v_{1}+c_{2} \lambda_{2}^{k} v_{2}=c_{1} 3^{k} v_{1}+c_{2} 1^{k} v_{2} . | |  |
| 1019 | 15.2 | 345 | \begin{aligned} & \text { (i) }\left|\lambda_{1}\right|>\left|\lambda_{2}\right| \geq\left|\lambda_{3}\right| \geq \cdots \geq\left|\lambda_{n}\right| \text {, } \\ & \text { (ii) } z_{0}^{T} v_{1} \neq 0 \end{aligned} | |  |
| 1020 | 15.3 | 345 | \boldsymbol{z}_{k}=c_{1} \lambda_{1}^{k} \boldsymbol{v}_{1}+c_{2} \lambda_{2}^{k} \boldsymbol{v}_{2}+\cdots+c_{n} \lambda_{n}^{k} \boldsymbol{v}_{n}, \quad k=0,1,2, \ldots . | |  |
| 1021 | 15.4 | 345 | \frac{z_{k}}{\lambda_{1}^{k}}=c_{1} \boldsymbol{v}_{1}+c_{2}\left(\frac{\lambda_{2}}{\lambda_{1}}\right)^{k} \boldsymbol{v}_{2}+\cdots+c_{n}\left(\frac{\lambda_{n}}{\lambda_{1}}\right)^{k} \boldsymbol{v}_{n}, \quad k=0,1,2, \ldots | |  |
| 1022 | 15.5 | 346 | \lim _{k \rightarrow \infty} \frac{z_{k}}{\lambda_{1}^{k}}=c_{1} \boldsymbol{v}_{1}, | |  |
| 1023 | 15.6 | 346 | \begin{array}{ll} \text { (i) } & \boldsymbol{y}_{k}=\boldsymbol{A} \boldsymbol{x}_{k-1} \\ \text { (ii) } & \boldsymbol{x}_{k}=\boldsymbol{y}_{k} /\left\|\boldsymbol{y}_{k}\right\| . \end{array} | |  |
| 1024 | | 346 | \lim _{k \rightarrow \infty}\left(\frac{\left|\lambda_{1}\right|}{\lambda_{1}}\right)^{k} \boldsymbol{x}_{k}=\frac{c_{1}}{\left|c_{1}\right|} \frac{\boldsymbol{v}_{1}}{\left\|\boldsymbol{v}_{1}\right\|} . | |  |
| 1025 | | 346 | \boldsymbol{x}_{k}=\frac{\boldsymbol{z}_{k}}{\left\|\boldsymbol{z}_{k}\right\|}=\frac{c_{1} \lambda_{1}^{k}}{\left|c_{1} \lambda_{1}^{k}\right|} \frac{\boldsymbol{v}_{1}+\frac{c_{2}}{c_{1}}\left(\frac{\lambda_{2}}{\lambda_{1}}\right)^{k} \boldsymbol{v}_{2}+\cdots+\frac{c_{n}}{c_{1}}\left(\frac{\lambda_{n}}{\lambda_{1}}\right)^{k} \boldsymbol{v}_{n}}{\left\|\boldsymbol{v}_{1}+\frac{c_{2}}{c_{1}}\left(\frac{\lambda_{2}}{\lambda_{1}}\right)^{k} \boldsymbol{v}_{2}+\cdots+\frac{c_{n}}{c_{1}}\left(\frac{\lambda_{n}}{\lambda_{1}}\right)^{k} \boldsymbol{v}_{n}\right\|}, \quad k=0,1,2, \ldots, | |  |
| 1026 | | 347 | \left[\begin{array}{c} \boldsymbol{u}^{*} \\ \boldsymbol{U}^{*} \end{array}\right](\boldsymbol{A} \boldsymbol{u}-\lambda \boldsymbol{u})=\left[\begin{array}{c} \boldsymbol{u}^{*} \boldsymbol{A} \boldsymbol{u}-\lambda \boldsymbol{u}^{*} \boldsymbol{u} \\ \boldsymbol{U}^{*} \boldsymbol{A} \boldsymbol{u}-\lambda \boldsymbol{U}^{*} \boldsymbol{u} \end{array}\right]=\left[\begin{array}{c} \boldsymbol{u}^{*} \boldsymbol{A} \boldsymbol{u}-\lambda \\ \boldsymbol{U}^{*} \boldsymbol{A} \boldsymbol{u} \end{array}\right] . | |  |
| 1027 | | 347 | \rho(\lambda)^{2}=\left|\boldsymbol{u}^{*} \boldsymbol{A} \boldsymbol{u}-\lambda\right|^{2}+\left\|\boldsymbol{U}^{*} \boldsymbol{A} \boldsymbol{u}\right\|_{2}^{2}, | |  |
| 1028 | | 347 | \frac{|\lambda-\mu|}{|\lambda|} \leq K_{2}(\boldsymbol{X}) K_{2}(\boldsymbol{A}) \frac{\|\boldsymbol{A} \boldsymbol{u}-\mu \boldsymbol{u}\|_{2}}{\|\boldsymbol{A}\|_{2}} . | |  |
| 1029 | | 348 | \boldsymbol{A}_{1}:=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right], \quad \boldsymbol{A}_{2}:=\left[\begin{array}{cc} 1.7 & -0.4 \\ 0.15 & 2.2 \end{array}\right], \quad \text { and } \boldsymbol{A}_{3}=\left[\begin{array}{cc} 1 & 2 \\ -3 & 4 \end{array}\right] . | |  |
| 1030 | | 348 | \mu_{1}(s)=\left(\lambda_{1}-s\right)^{-1}, \mu_{2}(s)=\left(\lambda_{2}-s\right)^{-1}, \ldots, \mu_{n}(s)=\left(\lambda_{n}-s\right)^{-1} . | |  |
| 1031 | 15.7 | 349 | \begin{array}{ll} \text { (i) } & (\boldsymbol{A}-s \boldsymbol{I}) \boldsymbol{y}_{k}=\boldsymbol{x}_{k-1} \\ \text { (ii) } & \boldsymbol{x}_{k}=\boldsymbol{y}_{k} /\left\|\boldsymbol{y}_{k}\right\| . \end{array} | |  |
| 1032 | | 349 | \begin{aligned} \text { (i) } & \left(\boldsymbol{A}-s_{k-1} \boldsymbol{I}\right) \boldsymbol{y}_{k}=\boldsymbol{x}_{k-1}, \\ \text { (ii) } & \boldsymbol{x}_{k}=\boldsymbol{y}_{k} /\left\|\boldsymbol{y}_{k}\right\|, \\ \text { (iii) } & \boldsymbol{s}_{k}=\boldsymbol{x}_{k}^{*} \boldsymbol{A} \boldsymbol{x}_{k}, \\ \text { (iv) } & \boldsymbol{r}_{k}=\boldsymbol{A} \boldsymbol{x}_{k}-s_{k} \boldsymbol{x}_{k} . \end{aligned} | |  |
| 1033 | | 349 | \rho_{k}:=\frac{\boldsymbol{y}_{k}^{*} \boldsymbol{x}_{k-1}}{\boldsymbol{y}_{k}^{*} \boldsymbol{y}_{k}}, \quad \boldsymbol{w}_{k}:=\frac{\boldsymbol{x}_{k-1}}{\left\|\boldsymbol{y}_{k}\right\|_{2}} . | |  |
| 1034 | | 349 | \begin{aligned} & s_{k}=\frac{\boldsymbol{y}_{k}^{*} \boldsymbol{A} \boldsymbol{y}_{k}}{\boldsymbol{y}_{k}^{*} \boldsymbol{y}_{k}}=s_{k-1}+\frac{\boldsymbol{y}_{k}^{*}\left(\boldsymbol{A}-s_{k-1} \boldsymbol{I}\right) \boldsymbol{y}_{k}}{\boldsymbol{y}_{k}^{*} \boldsymbol{y}_{k}}=s_{k-1}+\frac{\boldsymbol{y}_{k}^{*} \boldsymbol{x}_{k-1}}{\boldsymbol{y}_{k}^{*} \boldsymbol{y}_{k}}=s_{k-1}+\rho_{k}, \\ & \boldsymbol{r}_{k}=\boldsymbol{A} \boldsymbol{x}_{k}-s_{k} \boldsymbol{x}_{k}=\frac{\boldsymbol{A} \boldsymbol{y}_{k}-\left(s_{k-1}+\rho_{k}\right) \boldsymbol{y}_{k}}{\left\|\boldsymbol{y}_{k}\right\|_{2}}=\frac{\boldsymbol{x}_{k-1}-\rho_{k} \boldsymbol{y}_{k}}{\left\|\boldsymbol{y}_{k}\right\|_{2}}=\boldsymbol{w}_{k}-\rho_{k} \boldsymbol{x}_{k} . \end{aligned} | |  |
| 1035 | 15.8 | 351 | \begin{aligned} & \boldsymbol{A}_{1}=\boldsymbol{A} \\ & \text { for } k=1,2, \ldots \\ & \quad \boldsymbol{Q}_{k} \boldsymbol{R}_{k}=\boldsymbol{A}_{k} \quad\left(\text { QR factorization of } \boldsymbol{A}_{k}\right) \\ & \boldsymbol{A}_{k+1}=\boldsymbol{R}_{k} \boldsymbol{Q}_{k} . \end{aligned} | |  |
| 1036 | 15.9 | 351 | \boldsymbol{A}_{k+1}=\boldsymbol{R}_{k} \boldsymbol{Q}_{k}=\boldsymbol{Q}_{k}^{*} \boldsymbol{A}_{k} \boldsymbol{Q}_{k}, | |  |
| 1037 | | 351 | \boldsymbol{A}_{1}=\boldsymbol{A}=\left[\begin{array}{ll} 2 & 1 \\ 1 & 2 \end{array}\right]=\left(\frac{1}{\sqrt{5}}\left[\begin{array}{cc} -2 & -1 \\ -1 & 2 \end{array}\right]\right) *\left(\frac{1}{\sqrt{5}}\left[\begin{array}{cc} -5 & -4 \\ 0 & 3 \end{array}\right]\right)=\boldsymbol{Q}_{1} \boldsymbol{R}_{1} | |  |
| 1038 | | 351 | \boldsymbol{A}_{2}=\boldsymbol{R}_{1} \boldsymbol{Q}_{1}=\frac{1}{5}\left[\begin{array}{cc} -5 & -4 \\ 0 & 3 \end{array}\right] *\left[\begin{array}{cc} -2 & -1 \\ -1 & 2 \end{array}\right]=\frac{1}{5}\left[\begin{array}{cc} 14 & -3 \\ -3 & 6 \end{array}\right]=\left[\begin{array}{cc} 2.8 & -0.6 \\ -0.6 & 1.2 \end{array}\right] . | |  |
| 1039 | | 351 | \boldsymbol{A}_{4} \approx\left[\begin{array}{cc} 2.997 & -0.074 \\ -0.074 & 1.0027 \end{array}\right], \quad \boldsymbol{A}_{10} \approx\left[\begin{array}{cc} 3.0000 & -0.0001 \\ -0.0001 & 1.0000 \end{array}\right] | |  |
| 1040 | | 352 | A_{1}=A=\left[\begin{array}{llll} 0.9501 & 0.8913 & 0.8214 & 0.9218 \\ 0.2311 & 0.7621 & 0.4447 & 0.7382 \\ 0.6068 & 0.4565 & 0.6154 & 0.1763 \\ 0.4860 & 0.0185 & 0.7919 & 0.4057 \end{array}\right] | |  |
| 1041 | | 352 | A_{14}=\left[\begin{array}{r|rr|r} 2.323 & 0.047223 & -0.39232 & -0.65056 \\ \hline-2.1 e-10 & 0.13029 & 0.36125 & 0.15946 \\ -4.1 e-10 & -0.58622 & 0.052576 & -0.25774 \\ \hline 1.2 e-14 & 3.3 e-05 & -1.1 e-05 & 0.22746 \end{array}\right] . | |  |
| 1042 | 15.10 | 352 | \tilde{\boldsymbol{Q}}_{k}:=\boldsymbol{Q}_{1} \cdots \boldsymbol{Q}_{k} \text { and } \tilde{\boldsymbol{R}}_{k}:=\boldsymbol{R}_{k} \cdots \boldsymbol{R}_{1}, | |  |
| 1043 | | 353 | \begin{figure} \[ \boldsymbol{A}=\left[\begin{array}{llll} x & x & x & x \\ 0 & x & x & x \\ 0 & 0 & x & x \\ 0 & 0 & 0 & x \end{array}\right] \xrightarrow{\boldsymbol{P}_{12}^{*}}\left[\begin{array}{llll} x & x & x & x \\ \mathbf{x} & x & x & x \\ 0 & 0 & x & x \\ 0 & 0 & 0 & x \end{array}\right] \xrightarrow{\boldsymbol{P}_{23}^{*}}\left[\begin{array}{llll} x & x & x & x \\ x & x & x & x \\ 0 & \mathbf{x} & x & x \\ 0 & 0 & 0 & x \end{array}\right] \xrightarrow{\boldsymbol{P}_{34}^{*}}\left[\begin{array}{llll} x & x & x & x \\ x & x & x & x \\ 0 & x & x & x \\ 0 & 0 & \mathbf{x} & x \end{array}\right] . \] \captionsetup{labelformat=empty} \caption{Fig. 15.1 Post multiplication in a QR step} \end{figure} | |  |
| 1044 | 15.11 | 353 | \boldsymbol{A}_{k}=\boldsymbol{Q}_{k-1}^{*} \boldsymbol{A}_{k-1} \boldsymbol{Q}_{k-1}=\boldsymbol{Q}_{k-1}^{*} \boldsymbol{Q}_{k-2}^{*} \boldsymbol{A}_{k-2} \boldsymbol{Q}_{k-2} \boldsymbol{Q}_{k-1}=\cdots=\tilde{\boldsymbol{Q}}_{k-1}^{*} \boldsymbol{A} \tilde{\boldsymbol{Q}}_{k-1} . | |  |
| 1045 | | 353 | \tilde{\boldsymbol{Q}}_{k} \tilde{\boldsymbol{R}}_{k}=\tilde{\boldsymbol{Q}}_{k-1}\left(\boldsymbol{Q}_{k} \boldsymbol{R}_{k}\right) \tilde{\boldsymbol{R}}_{k-1}=\tilde{\boldsymbol{Q}}_{k-1} \boldsymbol{A}_{k} \tilde{\boldsymbol{R}}_{k-1}=\left(\tilde{\boldsymbol{Q}}_{k-1} \tilde{\boldsymbol{Q}}_{k-1}^{*}\right) \boldsymbol{A} \tilde{\boldsymbol{Q}}_{k-1} \tilde{\boldsymbol{R}}_{k-1}=\boldsymbol{A}^{k} . | |  |
| 1046 | | 353 | \boldsymbol{A}^{k} \boldsymbol{e}_{1}=\tilde{\boldsymbol{Q}}_{k} \tilde{\boldsymbol{R}}_{k} \boldsymbol{e}_{1}=\tilde{r}_{11}^{(k)} \tilde{\boldsymbol{Q}}_{k} \boldsymbol{e}_{1} \text { or } \tilde{\boldsymbol{q}}_{1}^{(k)}:=\tilde{\boldsymbol{Q}}_{k} \boldsymbol{e}_{1}=\frac{1}{\tilde{r}_{11}^{(k)}} \boldsymbol{A}^{k} \boldsymbol{e}_{1} . | |  |
| 1047 | | 354 | \hat{\boldsymbol{A}}_{k}=\tilde{\boldsymbol{Q}}_{k-1}^{*}(\boldsymbol{A}+\boldsymbol{E}) \tilde{\boldsymbol{Q}}_{k-1}, \quad \boldsymbol{E}=\tilde{\boldsymbol{Q}}_{k-1}\left(a_{i+1, i}^{(k)} \boldsymbol{e}_{i+1} \boldsymbol{e}_{i}^{T}\right) \tilde{\boldsymbol{Q}}_{k-1}^{*} . | |  |
| 1048 | | 355 | \boldsymbol{A}_{k+1}=\boldsymbol{Q}_{k}^{*}\left(\boldsymbol{A}_{k}-s_{k} \boldsymbol{I}\right) \boldsymbol{Q}_{k}+s_{k} \boldsymbol{I}=\boldsymbol{Q}_{k}^{*} \boldsymbol{A}_{k} \boldsymbol{Q}_{k} | |  |
| 1049 | | 358 | D_{i} f(\boldsymbol{x}):=\frac{\partial f(\boldsymbol{x})}{\partial x_{i}}:=\lim _{h \rightarrow \mathbf{0}} \frac{f\left(\boldsymbol{x}+h \boldsymbol{e}_{i}\right)-f(\boldsymbol{x})}{h}, \quad \boldsymbol{x} \in \mathbb{R}^{n}, | |  |
| 1050 | 16.1 | 358 | \nabla f:=\left[\begin{array}{c} D_{1} f \\ \vdots \\ D_{n} f \end{array}\right], \quad \boldsymbol{H} f:=\nabla \nabla^{T} f:=\left[\begin{array}{ccc} D_{1} D_{1} f & \cdots & D_{1} D_{n} f \\ \vdots & & \vdots \\ D_{n} D_{1} & \cdots & D_{n} D_{n} f \end{array}\right], | |  |
| 1051 | | 359 | \nabla^{T} \boldsymbol{f}:=\left[\begin{array}{ccc} D_{1} f_{1} & \cdots & D_{n} f_{1} \\ \vdots & & \vdots \\ D_{1} f_{m} & \cdots & D_{n} f_{m} \end{array}\right] . | |  |
| 1052 | | 359 | \begin{aligned} & \nabla f(x, y)=\left[\begin{array}{c} 2 x-y \\ -x+2 y \end{array}\right], \quad \nabla^{T} \boldsymbol{g}(x, y)=\left[\begin{array}{cc} 2 x-y-x+2 y \\ 1 & -1 \end{array}\right], \\ & \boldsymbol{H} f(x, y)=\left[\begin{array}{cc} \frac{\partial^{2} f}{\partial x^{2}} & \frac{\partial^{2} f}{\partial x \partial y} \\ \frac{\partial^{2} f}{\partial y \partial x} & \frac{\partial^{2} f}{\partial y^{2}} \end{array}\right]=\left[\begin{array}{rr} 2 & -1 \\ -1 & 2 \end{array}\right] . \end{aligned} | |  |
| 1053 | 16.2 | 359 | f(\boldsymbol{x}+\boldsymbol{h})=f(\boldsymbol{x})+\boldsymbol{h}^{T} \nabla f(\boldsymbol{x})+\frac{1}{2} \boldsymbol{h}^{T} \nabla \nabla^{T} f(\boldsymbol{c}) \boldsymbol{h}, \text { for some } \boldsymbol{c} \in L . | |  |
| 1054 | | 359 | \begin{aligned} g(0) & =f(\boldsymbol{x}) \quad g(1)=f(\boldsymbol{x}+\boldsymbol{h}), \\ g^{\prime}(t) & =\sum_{i=1}^{n} h_{i} \frac{\partial f(\boldsymbol{x}+t \boldsymbol{h})}{\partial x_{i}}=\boldsymbol{h}^{T} \nabla f(\boldsymbol{x}+t \boldsymbol{h}), \\ g^{\prime \prime}(t) & =\sum_{i=1}^{n} \sum_{j=1}^{n} h_{i} h_{j} \frac{\partial^{2} f(\boldsymbol{x}+t \boldsymbol{h})}{\partial x_{i} \partial x_{j}}=\boldsymbol{h}^{T} \nabla \nabla^{T} f(\boldsymbol{x}+t \boldsymbol{h}) \boldsymbol{h} . \end{aligned} | |  |
| 1055 | | 359 | g(1)=g(0)+g^{\prime}(0)+\frac{1}{2} g^{\prime \prime}(u), \text { for some } u \in(0,1), | |  |
| 1056 | | 360 | \begin{aligned} D_{i}\left(\boldsymbol{x}^{T} \boldsymbol{B} \boldsymbol{x}\right) & =\lim _{h \rightarrow 0} \frac{1}{h}\left(\left(\boldsymbol{x}+h \boldsymbol{e}_{i}\right)^{T} \boldsymbol{B}\left(\boldsymbol{x}+h \boldsymbol{e}_{i}\right)-\boldsymbol{x}^{T} \boldsymbol{B} \boldsymbol{x}\right) \\ & =\lim _{h \rightarrow 0}\left(\boldsymbol{e}_{i}^{T} \boldsymbol{B} \boldsymbol{x}+\boldsymbol{x}^{T} \boldsymbol{B} \boldsymbol{e}_{i}+h \boldsymbol{e}_{i}^{T} \boldsymbol{e}_{i}\right)=\boldsymbol{e}_{i}^{T}\left(\boldsymbol{B}+\boldsymbol{B}^{T}\right) \boldsymbol{x}, \end{aligned} | |  |
