lower is closer · 0 is exact
0707.4470
\mathscr{g}
\[\mathscr{g}\]
verified by source+ink
evidence {'source_span': '$\\mathcal{J}$', 'context_before': 'Given a discrete form and d
\mathscr{J}
\[\mathscr{J}\]
0902.0431
\begin{aligned} & A X A^{*} \times A Y A^{*}=A(X \times Y) A^{*} \\ & \left(P M A^{*}\right)^{*}\left(P N A^{*}\right)=A M^{*} P^{*} P N A=A\left(M^{*} N\right) A, \\ & \left(P M A^{*}\right)\left(A Y A^{*}\right)=P(M Y) A^{*}, \\ & \overline{P M A^{*} \times P N A^{*}}=\frac{\bar{t} \widetilde{P}(M \times N)^{t} \widetilde{A} *}{}=P \overline{M \times N} A^{*}, \text { etc. } \end{aligned}
\[\begin{aligned} & A X A^{*} \times A Y A^{*}=A(X \times Y) A^{*} \\ & \left(P M A^{*}\right)^{*}\left(P N A^{*}\right)=A M^{*} P^{*} P N A=A\left(M^{*} N\right) A, \\ & \left(P M A^{*}\right)\left(A Y A^{*}\right)=P(M Y) A^{*}, \\ & \overline{P M A^{*} \times P N A^{*}}=\frac{\bar{t} \widetilde{P}(M \times N)^{t} \widetilde{A} *}{}=P \overline{M \times N} A^{*}, \text { etc. } \end{aligned}\]
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\begin{aligned} & A X A^{*} \times A Y A^{*}=A(X \times Y) A^{*} \\ & \left(P M A^{*}\right)^{*}\left(P N A^{*}\right)=A M^{*} P^{*} P N A=A\left(M^{*} N\right) A, \\ & \left(P M A^{*}\right)\left(A Y A^{*}\right)=P(M Y) A^{*}, \\ & \overline{P M A^{*} \times P N A^{*}}=\bar{t} \widetilde{P}(M \times N)^{t} \widetilde{A}^{*}=P \overline{M \times N} A^{*}, \text{ etc. } \end{aligned}
\[\begin{aligned} & A X A^{*} \times A Y A^{*}=A(X \times Y) A^{*} \\ & \left(P M A^{*}\right)^{*}\left(P N A^{*}\right)=A M^{*} P^{*} P N A=A\left(M^{*} N\right) A, \\ & \left(P M A^{*}\right)\left(A Y A^{*}\right)=P(M Y) A^{*}, \\ & \overline{P M A^{*} \times P N A^{*}}=\bar{t} \widetilde{P}(M \times N)^{t} \widetilde{A}^{*}=P \overline{M \times N} A^{*}, \text{ etc. } \end{aligned}\]
0902.0431
\begin{aligned} & A X A^{*} \times A Y A^{*}=A(X \times Y) A^{*} \\ & \left(P M A^{*}\right)^{*}\left(P N A^{*}\right)=A M^{*} P^{*} P N A=A\left(M^{*} N\right) A, \\ & \left(P M A^{*}\right)\left(A Y A^{*}\right)=P(M Y) A^{*}, \\ & \overline{P M A^{*} \times P N A^{*}}=\frac{\bar{t} \widetilde{P}(M \times N)^{t} \widetilde{A} *}{}=P \overline{M \times N} A^{*}, \text { etc. } \end{aligned}
\[\begin{aligned} & A X A^{*} \times A Y A^{*}=A(X \times Y) A^{*} \\ & \left(P M A^{*}\right)^{*}\left(P N A^{*}\right)=A M^{*} P^{*} P N A=A\left(M^{*} N\right) A, \\ & \left(P M A^{*}\right)\left(A Y A^{*}\right)=P(M Y) A^{*}, \\ & \overline{P M A^{*} \times P N A^{*}}=\frac{\bar{t} \widetilde{P}(M \times N)^{t} \widetilde{A} *}{}=P \overline{M \times N} A^{*}, \text { etc. } \end{aligned}\]
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-49 · closer
\begin{aligned} &=\nu\Big(\sum_{k=0}^{3}\lambda_k H_k\Big)\\ &=\frac{1}{2}\big(\lambda_0(-H_0+H_1-H_2+H_3)+\lambda_1(-H_0+H_1+H_2-\\ &\quad+\lambda_2(H_0+H_1+H_2+H_3)+\lambda_3(-H_0-H_1+H_2+H_3) \end{aligned}
\[\begin{aligned}
&=\nu\Big(\sum_{k=0}^{3}\lambda_k H_k\Big)\\
&=\frac{1}{2}\big(\lambda_0(-H_0+H_1-H_2+H_3)+\lambda_1(-H_0+H_1+H_2-\\
&\quad+\lambda_2(H_0+H_1+H_2+H_3)+\lambda_3(-H_0-H_1+H_2+H_3)
\end{aligned}\]
0902.0431
\begin{aligned} & \lambda_{0}-\lambda_{1}=1 \quad 0 \quad 0 \quad 0 \quad 0 \quad 0 \\ & \lambda_{0}-\lambda_{2}=1 \quad 1 \quad 0 \quad 0 \quad 0 \quad \lambda_{0}+\lambda_{1}=1 \quad 2 \\ & \lambda_{0}-\lambda_{3}=1 \quad 1 \quad 1 \quad 0 \quad 2 \end{aligned} \quad \frac{1}{1} \text { = } \begin{array}{llllllll} 1 & 1 & 1 & 0 & 0 & \lambda_{0}+\lambda_{3}=1 & 1 & 2 \\ \lambda_{1}-\lambda_{2} &
\[\begin{aligned} & \lambda_{0}-\lambda_{1}=1 \quad 0 \quad 0 \quad 0 \quad 0 \quad 0 \\ & \lambda_{0}-\lambda_{2}=1 \quad 1 \quad 0 \quad 0 \quad 0 \quad \lambda_{0}+\lambda_{1}=1 \quad 2 \\ & \lambda_{0}-\lambda_{3}=1 \quad 1 \quad 1 \quad 0 \quad 2 \end{aligned} \quad \frac{1}{1} \text { = } \begin{array}{llllllll} 1 & 1 & 1 & 0 & 0 & \lambda_{0}+\lambda_{3}=1 & 1 & 2 \\ \lambda_{1}-\lambda_{2} & =0 & 1 & 0 & 0 & 0 & 0 & 1 \end{array}\]
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-91 · closer
\begin{aligned} \lambda_0 - \lambda_1 &= 1\ 0\ 0\ 0\ 0\ 0 & \lambda_0 + \lambda_1 &= 1\ 2\ 2\ 1\ 1\ 1 \\ \lambda_0 - \lambda_2 &= 1\ 1\ 0\ 0\ 0\ 0 & \lambda_0 + \lambda_2 &= 1\ 1\ 2\ 1\ 1\ 1 \\ \lambda_0 - \lambda_3 &= 1\ 1\ 1\ 0\ 0\ 0 & \lambda_0 + \lambda_3 &= 1\ 1\ 1\ 1\ 1\ 1 \\ \lambda_1 - \lambda_2 &= 0\ 1\ 0\ 0\ 0\ 0 & \lambda_1 + \lambda_2 &= 0\ 1\ 2\ 1\ 1\ 1 \\ \lambda_1 - \lambda_3 &= 0\
\[\begin{aligned}
\lambda_0 - \lambda_1 &= 1\ 0\ 0\ 0\ 0\ 0 &
\lambda_0 + \lambda_1 &= 1\ 2\ 2\ 1\ 1\ 1 \\
\lambda_0 - \lambda_2 &= 1\ 1\ 0\ 0\ 0\ 0 &
\lambda_0 + \lambda_2 &= 1\ 1\ 2\ 1\ 1\ 1 \\
\lambda_0 - \lambda_3 &= 1\ 1\ 1\ 0\ 0\ 0 &
\lambda_0 + \lambda_3 &= 1\ 1\ 1\ 1\ 1\ 1 \\
\lambda_1 - \lambda_2 &= 0\ 1\ 0\ 0\ 0\ 0 &
\lambda_1 + \lambda_2 &= 0\ 1\ 2\ 1\ 1\ 1 \\
\lambda_1 - \lambda_3 &= 0\ 1\ 1\ 0\ 0\ 0 &
\lambda_1 + \lambda_3 &= 0\ 1\ 1\ 1\ 1\ 1 \\
\lambda_2 - \lambda_3 &= 0\ 0\ 1\ 0\ 0\ 0 &
\lambda_2 + \lambda_3 &= 0\ 0\ 1\ 1\ 1\ 1
\end{aligned}\]
0902.0431
\begin{aligned} & \lambda_{0}-\lambda_{1}=1 \quad 0 \quad 0 \quad 0 \quad 0 \quad 0 \\ & \lambda_{0}-\lambda_{2}=1 \quad 1 \quad 0 \quad 0 \quad 0 \quad \lambda_{0}+\lambda_{1}=1 \quad 2 \\ & \lambda_{0}-\lambda_{3}=1 \quad 1 \quad 1 \quad 0 \quad 2 \end{aligned} \quad \frac{1}{1} \text { = } \begin{array}{llllllll} 1 & 1 & 1 & 0 & 0 & \lambda_{0}+\lambda_{3}=1 & 1 & 2 \\ \lambda_{1}-\lambda_{2} &
\[\begin{aligned} & \lambda_{0}-\lambda_{1}=1 \quad 0 \quad 0 \quad 0 \quad 0 \quad 0 \\ & \lambda_{0}-\lambda_{2}=1 \quad 1 \quad 0 \quad 0 \quad 0 \quad \lambda_{0}+\lambda_{1}=1 \quad 2 \\ & \lambda_{0}-\lambda_{3}=1 \quad 1 \quad 1 \quad 0 \quad 2 \end{aligned} \quad \frac{1}{1} \text { = } \begin{array}{llllllll} 1 & 1 & 1 & 0 & 0 & \lambda_{0}+\lambda_{3}=1 & 1 & 2 \\ \lambda_{1}-\lambda_{2} & =0 & 1 & 0 & 0 & 0 & 0 & 1 \end{array}\]
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-59 · closer
\begin{aligned} \nu H &= \nu\bigg(\sum_{k=0}^{3} \lambda_k H_k\bigg) \\ &= \frac{1}{2}\big(\lambda_0(-H_0 + H_1 - H_2 + H_3) + \lambda_1(-H_0 + H_1 + H_2 - H_3) \\ &\quad + \lambda_2(H_0 + H_1 + H_2 + H_3) + \lambda_3(-H_0 - H_1 + H_2 + H_3)\big) \\ &= \frac{1}{2}(-\lambda_0 - \lambda_1 + \lambda_2 - \lambda_3)H_0 + \frac{1}{2}(\lambda_0 + \lambda_1 + \lambda_2 - \lambda_3)H_1 \end{aligned}
\[\begin{aligned}
\nu H &= \nu\bigg(\sum_{k=0}^{3} \lambda_k H_k\bigg) \\
&= \frac{1}{2}\big(\lambda_0(-H_0 + H_1 - H_2 + H_3) + \lambda_1(-H_0 + H_1 + H_2 - H_3) \\
&\quad + \lambda_2(H_0 + H_1 + H_2 + H_3) + \lambda_3(-H_0 - H_1 + H_2 + H_3)\big) \\
&= \frac{1}{2}(-\lambda_0 - \lambda_1 + \lambda_2 - \lambda_3)H_0 + \frac{1}{2}(\lambda_0 + \lambda_1 + \lambda_2 - \lambda_3)H_1
\end{aligned}\]
0902.0431
\begin{array}{lllllllllllll} \lambda_{0}-\lambda_{1}=1 & 0 & 0 & 0 & 0 & 0 & 0 & \lambda_{0}+\lambda_{1}=1 & 2 & 2 & 2 & 2 & 1 \\ \lambda_{0}-\lambda_{2}=1 & 1 & 0 & 0 & 0 & 0 & 0 & \lambda_{0}+\lambda_{2}=1 & 1 & 2 & 2 & 2 & 1 \\ \lambda_{0}-\lambda_{3}=1 & 1 & 1 & 0 & 0 & 0 & 0 & \lambda_{0}+\lambda_{3}=1 & 1 & 1 & 2 & 2 & 1 \\ \lambda_{1}-\lambda_{2}=0 & 1 & 0 & 0 & 0 & 0 & 0 & \lambda_{1}+\lam
\[\begin{array}{lllllllllllll} \lambda_{0}-\lambda_{1}=1 & 0 & 0 & 0 & 0 & 0 & 0 & \lambda_{0}+\lambda_{1}=1 & 2 & 2 & 2 & 2 & 1 \\ \lambda_{0}-\lambda_{2}=1 & 1 & 0 & 0 & 0 & 0 & 0 & \lambda_{0}+\lambda_{2}=1 & 1 & 2 & 2 & 2 & 1 \\ \lambda_{0}-\lambda_{3}=1 & 1 & 1 & 0 & 0 & 0 & 0 & \lambda_{0}+\lambda_{3}=1 & 1 & 1 & 2 & 2 & 1 \\ \lambda_{1}-\lambda_{2}=0 & 1 & 0 & 0 & 0 & 0 & 0 & \lambda_{1}+\lambda_{2}=0 & 1 & 2 & 2 & 2 & 1 \\ \lambda_{1}-\lambda_{3}=0 & 1 & 1 & 0 & 0 & 0 & 0 & \lambda_{1}+\lambda_{3}=0 & 1 & 1 & 2 & 2 & 1 \\ \lambda_{2}-\lambda_{3}=0 & 0 & 1 & 0 & 0 & 0 & 0 & \lambda_{2}+\lambda_{3}=0 & 0 & 1 & 2 & 2 & 1 \\ & & \lambda_{0}+\frac{1}{2}\left(\mu_{2}-\mu_{3}\right)=1 & 1 & 1 & 1 & 1 & 1 & 1 & & & \\ & & \lambda_{1}+\frac{1}{2}\left(\mu_{2}-\mu_{3}\right)=0 & 1 & 1 & 1 & 1 & 1 & 1 & & & \\ & & \lambda_{2}+\frac{1}{2}\left(\mu_{2}-\mu_{3}\right)=0 & 0 & 1 & 1 & 1 & 1 & 1 & & & \\ & & +\frac{1}{2}\left(\mu_{2}-\mu_{3}\right)=0 & 0 & 0 & 1 & 1 & 1 & 1 & & & \end{array}\]
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\begin{array}{llllllllllllll} \lambda_{0}-\lambda_{1}=1 & 0 & 0 & 0 & 0 & 0 & 0 & \lambda_{0}+\lambda_{1}=1 & 2 & 2 & 2 & 2 & 1 \\ \lambda_{0}-\lambda_{2}=1 & 1 & 0 & 0 & 0 & 0 & 0 & \lambda_{0}+\lambda_{2}=1 & 1 & 2 & 2 & 2 & 1 \\ \lambda_{0}-\lambda_{3}=1 & 1 & 1 & 0 & 0 & 0 & 0 & \lambda_{0}+\lambda_{3}=1 & 1 & 1 & 2 & 2 & 1 \\ \lambda_{1}-\lambda_{2}=0 & 1 & 0 & 0 & 0 & 0 & 0 & \lambda_{1}+\la
\[\begin{array}{llllllllllllll}
\lambda_{0}-\lambda_{1}=1 & 0 & 0 & 0 & 0 & 0 & 0 & \lambda_{0}+\lambda_{1}=1 & 2 & 2 & 2 & 2 & 1 \\
\lambda_{0}-\lambda_{2}=1 & 1 & 0 & 0 & 0 & 0 & 0 & \lambda_{0}+\lambda_{2}=1 & 1 & 2 & 2 & 2 & 1 \\
\lambda_{0}-\lambda_{3}=1 & 1 & 1 & 0 & 0 & 0 & 0 & \lambda_{0}+\lambda_{3}=1 & 1 & 1 & 2 & 2 & 1 \\
\lambda_{1}-\lambda_{2}=0 & 1 & 0 & 0 & 0 & 0 & 0 & \lambda_{1}+\lambda_{2}=0 & 1 & 2 & 2 & 2 & 1 \\
\lambda_{1}-\lambda_{3}=0 & 1 & 1 & 0 & 0 & 0 & 0 & \lambda_{1}+\lambda_{3}=0 & 1 & 1 & 2 & 2 & 1 \\
\lambda_{2}-\lambda_{3}=0 & 0 & 1 & 0 & 0 & 0 & 0 & \lambda_{2}+\lambda_{3}=0 & 0 & 1 & 2 & 2 & 1 \\
& & \lambda_{0}+\frac{1}{2}\left(\mu_{2}-\mu_{3}\right)=1 & 1 & 1 & 1 & 1 & 1 & 1 & & & & \\
& & \lambda_{1}+\frac{1}{2}\left(\mu_{2}-\mu_{3}\right)=0 & 1 & 1 & 1 & 1 & 1 & 1 & & & & \\
& & \lambda_{2}+\frac{1}{2}\left(\mu_{2}-\mu_{3}\right)=0 & 0 & 1 & 1 & 1 & 1 & 1 & & & & \\
& & \lambda_{3}+\frac{1}{2}\left(\mu_{2}-\mu_{3}\right)=0 & 0 & 0 & 1 & 1 & 1 & 1 & & & &
\end{array}\]
0902.0431
f^{-1}\left(\sum_{i<j<k} x_{i j k} \boldsymbol{e}_{i} \wedge \boldsymbol{e}_{j} \wedge \boldsymbol{e}_{k}\right)=\left(\begin{array}{c} \left(\begin{array}{ccc} h\left(x_{156}\right) & h\left(x_{164}, \bar{x}_{256}\right) & h\left(x_{145}, \bar{x}_{356}\right) \\ h\left(x_{256}, \bar{x}_{164}\right) & h\left(x_{264}\right) & h\left(x_{245}, \bar{x}_{364}\right) \\ h\left(x_{356}, \bar{x}_{145}\rig
\[f^{-1}\left(\sum_{i<j<k} x_{i j k} \boldsymbol{e}_{i} \wedge \boldsymbol{e}_{j} \wedge \boldsymbol{e}_{k}\right)=\left(\begin{array}{c} \left(\begin{array}{ccc} h\left(x_{156}\right) & h\left(x_{164}, \bar{x}_{256}\right) & h\left(x_{145}, \bar{x}_{356}\right) \\ h\left(x_{256}, \bar{x}_{164}\right) & h\left(x_{264}\right) & h\left(x_{245}, \bar{x}_{364}\right) \\ h\left(x_{356}, \bar{x}_{145}\right) & h\left(x_{364}, \bar{x}_{245}\right) & h\left(x_{345}\right) \end{array}\right) \\ \left(\begin{array}{ccc} h\left(x_{423}\right) & h\left(x_{431}, \bar{x}_{523}\right) & h\left(x_{412}, \bar{x}_{623}\right) \\ h\left(x_{523}, \bar{x}_{431}\right) & h\left(x_{531}\right) & h\left(x_{512}, \bar{x}_{631}\right) \\ h\left(x_{623}, \bar{x}_{412}\right) & h\left(x_{631}, \bar{x}_{512}\right) & h\left(x_{612}\right) \end{array}\right) \\ \\ \\ \left(\begin{array}{cc} h\left(x_{123}\right) & \\ h\left(x_{456}\right) \end{array}\right. \end{array}\right),\]
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f^{-1}\left(\sum_{i<j<k} x_{ijk} \boldsymbol{e}_{i} \wedge \boldsymbol{e}_{j} \wedge \boldsymbol{e}_{k}\right)=\left(\begin{array}{c} \left(\begin{array}{ccc} h\left(x_{156}\right) & h\left(x_{164}, \bar{x}_{256}\right) & h\left(x_{145}, \bar{x}_{356}\right) \\ h\left(x_{256}, \bar{x}_{164}\right) & h\left(x_{264}\right) & h\left(x_{245}, \bar{x}_{364}\right) \\ h\left(x_{356}, \bar{x}_{145}\right
\[f^{-1}\left(\sum_{i<j<k} x_{ijk} \boldsymbol{e}_{i} \wedge \boldsymbol{e}_{j} \wedge \boldsymbol{e}_{k}\right)=\left(\begin{array}{c}
\left(\begin{array}{ccc}
h\left(x_{156}\right) & h\left(x_{164}, \bar{x}_{256}\right) & h\left(x_{145}, \bar{x}_{356}\right) \\
h\left(x_{256}, \bar{x}_{164}\right) & h\left(x_{264}\right) & h\left(x_{245}, \bar{x}_{364}\right) \\
h\left(x_{356}, \bar{x}_{145}\right) & h\left(x_{364}, \bar{x}_{245}\right) & h\left(x_{345}\right)
\end{array}\right) \\
\left(\begin{array}{ccc}
h\left(x_{423}\right) & h\left(x_{431}, \bar{x}_{523}\right) & h\left(x_{412}, \bar{x}_{623}\right) \\
h\left(x_{523}, \bar{x}_{431}\right) & h\left(x_{531}\right) & h\left(x_{512}, \bar{x}_{631}\right) \\
h\left(x_{623}, \bar{x}_{412}\right) & h\left(x_{631}, \bar{x}_{512}\right) & h\left(x_{612}\right)
\end{array}\right) \\
\\
\\
\left(\begin{array}{cc}
h\left(x_{123}\right) & \\
& h\left(x_{456}\right)
\end{array}\right)
\end{array}\right),\]
0902.0431
\begin{aligned} & \frac{1}{2}\left(\lambda_{0}+\lambda_{1}+\lambda_{2}-\lambda_{3}\right)+\frac{1}{2} \mu_{2}-\frac{2}{3} \nu=1 \quad 2 \quad 3 \quad 3 \quad 2 \quad 2 \end{aligned} \quad 1 \quad ⿱=\frac{1}{2}\left(\lambda_{0}+\lambda_{1}-\lambda_{2}+\lambda_{3}\right)+\frac{1}{2} \mu_{2}-\frac{2}{3} \nu=1 \quad 2 \quad 3 \quad 3 \quad 2 \quad 1 \quad 1 \quad 1 .
\[\begin{aligned} & \frac{1}{2}\left(\lambda_{0}+\lambda_{1}+\lambda_{2}-\lambda_{3}\right)+\frac{1}{2} \mu_{2}-\frac{2}{3} \nu=1 \quad 2 \quad 3 \quad 3 \quad 2 \quad 2 \end{aligned} \quad 1 \quad ⿱=\frac{1}{2}\left(\lambda_{0}+\lambda_{1}-\lambda_{2}+\lambda_{3}\right)+\frac{1}{2} \mu_{2}-\frac{2}{3} \nu=1 \quad 2 \quad 3 \quad 3 \quad 2 \quad 1 \quad 1 \quad 1 .\]
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\begin{aligned} \kappa\pi H &= \tfrac{1}{2}(-\lambda_0+\lambda_1-\lambda_2+\lambda_3)H_0 + \tfrac{1}{2}(-\lambda_0+\lambda_1+\lambda_2-\lambda_3)H_1 \\ &\quad +\tfrac{1}{2}(\lambda_0+\lambda_1+\lambda_2+\lambda_3)H_2 + \tfrac{1}{2}(-\lambda_0-\lambda_1+\lambda_2+\lambda_3)H_3, \end{aligned}
\[\begin{aligned}
\kappa\pi H &= \tfrac{1}{2}(-\lambda_0+\lambda_1-\lambda_2+\lambda_3)H_0 + \tfrac{1}{2}(-\lambda_0+\lambda_1+\lambda_2-\lambda_3)H_1 \\
&\quad +\tfrac{1}{2}(\lambda_0+\lambda_1+\lambda_2+\lambda_3)H_2 + \tfrac{1}{2}(-\lambda_0-\lambda_1+\lambda_2+\lambda_3)H_3,
\end{aligned}\]
0902.0431
\begin{aligned} & =\left\{\alpha \in\left(E_{8}{ }^{C}\right)_{1,1-, 1-} \mid \tau \widetilde{\lambda} \alpha=\alpha \tau \widetilde{\lambda}\right\} \\ & =\left\{\alpha \in E_{7}{ }^{C} \mid \tau \widetilde{\lambda} \alpha=\alpha \tau \widetilde{\lambda}\right\} \text { (Proposition 5.7.1) } \\ & =\left\{\alpha \in E_{7}{ }^{C} \mid \tau \lambda \alpha=\alpha \tau \lambda\right\} \text { (by the
\[\begin{aligned} & =\left\{\alpha \in\left(E_{8}{ }^{C}\right)_{1,1-, 1-} \mid \tau \widetilde{\lambda} \alpha=\alpha \tau \widetilde{\lambda}\right\} \\ & =\left\{\alpha \in E_{7}{ }^{C} \mid \tau \widetilde{\lambda} \alpha=\alpha \tau \widetilde{\lambda}\right\} \text { (Proposition 5.7.1) } \\ & =\left\{\alpha \in E_{7}{ }^{C} \mid \tau \lambda \alpha=\alpha \tau \lambda\right\} \text { (by the correspondence to Proposition 5.7.1) } \\ & =E_{7} \text { (Lemma 4.3.3.(4)). } \end{aligned}\]
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\begin{aligned} \kappa\pi H ={}& \frac{1}{2}(-\lambda_0 + \lambda_1 - \lambda_2 + \lambda_3)H_0 + \frac{1}{2}(-\lambda_0 + \lambda_1 + \lambda_2 - \lambda_3)H_1 \\ & + \frac{1}{2}(\lambda_0 + \lambda_1 + \lambda_2 + \lambda_3)H_2 + \frac{1}{2}(-\lambda_0 - \lambda_1 + \lambda_2 + \lambda_3)H_3, \end{aligned}
\[\begin{aligned}
\kappa\pi H ={}& \frac{1}{2}(-\lambda_0 + \lambda_1 - \lambda_2 + \lambda_3)H_0 + \frac{1}{2}(-\lambda_0 + \lambda_1 + \lambda_2 - \lambda_3)H_1 \\
& + \frac{1}{2}(\lambda_0 + \lambda_1 + \lambda_2 + \lambda_3)H_2 + \frac{1}{2}(-\lambda_0 - \lambda_1 + \lambda_2 + \lambda_3)H_3,
\end{aligned}\]
0902.0431
\begin{aligned} & (\boldsymbol{X}, \boldsymbol{Y})=\left(X_{1}, Y_{1}\right)+\left(X_{2}, Y_{2}\right)+\left(X_{3}, Y_{3}\right) \in C, \\ & \langle\boldsymbol{X}, \boldsymbol{Y}\rangle=\left\langle X_{1}, Y_{1}\right\rangle+\left\langle X_{2}, Y_{2}\right\rangle+\left\langle X_{3}, Y_{3}\right\rangle \in C, \\ & \boldsymbol{X} \times \boldsymbol{Y}=\left(\begin{array}{ll} X_{2} \times Y_{3}-Y_{2}
\[\begin{aligned} & (\boldsymbol{X}, \boldsymbol{Y})=\left(X_{1}, Y_{1}\right)+\left(X_{2}, Y_{2}\right)+\left(X_{3}, Y_{3}\right) \in C, \\ & \langle\boldsymbol{X}, \boldsymbol{Y}\rangle=\left\langle X_{1}, Y_{1}\right\rangle+\left\langle X_{2}, Y_{2}\right\rangle+\left\langle X_{3}, Y_{3}\right\rangle \in C, \\ & \boldsymbol{X} \times \boldsymbol{Y}=\left(\begin{array}{ll} X_{2} \times Y_{3}-Y_{2} \times X_{3} \\ X_{3} \times Y_{1}-Y_{3} \times X_{1} \\ X_{1} \times Y_{2}-Y_{1} \times X_{2} \end{array}\right) \in\left(\mathfrak{J}^{C}\right)^{3}, \\ & \boldsymbol{X} \cdot \boldsymbol{Y}=\left(\begin{array}{ll} \left(X_{1}, Y_{1}\right) & \left(X_{1}, Y_{2}\right) \\ \left(X_{2}, Y_{1}\right) & \left(X_{2}, Y_{2}\right) \\ \left(X_{3}, Y_{1}\right) & \left(X_{2}, Y_{3}\right) \\ X \vee \boldsymbol{Y} & \left(X_{3}, Y_{3}\right) \end{array}\right)-\frac{1}{3}(\boldsymbol{X}, \boldsymbol{Y}) E \in \mathfrak{s} \mathfrak{l}(3, C), \\ & \boldsymbol{X} Y_{1}+X_{2} \vee Y_{2}+X_{3} \vee Y_{3} \in \mathfrak{e}_{6}{ }^{C}, \end{aligned}\]
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\begin{aligned} & (\boldsymbol{X}, \boldsymbol{Y})=\left(X_{1}, Y_{1}\right)+\left(X_{2}, Y_{2}\right)+\left(X_{3}, Y_{3}\right) \in C, \\ & \langle\boldsymbol{X}, \boldsymbol{Y}\rangle=\left\langle X_{1}, Y_{1}\right\rangle+\left\langle X_{2}, Y_{2}\right\rangle+\left\langle X_{3}, Y_{3}\right\rangle \in C, \\ & \boldsymbol{X} \wedge \boldsymbol{Y}=\left(\begin{array}{l} X_{2} \wedge Y_{3}-Y_{2}
\[\begin{aligned} & (\boldsymbol{X}, \boldsymbol{Y})=\left(X_{1}, Y_{1}\right)+\left(X_{2}, Y_{2}\right)+\left(X_{3}, Y_{3}\right) \in C, \\ & \langle\boldsymbol{X}, \boldsymbol{Y}\rangle=\left\langle X_{1}, Y_{1}\right\rangle+\left\langle X_{2}, Y_{2}\right\rangle+\left\langle X_{3}, Y_{3}\right\rangle \in C, \\ & \boldsymbol{X} \wedge \boldsymbol{Y}=\left(\begin{array}{l} X_{2} \wedge Y_{3}-Y_{2} \wedge X_{3} \\ X_{3} \wedge Y_{1}-Y_{3} \wedge X_{1} \\ X_{1} \wedge Y_{2}-Y_{1} \wedge X_{2} \end{array}\right) \in\left(\mathfrak{J}^{C}\right)^{3}, \\ & \boldsymbol{X} \cdot \boldsymbol{Y}=\left(\begin{array}{ccc} \left(X_{1}, Y_{1}\right) & \left(X_{1}, Y_{2}\right) & \left(X_{1}, Y_{3}\right) \\ \left(X_{2}, Y_{1}\right) & \left(X_{2}, Y_{2}\right) & \left(X_{2}, Y_{3}\right) \\ \left(X_{3}, Y_{1}\right) & \left(X_{3}, Y_{2}\right) & \left(X_{3}, Y_{3}\right) \end{array}\right)-\frac{1}{3}(\boldsymbol{X}, \boldsymbol{Y}) E \in \mathfrak{sl}(3, C), \\ & \boldsymbol{X} \vee \boldsymbol{Y}=X_{1} \vee Y_{1}+X_{2} \vee Y_{2}+X_{3} \vee Y_{3} \in \mathfrak{e}_{6}{ }^{C}, \end{aligned}\]
0902.0431
\begin{aligned} &=(\boldsymbol{v}, \boldsymbol{w})(\boldsymbol{a}, *(\boldsymbol{b} \wedge \boldsymbol{x}))-\left(\boldsymbol{a}_{1}, \boldsymbol{w}\right)\left(\boldsymbol{w} \wedge \boldsymbol{a}_{2}, *(\boldsymbol{b} \wedge \boldsymbol{x})\right) \\ &+\left(\boldsymbol{a}_{2}, \boldsymbol{w}\right)\left(\boldsymbol{w} \wedge \boldsymbol{a}_{1}, *(\boldsymbol{b} \wedge \boldsymbol{x})\right)-\fr
\[\begin{aligned} &=(\boldsymbol{v}, \boldsymbol{w})(\boldsymbol{a}, *(\boldsymbol{b} \wedge \boldsymbol{x}))-\left(\boldsymbol{a}_{1}, \boldsymbol{w}\right)\left(\boldsymbol{w} \wedge \boldsymbol{a}_{2}, *(\boldsymbol{b} \wedge \boldsymbol{x})\right) \\ &+\left(\boldsymbol{a}_{2}, \boldsymbol{w}\right)\left(\boldsymbol{w} \wedge \boldsymbol{a}_{1}, *(\boldsymbol{b} \wedge \boldsymbol{x})\right)-\frac{3}{5}(*(\boldsymbol{a} \wedge \boldsymbol{b}), \boldsymbol{x})(\boldsymbol{v}, \boldsymbol{w}) \\ &= \frac{2}{5}(*(\boldsymbol{a} \wedge \boldsymbol{b}), \boldsymbol{x})(\boldsymbol{v}, \boldsymbol{w})-\left(\boldsymbol{a}_{1}, \boldsymbol{w}\right)\left(\boldsymbol{b} \wedge \boldsymbol{x} \wedge \boldsymbol{w} \wedge \boldsymbol{a}_{2}, \boldsymbol{e}_{1} \wedge \cdots \wedge \boldsymbol{e}_{5}\right) \\ &+\left(\boldsymbol{a}_{2}, \boldsymbol{w}\right)\left(\boldsymbol{b} \wedge \boldsymbol{x} \wedge \boldsymbol{w} \wedge \boldsymbol{a}_{1}, \boldsymbol{e}_{1} \wedge \cdots \wedge \boldsymbol{e}_{5}\right) \\ &((\boldsymbol{b} \times ⿻ ⿱ 一 ⿱ 日 一 \zh20 \\ &=(\boldsymbol{a} \wedge \boldsymbol{x})) \boldsymbol{v}, \boldsymbol{w}) \\ &=(\boldsymbol{v}, \boldsymbol{w})(*(\boldsymbol{a} \wedge \boldsymbol{b}), \boldsymbol{v})+\left(\boldsymbol{a}_{1}, \boldsymbol{w}\right)\left(\boldsymbol{b} \wedge \boldsymbol{x} \wedge \boldsymbol{w} \wedge \boldsymbol{a}_{2}, \boldsymbol{e}_{1} \wedge \cdots \wedge \boldsymbol{e}_{5}\right) \\ &-\left(\boldsymbol{a}_{2}, \boldsymbol{w}\right)\left(\boldsymbol{b} \wedge \boldsymbol{x} \wedge \boldsymbol{w} \wedge \boldsymbol{a}_{1}, \boldsymbol{e}_{1} \wedge \cdots \wedge \boldsymbol{e}_{5}\right)-\frac{3}{5}(*(\boldsymbol{a} \wedge \boldsymbol{b}), \boldsymbol{x})(\boldsymbol{v}, \boldsymbol{w}) . \end{aligned}\]
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\begin{aligned} H &= \nu\Big(\sum_{k=0}^{3} \lambda_k H_k\Big) \\ &= \tfrac{1}{2}\big(\lambda_0(-H_0 + H_1 - H_2 + H_3) + \lambda_1(-H_0 + H_1 + H_2 - H_3) \\ &\quad + \lambda_2(H_0 + H_1 + H_2 + H_3) + \lambda_3(-H_0 - H_1 + H_2 + H_3)\big) \\ &= \tfrac{1}{2}(-\lambda_0 - \lambda_1 + \lambda_2 - \lambda_3)H_0 + \tfrac{1}{2}(\lambda_0 + \lambda_1 + \lambda_2 - \lambda_3)H_1 \\ &\quad + \tfrac{1}{2
\[\begin{aligned}
H &= \nu\Big(\sum_{k=0}^{3} \lambda_k H_k\Big) \\
&= \tfrac{1}{2}\big(\lambda_0(-H_0 + H_1 - H_2 + H_3) + \lambda_1(-H_0 + H_1 + H_2 - H_3) \\
&\quad + \lambda_2(H_0 + H_1 + H_2 + H_3) + \lambda_3(-H_0 - H_1 + H_2 + H_3)\big) \\
&= \tfrac{1}{2}(-\lambda_0 - \lambda_1 + \lambda_2 - \lambda_3)H_0 + \tfrac{1}{2}(\lambda_0 + \lambda_1 + \lambda_2 - \lambda_3)H_1 \\
&\quad + \tfrac{1}{2}(-\lambda_0 + \lambda_1 + \lambda_2 + \lambda_3)H_2 + \tfrac{1}{2}(\lambda_0 - \lambda_1 + \lambda_2 + \lambda_3)H_3,
\end{aligned}\]
Geometrodynamics of Gauge Fi
D D \vartheta^{\alpha}=R_{\beta}{ }^{\alpha} \wedge \vartheta^{\beta}=0, \stackrel{( \pm)}{D} \stackrel{( \pm)}{D} \stackrel{( \pm)}{\Pi}_{\alpha}=-\stackrel{( \pm)}{R}_{\alpha}{ }^{\beta} \wedge \stackrel{( \pm)}{\Pi}_{\beta}= \pm \frac{i}{2} \ell^{2}\left(e_{\alpha} \downharpoonleft \stackrel{( \pm)}{\Pi^{\beta}}\right) \wedge \Sigma_{\beta} \cong 0 .
\[D D \vartheta^{\alpha}=R_{\beta}{ }^{\alpha} \wedge \vartheta^{\beta}=0, \stackrel{( \pm)}{D} \stackrel{( \pm)}{D} \stackrel{( \pm)}{\Pi}_{\alpha}=-\stackrel{( \pm)}{R}_{\alpha}{ }^{\beta} \wedge \stackrel{( \pm)}{\Pi}_{\beta}= \pm \frac{i}{2} \ell^{2}\left(e_{\alpha} \downharpoonleft \stackrel{( \pm)}{\Pi^{\beta}}\right) \wedge \Sigma_{\beta} \cong 0 .\]
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preserve the term Pontryagin index for the classification of real associated vector bundles (\textsc{Mayer} \& \textsc{Drechsler} 1977). Nevertheless, the denotation of \textsc{Belavin}
\[preserve the term Pontryagin index for the classification of real associated vector bundles (\textsc{Mayer} \& \textsc{Drechsler} 1977). Nevertheless, the denotation of \textsc{Belavin}\]
Geometrodynamics of Gauge Fi
\begin{array}{ll} q: & \underline{*}^{\vartheta^{\beta}} \Psi_{\|}(\vartheta)=\underline{*}^{\prime} \underline{\vartheta}^{\beta} \Psi_{\|}(\vartheta), \\ \stackrel{ \pm}{p}: & \stackrel{*}{+}_{\underline{\Pi}} \Psi_{\|}(\vartheta)=-i \ell^{2} \frac{\delta}{\delta \underline{*}_{\alpha}} \Psi_{\|}(\vartheta), \end{array}
\[\begin{array}{ll} q: & \underline{*}^{\vartheta^{\beta}} \Psi_{\|}(\vartheta)=\underline{*}^{\prime} \underline{\vartheta}^{\beta} \Psi_{\|}(\vartheta), \\ \stackrel{ \pm}{p}: & \stackrel{*}{+}_{\underline{\Pi}} \Psi_{\|}(\vartheta)=-i \ell^{2} \frac{\delta}{\delta \underline{*}_{\alpha}} \Psi_{\|}(\vartheta), \end{array}\]
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\begin{array}{ll} q: & \stackrel{*}{\underline{\vartheta}}^{\beta} \Psi_{\|}(\vartheta) = \stackrel{*'}{\underline{\vartheta}}^{\beta} \Psi_{\|}(\vartheta), \\ \stackrel{\pm}{p}: & \stackrel{*^{(\pm)}}{\underline{\Pi}_{\alpha}} \Psi_{\|}(\vartheta) = -i\ell^{2} \dfrac{\delta}{\delta \stackrel{*}{\underline{\vartheta}}_{\alpha}} \Psi_{\|}(\vartheta), \end{array}
\[\begin{array}{ll}
q: & \stackrel{*}{\underline{\vartheta}}^{\beta} \Psi_{\|}(\vartheta) = \stackrel{*'}{\underline{\vartheta}}^{\beta} \Psi_{\|}(\vartheta), \\
\stackrel{\pm}{p}: & \stackrel{*^{(\pm)}}{\underline{\Pi}_{\alpha}} \Psi_{\|}(\vartheta) = -i\ell^{2} \dfrac{\delta}{\delta \stackrel{*}{\underline{\vartheta}}_{\alpha}} \Psi_{\|}(\vartheta),
\end{array}\]
Geometrodynamics of Gauge Fi
\stackrel{\text { (১) }}{D} \alpha:=\stackrel{\text { (০) }}{d} \alpha-\omega^{g} \wedge \alpha-\alpha \wedge \stackrel{\circ}{\circ}
\[\stackrel{\text { (১) }}{D} \alpha:=\stackrel{\text { (০) }}{d} \alpha-\omega^{g} \wedge \alpha-\alpha \wedge \stackrel{\circ}{\circ}\]
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\overset{(1)}{D}\alpha:=\overset{(0)}{d}\alpha-\omega^{g}\wedge\alpha-\alpha\wedge\omega^{g}
\[\overset{(1)}{D}\alpha:=\overset{(0)}{d}\alpha-\omega^{g}\wedge\alpha-\alpha\wedge\omega^{g}\]
Introduction to Graph and Hy
(\mathcal{H})_{2}=(X, \mathcal{E}) \text { where }\left\{x_{i}, x_{j}\right\} \in \mathcal{E} \Leftrightarrow \mathcal{D}\left(x_{i}\right) \cap \mathcal{D}\left(x_{j}\right) \neq \mathbb{O} \text {. }
\[(\mathcal{H})_{2}=(X, \mathcal{E}) \text { where }\left\{x_{i}, x_{j}\right\} \in \mathcal{E} \Leftrightarrow \mathcal{D}\left(x_{i}\right) \cap \mathcal{D}\left(x_{j}\right) \neq \mathbb{O} \text {. }\]
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(\mathcal{H})_{2}=(X, \mathcal{E}) \text { where }\left\{x_{i}, x_{j}\right\} \in \mathcal{E} \Leftrightarrow \mathcal{D}\left(x_{i}\right) \cap \mathcal{D}\left(x_{j}\right) \neq \varnothing \text {. }
\[(\mathcal{H})_{2}=(X, \mathcal{E}) \text { where }\left\{x_{i}, x_{j}\right\} \in \mathcal{E} \Leftrightarrow \mathcal{D}\left(x_{i}\right) \cap \mathcal{D}\left(x_{j}\right) \neq \varnothing \text {. }\]
Introduction to Graph and Hy
\tau(\mathcal{H})=\nu(\mathcal{H}) .
\[\tau(\mathcal{H})=\nu(\mathcal{H}) .\]
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\tau(\mathcal{H})=\nu(\mathcal{H}) . a hypergraph, draw
\[\tau(\mathcal{H})=\nu(\mathcal{H}) .
a hypergraph, draw\]
Introduction to Graph and Hy
\chi(\mathcal{H})=\bar{\chi}(\mathcal{H})=0, \quad R(\mathcal{H})=(0,0, \ldots, 0), \quad P(\mathcal{H}, \lambda)=0 .
\[\chi(\mathcal{H})=\bar{\chi}(\mathcal{H})=0, \quad R(\mathcal{H})=(0,0, \ldots, 0), \quad P(\mathcal{H}, \lambda)=0 .\]
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en the incidence matrix of a hypergraph, construct the edge list.
\[en the incidence matrix of a hypergraph, construct the edge list.\]
Introduction to Linear and M
\begin{array}{rlrl} & & \mathbf{x}+2(\mathbf{v}+\mathbf{w}) & =-\mathbf{v}-3(\mathbf{x}-\mathbf{w}) \\ \Longrightarrow & \mathbf{x}+2 \mathbf{v}+2 \mathbf{w} & =-\mathbf{v}-3 \mathbf{x}+3 \mathbf{w} & \\ \Longrightarrow & 4 \mathbf{x} & =-3 \mathbf{v}+\mathbf{w} & \\ \Longrightarrow & \mathbf{x} & =\frac{1}{4}(\mathbf{w}-3 \mathbf{v}) . & \\ \Longrightarrow & \text { (divide both sides by } 4) \en
\[\begin{array}{rlrl} & & \mathbf{x}+2(\mathbf{v}+\mathbf{w}) & =-\mathbf{v}-3(\mathbf{x}-\mathbf{w}) \\ \Longrightarrow & \mathbf{x}+2 \mathbf{v}+2 \mathbf{w} & =-\mathbf{v}-3 \mathbf{x}+3 \mathbf{w} & \\ \Longrightarrow & 4 \mathbf{x} & =-3 \mathbf{v}+\mathbf{w} & \\ \Longrightarrow & \mathbf{x} & =\frac{1}{4}(\mathbf{w}-3 \mathbf{v}) . & \\ \Longrightarrow & \text { (divide both sides by } 4) \end{array}\]
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\begin{array}{rll} & \mathbf{x}+2(\mathbf{v}+\mathbf{w}) = -\mathbf{v}-3(\mathbf{x}-\mathbf{w}) & \\ \Longrightarrow & \mathbf{x}+2\mathbf{v}+2\mathbf{w} = -\mathbf{v}-3\mathbf{x}+3\mathbf{w} & \text{(expand parentheses)} \\ \Longrightarrow & 4\mathbf{x} = -3\mathbf{v}+\mathbf{w} & \text{(add $3\mathbf{x}$, subtract $2\mathbf{v}+2\mathbf{w}$)} \\ \Longrightarrow & \mathbf{x} = \frac{1}{4}(\mathbf
\[\begin{array}{rll}
& \mathbf{x}+2(\mathbf{v}+\mathbf{w}) = -\mathbf{v}-3(\mathbf{x}-\mathbf{w}) & \\
\Longrightarrow & \mathbf{x}+2\mathbf{v}+2\mathbf{w} = -\mathbf{v}-3\mathbf{x}+3\mathbf{w} & \text{(expand parentheses)} \\
\Longrightarrow & 4\mathbf{x} = -3\mathbf{v}+\mathbf{w} & \text{(add $3\mathbf{x}$, subtract $2\mathbf{v}+2\mathbf{w}$)} \\
\Longrightarrow & \mathbf{x} = \frac{1}{4}(\mathbf{w}-3\mathbf{v}). & \text{(divide both sides by 4)}
\end{array}\]
Introduction to Linear and M
\left.\begin{array}{rl} (\mathbf{v}+\mathbf{w}) \cdot(\mathbf{x}+\mathbf{y}) & =(\mathbf{v}+\mathbf{w}) \cdot \mathbf{x}+(\mathbf{v}+\mathbf{w}) \cdot \mathbf{y} \\ & =\mathbf{x} \cdot(\mathbf{v}+\mathbf{w})+\mathbf{y} \cdot(\mathbf{v}+\mathbf{w}) \\ & =\mathbf{x} \cdot \mathbf{v}+\mathbf{x} \cdot \mathbf{w}+\mathbf{y} \cdot \mathbf{v}+\mathbf{y} \cdot \mathbf{w} \\ & =\mathbf{v} \cdot \mathbf{x}+
\[\left.\begin{array}{rl} (\mathbf{v}+\mathbf{w}) \cdot(\mathbf{x}+\mathbf{y}) & =(\mathbf{v}+\mathbf{w}) \cdot \mathbf{x}+(\mathbf{v}+\mathbf{w}) \cdot \mathbf{y} \\ & =\mathbf{x} \cdot(\mathbf{v}+\mathbf{w})+\mathbf{y} \cdot(\mathbf{v}+\mathbf{w}) \\ & =\mathbf{x} \cdot \mathbf{v}+\mathbf{x} \cdot \mathbf{w}+\mathbf{y} \cdot \mathbf{v}+\mathbf{y} \cdot \mathbf{w} \\ & =\mathbf{v} \cdot \mathbf{x}+\mathbf{w} \cdot \mathbf{x}+\mathbf{v} \cdot \mathbf{y}+\mathbf{w} \cdot \mathbf{y} \end{array}(\text { property (broperty (a) })\right)\]
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\begin{array}{rl} (\mathbf{v}+\mathbf{w}) \cdot(\mathbf{x}+\mathbf{y}) =(\mathbf{v}+\mathbf{w}) \cdot \mathbf{x}+(\mathbf{v}+\mathbf{w}) \cdot \mathbf{y} & (\text{property (b)})\\ = \mathbf{x} \cdot(\mathbf{v}+\mathbf{w})+\mathbf{y} \cdot(\mathbf{v}+\mathbf{w}) & (\text{property (a)})\\ = \mathbf{x} \cdot \mathbf{v}+\mathbf{x} \cdot \mathbf{w}+\mathbf{y} \cdot \mathbf{v}+\mathbf{y} \cdot \mathbf{w
\[\begin{array}{rl}
(\mathbf{v}+\mathbf{w}) \cdot(\mathbf{x}+\mathbf{y}) =(\mathbf{v}+\mathbf{w}) \cdot \mathbf{x}+(\mathbf{v}+\mathbf{w}) \cdot \mathbf{y} & (\text{property (b)})\\
= \mathbf{x} \cdot(\mathbf{v}+\mathbf{w})+\mathbf{y} \cdot(\mathbf{v}+\mathbf{w}) & (\text{property (a)})\\
= \mathbf{x} \cdot \mathbf{v}+\mathbf{x} \cdot \mathbf{w}+\mathbf{y} \cdot \mathbf{v}+\mathbf{y} \cdot \mathbf{w} & (\text{property (b)})\\
= \mathbf{v} \cdot \mathbf{x}+\mathbf{w} \cdot \mathbf{x}+\mathbf{v} \cdot \mathbf{y}+\mathbf{w} \cdot \mathbf{y}. & (\text{property (a)})
\end{array}\]
Introduction to Linear and M
\begin{aligned} A B & =\left[\begin{array}{ll} I_{3} & I_{3} \\ O & C \end{array}\right]\left[\begin{array}{ll} D & O \\ O & D \end{array}\right] \\ & \left.=\left[\begin{array}{ll} I_{3} & D+I_{3} \end{array}\right] \quad \text { O } \quad \begin{array}{ll} I_{3} & O+I_{3} \\ O & D+C \\ O \end{array} \quad \begin{array}{ll} O & O+C \\ O \end{array}\right]=\left[\begin{array}{cc} D & D \\ O & C D
\[\begin{aligned} A B & =\left[\begin{array}{ll} I_{3} & I_{3} \\ O & C \end{array}\right]\left[\begin{array}{ll} D & O \\ O & D \end{array}\right] \\ & \left.=\left[\begin{array}{ll} I_{3} & D+I_{3} \end{array}\right] \quad \text { O } \quad \begin{array}{ll} I_{3} & O+I_{3} \\ O & D+C \\ O \end{array} \quad \begin{array}{ll} O & O+C \\ O \end{array}\right]=\left[\begin{array}{cc} D & D \\ O & C D \end{array}\right] . \end{aligned}\]
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\begin{aligned} \Longrightarrow \quad 4\mathbf{x} &= -3\mathbf{v} + \mathbf{w} \qquad &&\text{(add $3\mathbf{x}$, subtract $2\mathbf{v}$)} \\ \Longrightarrow \quad \mathbf{x} &= \tfrac{1}{4}(\mathbf{w} - 3\mathbf{v}). \qquad &&\text{(divide both sides by $4$)} \end{aligned}
\[\begin{aligned}
\Longrightarrow \quad 4\mathbf{x} &= -3\mathbf{v} + \mathbf{w} \qquad &&\text{(add $3\mathbf{x}$, subtract $2\mathbf{v}$)} \\
\Longrightarrow \quad \mathbf{x} &= \tfrac{1}{4}(\mathbf{w} - 3\mathbf{v}). \qquad &&\text{(divide both sides by $4$)}
\end{aligned}\]
Introduction to Linear and M
\begin{array}{rrrrrrrr} v & + & w & + & x & + & y & - \\ v & - & 2 w & + & 2 x & + & 2 y & + \\ -v & + & w & + & x & & & + \\ & & & + & z & \\ 2 w & & - & x & + & y & - & 3 z \\ 2 v & & & - & 2 y & + & z & - \\ & & & & \end{array}
\[\begin{array}{rrrrrrrr} v & + & w & + & x & + & y & - \\ v & - & 2 w & + & 2 x & + & 2 y & + \\ -v & + & w & + & x & & & + \\ & & & + & z & \\ 2 w & & - & x & + & y & - & 3 z \\ 2 v & & & - & 2 y & + & z & - \\ & & & & \end{array}\]
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\begin{array}{r c r c r c r c r c r} v & + & w & + & x & + & y & - & z & - & 1 \\ v & - & 2w & + & 2x & + & 2y & + & 2z & + & 2 \\ -v & + & w & + & x & & & + & z & & \\ & & 2w & - & x & + & y & - & 3z & + & 3 \\ 2v & + & w & & & - & 2y & + & z & - & 4 \\ \end{array}
\[\begin{array}{r c r c r c r c r c r}
v & + & w & + & x & + & y & - & z & - & 1 \\
v & - & 2w & + & 2x & + & 2y & + & 2z & + & 2 \\
-v & + & w & + & x & & & + & z & & \\
& & 2w & - & x & + & y & - & 3z & + & 3 \\
2v & + & w & & & - & 2y & + & z & - & 4 \\
\end{array}\]
Introduction to Linear and M
\begin{array}{r} \\ \left.\xrightarrow{\begin{array}{c} z \\ R_{1} \end{array}} \begin{array}{c} S_{2} \quad S_{3} \\ R_{3}+R_{4} \\ R_{4}+R_{4} \\ \hline 0 \\ 0 \\ 0 \end{array} \left\lvert\, \begin{array}{c|cccc|c} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 \\ \hline \end{array}\left[\begin{array}{c|ccc|ccc} 1 & 0 & 0 & 1 & -2 & 1 & 0 \\ \hline 0 & 1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & -1 & 1
\[\begin{array}{r} \\ \left.\xrightarrow{\begin{array}{c} z \\ R_{1} \end{array}} \begin{array}{c} S_{2} \quad S_{3} \\ R_{3}+R_{4} \\ R_{4}+R_{4} \\ \hline 0 \\ 0 \\ 0 \end{array} \left\lvert\, \begin{array}{c|cccc|c} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 \\ \hline \end{array}\left[\begin{array}{c|ccc|ccc} 1 & 0 & 0 & 1 & -2 & 1 & 0 \\ \hline 0 & 1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & -1 & 1 & -2 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 1 \end{array}\right] \begin{array}{c} 2 \\ \text { pivot } \end{array}\right.\right] \end{array}\]
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\xrightarrow{\begin{array}{c}z\\R_1\end{array}}\begin{array}{c}S_2\quad S_3\\R_3+R_4\\R_4+R_4\end{array}\left[\begin{array}{c|ccc|ccc|c}1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 & 1 & 0 & 2 \\ 0 & 0 & 1 & 0 & -1 & -1 & -1 & -1 \\ 0 & 0 & 0 & 1 & -2 & 1 & -1 & -2\end{array}\right]\curvearrowright\text{pivot here}\xrightarrow{\begin{array}{c}-R_4\\R_1-R_4\\R_3+R_4\end{array}}\left[\begin{ar
\[\xrightarrow{\begin{array}{c}z\\R_1\end{array}}\begin{array}{c}S_2\quad S_3\\R_3+R_4\\R_4+R_4\end{array}\left[\begin{array}{c|ccc|ccc|c}1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 & 1 & 0 & 2 \\ 0 & 0 & 1 & 0 & -1 & -1 & -1 & -1 \\ 0 & 0 & 0 & 1 & -2 & 1 & -1 & -2\end{array}\right]\curvearrowright\text{pivot here}\xrightarrow{\begin{array}{c}-R_4\\R_1-R_4\\R_3+R_4\end{array}}\left[\begin{array}{c|ccc|ccc|c}1 & 0 & 0 & 1 & -2 & 1 & 0 & -2 \\ 0 & 1 & 0 & 0 & 1 & 1 & 0 & 2 \\ 0 & 0 & 1 & -1 & 1 & -2 & 0 & 1 \\ 0 & 0 & 0 & -1 & 2 & -1 & 1 & 2\end{array}\right]\]
Introduction to Linear and M
\begin{array}{rlrl} \operatorname{minimize}: & 3 y_{1}+3 y_{2}+3 y_{3}+3 y_{4}+3 y_{5} & \\ \text { subject to: } & y_{1}+y_{4}+y_{5} & \geq 1 \\ & y_{1}+y_{2}+y_{5} & \geq 1 \\ & y_{1}+y_{2}+y_{3} & \geq 1 \\ & y_{2}+y_{3}+y_{4} & \geq 1 \\ & y_{3}+y_{4}+y_{5} \geq 1 \\ y_{2}, \quad y_{3}, \quad y_{4}, \quad y_{5} & \geq 0 \end{array}
\[\begin{array}{rlrl} \operatorname{minimize}: & 3 y_{1}+3 y_{2}+3 y_{3}+3 y_{4}+3 y_{5} & \\ \text { subject to: } & y_{1}+y_{4}+y_{5} & \geq 1 \\ & y_{1}+y_{2}+y_{5} & \geq 1 \\ & y_{1}+y_{2}+y_{3} & \geq 1 \\ & y_{2}+y_{3}+y_{4} & \geq 1 \\ & y_{3}+y_{4}+y_{5} \geq 1 \\ y_{2}, \quad y_{3}, \quad y_{4}, \quad y_{5} & \geq 0 \end{array}\]
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v = (6, 3, 3), (-2, 0, 6) = (8, 3, 0)
\[v = (6, 3, 3), (-2, 0, 6) = (8, 3, 0)\]
Introduction to Linear and M
\begin{aligned} {\left[\begin{array}{lll} 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 3 & 9 \end{array}\right] } & \xrightarrow{\substack{R_{2}-R_{1} \\ R_{3}-R_{1}}}\left[\begin{array}{lll} 1 & 1 & 1 \\ 0 & 1 & 3 \\ 0 & 2 & 8 \end{array}\right] \\ & \xrightarrow{\substack{R_{1}-R_{2} \\ R_{3}-2 R_{2}}}\left[\begin{array}{ccc} 1 & 0 & -2 \\ 0 & 1 & 3 \\ 0 & 0 & 2 \end{array}\right] \\ \frac{\frac{1}{2} R_{3}}{}
\[\begin{aligned} {\left[\begin{array}{lll} 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 3 & 9 \end{array}\right] } & \xrightarrow{\substack{R_{2}-R_{1} \\ R_{3}-R_{1}}}\left[\begin{array}{lll} 1 & 1 & 1 \\ 0 & 1 & 3 \\ 0 & 2 & 8 \end{array}\right] \\ & \xrightarrow{\substack{R_{1}-R_{2} \\ R_{3}-2 R_{2}}}\left[\begin{array}{ccc} 1 & 0 & -2 \\ 0 & 1 & 3 \\ 0 & 0 & 2 \end{array}\right] \\ \frac{\frac{1}{2} R_{3}}{}\left[\begin{array}{ccc} 1 & 0 & -2 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{array}\right] & \xrightarrow{\substack{R_{1}+2 R_{3} \\ R_{2}-3 R_{3}}}\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] . \end{aligned}\]
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\mathbf{w} = (6,3,-3) - (-2,0,6) = (8,3,-9).
\[\mathbf{w} = (6,3,-3) - (-2,0,6) = (8,3,-9).\]
Introduction to Linear and M
\left.3 x ^ { 2 } - 5 x - 2 \longdiv { \frac { x ^ { 2 } + 3 x + 1 } { 3 x ^ { 4 } + 4 x ^ { 3 } - 1 4 x ^ { 2 } - 1 1 x - 2 } }, \frac{3 x^{4}-5 x^{3}-2 x^{2}}{9 x^{3}-12 x^{2}-11 x-2}, 2, \frac{9 x^{3}-15 x^{2}-6 x}{3 x^{2}-5 x-2}, 3 \frac{3 x^{2}-5 x-2}{0}\right)
\[\left.3 x ^ { 2 } - 5 x - 2 \longdiv { \frac { x ^ { 2 } + 3 x + 1 } { 3 x ^ { 4 } + 4 x ^ { 3 } - 1 4 x ^ { 2 } - 1 1 x - 2 } }, \frac{3 x^{4}-5 x^{3}-2 x^{2}}{9 x^{3}-12 x^{2}-11 x-2}, 2, \frac{9 x^{3}-15 x^{2}-6 x}{3 x^{2}-5 x-2}, 3 \frac{3 x^{2}-5 x-2}{0}\right)\]
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\begin{array}{r} x^2+3x+1\phantom{xxxxxxxxx}\\ 3x^2-5x-2\overline{)3x^4+4x^3-14x^2-11x-2}\\ \underline{3x^4-5x^3-2x^2\phantom{xxxxxxxxxx}}\\ 9x^3-12x^2-11x-2\\ \underline{9x^3-15x^2-6x\phantom{xxx}}\\ 3x^2-5x-2\\ \underline{3x^2-5x-2}\\ 0 \end{array}
\[\begin{array}{r}
x^2+3x+1\phantom{xxxxxxxxx}\\
3x^2-5x-2\overline{)3x^4+4x^3-14x^2-11x-2}\\
\underline{3x^4-5x^3-2x^2\phantom{xxxxxxxxxx}}\\
9x^3-12x^2-11x-2\\
\underline{9x^3-15x^2-6x\phantom{xxx}}\\
3x^2-5x-2\\
\underline{3x^2-5x-2}\\
0
\end{array}\]
Numerical Linear Algebra and
\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} &
\[\begin{aligned} & \boldsymbol{Q}_{1}:=\left[\begin{array}{llll} \boldsymbol{q}_{1}, \ldots, \boldsymbol{q}_{n} \end{array}\right], \\ & \boldsymbol{R}_{1}:= \\ & \boldsymbol{q}_{j}:=\frac{\boldsymbol{v}_{j}}{\left\|\boldsymbol{v}_{j}\right\|_{2}}, \quad j=1, \ldots, n \text { and } \\ & {\left[\begin{array}{ccccc} \left\|\boldsymbol{v}_{1}\right\|_{2} & \boldsymbol{a}_{2}^{T} \boldsymbol{q}_{1} & \boldsymbol{a}_{3}^{T} \boldsymbol{q}_{1} & \cdots & \boldsymbol{a}_{n-1}^{T} \boldsymbol{q}_{1} \\ 0 & \left\|\boldsymbol{v}_{2}\right\|_{2} & \boldsymbol{a}_{3}^{T} \boldsymbol{q}_{2} & \cdots & \boldsymbol{a}_{n-1}^{T} \boldsymbol{q}_{2} \\ & 0 & \left\|\boldsymbol{v}_{3}\right\|_{2} & \cdots & \boldsymbol{a}_{n-1}^{T} \boldsymbol{q}_{2} \\ & & \ddots & \ddots & \boldsymbol{a}_{n}^{T} \boldsymbol{q}_{3} \\ & & & \ddots & \vdots \\ & & & \left\|\boldsymbol{v}_{n-1}\right\|_{2} & \boldsymbol{a}_{n}^{T} \boldsymbol{q}_{n-1} \\ & & & 0 & \left\|\boldsymbol{v}_{n}\right\|_{2} \end{array}\right] .} \end{aligned}\]
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\begin{aligned} & \boldsymbol{Q}_{1}:=\left[\boldsymbol{q}_{1}, \ldots, \boldsymbol{q}_{n}\right], \quad \boldsymbol{q}_{j}:=\frac{\boldsymbol{v}_{j}}{\left\|\boldsymbol{v}_{j}\right\|_{2}}, \quad j=1, \ldots, n \text{ and} \\ & \boldsymbol{R}_{1}:=\left[\begin{array}{cccccc} \left\|\boldsymbol{v}_{1}\right\|_{2} & \boldsymbol{a}_{2}^{T} \boldsymbol{q}_{1} & \boldsymbol{a}_{3}^{T} \boldsymbol{q}_{
\[\begin{aligned}
& \boldsymbol{Q}_{1}:=\left[\boldsymbol{q}_{1}, \ldots, \boldsymbol{q}_{n}\right], \quad \boldsymbol{q}_{j}:=\frac{\boldsymbol{v}_{j}}{\left\|\boldsymbol{v}_{j}\right\|_{2}}, \quad j=1, \ldots, n \text{ and} \\
& \boldsymbol{R}_{1}:=\left[\begin{array}{cccccc}
\left\|\boldsymbol{v}_{1}\right\|_{2} & \boldsymbol{a}_{2}^{T} \boldsymbol{q}_{1} & \boldsymbol{a}_{3}^{T} \boldsymbol{q}_{1} & \cdots & \boldsymbol{a}_{n-1}^{T} \boldsymbol{q}_{1} & \boldsymbol{a}_{n}^{T} \boldsymbol{q}_{1} \\
0 & \left\|\boldsymbol{v}_{2}\right\|_{2} & \boldsymbol{a}_{3}^{T} \boldsymbol{q}_{2} & \cdots & \boldsymbol{a}_{n-1}^{T} \boldsymbol{q}_{2} & \boldsymbol{a}_{n}^{T} \boldsymbol{q}_{2} \\
& 0 & \left\|\boldsymbol{v}_{3}\right\|_{2} & \cdots & \boldsymbol{a}_{n-1}^{T} \boldsymbol{q}_{3} & \boldsymbol{a}_{n}^{T} \boldsymbol{q}_{3} \\
& & \ddots & \ddots & \vdots & \vdots \\
& & & \ddots & \left\|\boldsymbol{v}_{n-1}\right\|_{2} & \boldsymbol{a}_{n}^{T} \boldsymbol{q}_{n-1} \\
& & & & 0 & \left\|\boldsymbol{v}_{n}\right\|_{2}
\end{array}\right].
\end{aligned}\]
Numerical Linear Algebra and
\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} .
\[\boldsymbol{J}:=\operatorname{diag}\left(\boldsymbol{U}_{1}, \boldsymbol{U}_{2}\right)=\left[\begin{array}{ccccccc} 2 & 1 & 0 & & & & \\ 0 & 2 & 1 & & & & \\ 0 & 0 & 2 & & 1 & & \\ & & 0 & 1 & & \\ & & 0 & 2 & & & \\ & & & & & 0 & \\ & & & & & 0 & 1 \end{array}\right] \in \mathbb{R}^{8 \times 8} .\]
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d_i for i = 1, \ldots, n, b_{i+1,i} := a_i, b_{i,i+1} := c_i for i \text{otherwise.} \in \mathbb{C}^{m \times n} and 1 \le i_1 < i_2 < \cdots < i_r \le m, 1 \le j \text{matrix } A(\boldsymbol{i}, \boldsymbol{j}) \in \mathbb{C}^{r \times c} \text{ is the submatrix of } A \text{ c}
\[d_i for i = 1, \ldots, n, b_{i+1,i} := a_i, b_{i,i+1} := c_i for i
\text{otherwise.}
\in \mathbb{C}^{m \times n} and 1 \le i_1 < i_2 < \cdots < i_r \le m, 1 \le j
\text{matrix } A(\boldsymbol{i}, \boldsymbol{j}) \in \mathbb{C}^{r \times c} \text{ is the submatrix of } A \text{ c}\]
Numerical Linear Algebra and
\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_{
\[\begin{aligned} \boldsymbol{A}^{*} \boldsymbol{A} \boldsymbol{x} & =\left[\begin{array}{ccc} 1 & \cdots & 1 \\ t_{1} & \cdots & t_{m} \end{array}\right]\left[\begin{array}{c} 1 \\ \vdots \\ 1 \\ 1 \\ m \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]=\left[\begin{array}{cc} m & \sum t_{k} \\ \sum t_{k} & \sum t_{k}^{2} \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right], \\ & =\left[\begin{array}{ccc} 1 & \cdots & 1 \\ t_{1} & \cdots & t_{m} \end{array}\right]\left[\begin{array}{c} y_{1} \\ \vdots \\ y_{m} \end{array}\right]=\left[\begin{array}{c} \sum y_{k} \\ \sum t_{k} y_{k} \end{array}\right]=\boldsymbol{A}^{*} \boldsymbol{b}, \end{aligned}\]
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\text{nal if } a_{ij} = 0 \text{ for } i \neq j. \\ \text{triangular or \textbf{right triangular} if } a_{ij} = 0 \text{ for } i > j. \\ \text{triangular or \textbf{left triangular} if } a_{ij} = 0 \text{ for } i < j. \\ \textbf{Hessenberg if } a_{ij} = 0 \text{ for } i > j + 1. \\ \textbf{Hessenberg if } a_{ij} = 0 \text{ for } i < j + 1. \\ \text{gonal if } a_{ij} = 0 \text{ for } |i - j| > 1. \
\[\text{nal if } a_{ij} = 0 \text{ for } i \neq j. \\
\text{triangular or \textbf{right triangular} if } a_{ij} = 0 \text{ for } i > j. \\
\text{triangular or \textbf{left triangular} if } a_{ij} = 0 \text{ for } i < j. \\
\textbf{Hessenberg if } a_{ij} = 0 \text{ for } i > j + 1. \\
\textbf{Hessenberg if } a_{ij} = 0 \text{ for } i < j + 1. \\
\text{gonal if } a_{ij} = 0 \text{ for } |i - j| > 1. \\
\text{ded if } a_{ij} = 0 \text{ for } |i - j| > d. \\[1ex]
\text{the following notations for diagonal- and tridiagonal } n \times n\]
Numerical Linear Algebra and
\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
\[\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}\]
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\begin{aligned} a_{11}x_{1} + a_{12}x_{2} + \cdots + a_{1n}x_{n} &= b_{1} \\ a_{21}x_{1} + a_{22}x_{2} + \cdots + a_{2n}x_{n} &= b_{2} \\ \vdots \qquad \vdots \qquad\qquad\vdots \qquad &\vdots \\ a_{m1}x_{1} + a_{m2}x_{2} + \cdots + a_{mn}x_{n} &= b_{m} \end{aligned}
\[\begin{aligned}
a_{11}x_{1} + a_{12}x_{2} + \cdots + a_{1n}x_{n} &= b_{1} \\
a_{21}x_{1} + a_{22}x_{2} + \cdots + a_{2n}x_{n} &= b_{2} \\
\vdots \qquad \vdots \qquad\qquad\vdots \qquad &\vdots \\
a_{m1}x_{1} + a_{m2}x_{2} + \cdots + a_{mn}x_{n} &= b_{m}
\end{aligned}\]
Numerical Linear Algebra and
\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}
\[\begin{aligned} a_{i i} & =2 d, i=1, \ldots, n, \\ a_{i, i+1}=a_{i+1, i} & =a, \quad i=1, \ldots, n-1, \quad i \neq m, 2 m, \ldots,(m-1) m, \\ a_{i, i+m}=a_{i+m, i} & =a, \quad i=1, \ldots, n-m, \\ a_{i j} & =0, \text { otherwise } \end{aligned}\]
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\begin{bmatrix} \vdots & \vdots & \cdots & \vdots \\ a_{r_2, c_1} & a_{r_2, c_1+1} & \cdots & a_{r_2, c_2} \end{bmatrix}
\[\begin{bmatrix} \vdots & \vdots & \cdots & \vdots \\ a_{r_2, c_1} & a_{r_2, c_1+1} & \cdots & a_{r_2, c_2} \end{bmatrix}\]
Numerical Linear Algebra and
\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} \boldsy
\[\begin{array}{rl} \boldsymbol{T} \boldsymbol{V}+\boldsymbol{V} \boldsymbol{T} & =h^{2} \boldsymbol{F} \\ \boldsymbol{V}^{\boldsymbol{V}=\boldsymbol{S} \boldsymbol{X} \boldsymbol{S}} \boldsymbol{T} \boldsymbol{S} \boldsymbol{X} \boldsymbol{S}+\boldsymbol{S} \boldsymbol{X} \boldsymbol{S} \boldsymbol{T} & =h^{2} \boldsymbol{F} \\ \stackrel{\boldsymbol{S}() \boldsymbol{S}}{\Longleftrightarrow} \boldsymbol{S} \boldsymbol{T} \boldsymbol{S} \boldsymbol{X} \boldsymbol{S}^{2}+\boldsymbol{S}^{2} \boldsymbol{X} \boldsymbol{S} \boldsymbol{T} \boldsymbol{S} & =h^{2} \boldsymbol{S} \boldsymbol{F} \boldsymbol{S}=h^{2} \boldsymbol{G} \\ \stackrel{\boldsymbol{T}}{\Leftrightarrow} \boldsymbol{S}^{\mathbf{2}} \boldsymbol{D} & \boldsymbol{S} \boldsymbol{X} \boldsymbol{S}^{2}+\boldsymbol{S}^{2} \boldsymbol{X} \boldsymbol{S}^{2} \boldsymbol{D} \\ \boldsymbol{S}^{2} & =h^{2} \boldsymbol{G} \\ \Longleftrightarrow & \boldsymbol{I} /(2 h) \end{array}\]
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\begin{array}{rl} \boldsymbol{TV}+\boldsymbol{VT} & = h^{2}\boldsymbol{F} \\ \stackrel{V=SXS}{\Longleftrightarrow} \boldsymbol{TSXS}+\boldsymbol{SXST} & = h^{2}\boldsymbol{F} \\ \stackrel{S()S}{\Longleftrightarrow} \boldsymbol{STSXS^{2}}+\boldsymbol{S^{2}XSTS} & = h^{2}\boldsymbol{SFS}=h^{2}\boldsymbol{G} \\ \stackrel{TS=SD}{\Longleftrightarrow} \boldsymbol{S^{2}DXS^{2}}+\boldsymbol{S^{2}XS^{2}D}
\[\begin{array}{rl}
\boldsymbol{TV}+\boldsymbol{VT} & = h^{2}\boldsymbol{F} \\
\stackrel{V=SXS}{\Longleftrightarrow} \boldsymbol{TSXS}+\boldsymbol{SXST} & = h^{2}\boldsymbol{F} \\
\stackrel{S()S}{\Longleftrightarrow} \boldsymbol{STSXS^{2}}+\boldsymbol{S^{2}XSTS} & = h^{2}\boldsymbol{SFS}=h^{2}\boldsymbol{G} \\
\stackrel{TS=SD}{\Longleftrightarrow} \boldsymbol{S^{2}DXS^{2}}+\boldsymbol{S^{2}XS^{2}D} & = h^{2}\boldsymbol{G} \\
\stackrel{S^{2}=I/(2h)}{\Longleftrightarrow} \boldsymbol{DX}+\boldsymbol{XD} & = 4h^{4}\boldsymbol{G}.
\end{array}\]
Numerical Linear Algebra and
\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
\[\begin{array}{lll} \left(\boldsymbol{F}_{2 m} \boldsymbol{P}_{2 m}\right)_{p, q} & =\omega_{2 m}^{j(2 k)} & =\omega_{m}^{j k}, \\ \left(\boldsymbol{F}_{2 m} \boldsymbol{P}_{2 m}\right)_{p+m, q} & =\omega_{2 m}^{(j+m)(2 k)} & =\omega_{m}^{j+m) k} \\ \left(\boldsymbol{F}_{2 m} \boldsymbol{P}_{2 m}\right)_{p, q+m} & =\omega_{2 m}^{j(2 k+1)} & =\omega_{2 m}^{j} \omega_{m}^{j k}, \\ \left(\boldsymbol{F}_{2 m} \boldsymbol{P}_{2 m}\right)_{p+m, q+m} & \left.=\omega_{2 m}^{j+m}\right)(2 k+1) & \left.=\omega_{2 m}^{j+m} \omega_{m}^{j+m}\right) k \\ & \\ (j+m & & \end{array}\]
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(DX + XD)_{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} \quad \text{for all } j, k
\[(DX + XD)_{j,k} = \sum_{\ell=1}^{m} d_{j,\ell} x_{\ell,k} + \sum_{\ell=1}^{m} x_{j,\ell} d_{\ell,k} = \lambda_j x_{j,k} + \lambda_k x_{j,k} \quad \text{for all } j, k\]
Numerical Linear Algebra and
\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] .
\[\boldsymbol{G}_{\omega}=\left[\begin{array}{ll} l & 0 \\ 0 & 1 \end{array}\right]-\left[\begin{array}{cc} \omega / 2 & 0 \\ \omega^{2} / 4 & \omega / 2 \end{array}\right]\left[\begin{array}{cc} 2 & -1 \\ -1 & 2 \end{array}\right]=\left[\begin{array}{cc} 1-\omega & \omega / 2 \\ \omega(1-\omega) / 2 & 1-\omega+\omega^{2} / 4 \end{array}\right] .\]
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\langle j_1, j_2 \mid j_c \rangle \qquad \begin{bmatrix} \cdot & \cdot & \cdots & \cdot \\ a_{i_r, j_1} & a_{i_r, j_2} & \cdots & a_{i_r, j_c} \end{bmatrix}
\[\langle j_1, j_2 \mid j_c \rangle \qquad \begin{bmatrix} \cdot & \cdot & \cdots & \cdot \\ a_{i_r, j_1} & a_{i_r, j_2} & \cdots & a_{i_r, j_c} \end{bmatrix}\]