第14章 14.9

$X = \begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix}$ $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ とおくと

$\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix} = \begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix}\begin{pmatrix} a & b \\ c & d \end{pmatrix}$

$\begin{pmatrix} ax_{11}+bx_{21} & ax_{12}+bx_{22} \\ cx_{11}+dx_{21} & cx_{12}+dx_{22} \end{pmatrix} = \begin{pmatrix} ax_{11}+cx_{12} & bx_{11}+dx_{12} \\ ax_{21}+cx_{22} & bx_{21}+dx_{22} \end{pmatrix}$

これより

$ax_{11}+bx_{21} = ax_{11}+cx_{12}$ ① $ax_{12}+bx_{22} = bx_{11}+dx_{12}$ ②

$cx_{11}+dx_{21} = ax_{21}+cx_{22}$ ③ $cx_{12}+dx_{22} = bx_{21}+dx_{22}$ ④

これら４式を行列に書き直して

$\begin{pmatrix} 0 & x_{21} & -x_{12} & 0 \\ x_{12} & -x_{11}+x_{22} & 0 & -x_{12} \\ x_{21} & 0 & -x_{11}+x_{22} & -x_{21} \\ 0 & x_{21} & -x_{12} & 0 \end{pmatrix}\begin{pmatrix} a \\ b \\ c \\ d \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}$

$\begin{pmatrix} 0 & x_{21} & -x_{12} & 0 \\ x_{12} & -x_{11}+x_{22} & 0 & -x_{12} \\ x_{21} & 0 & -x_{11}+x_{22} & -x_{21} \\ 0 & x_{21} & -x_{12} & 0 \end{pmatrix} \rightarrow \begin{pmatrix} 0 & x_{21} & -x_{12} & 0 \\ x_{12} & -x_{11}+x_{22} & 0 & -x_{12} \\ x_{21} & 0 & -x_{11}+x_{22} & -x_{21} \\ 0 & 0 & 0 & 0 \end{pmatrix}$

$\rightarrow \begin{pmatrix} 0 & x_{21} & -x_{12} & 0 \\ x_{12} & -x_{11}+x_{22} & 0 & -x_{12} \\ 0 & -x_{21}(-x_{11}+x_{22}) & -x_{11}+x_{22} & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \rightarrow \begin{pmatrix} 0 & x_{21} & -x_{12} & 0 \\ x_{12} & -x_{11}+x_{22} & 0 & -x_{12} \\ 0 & -x_{21} & 1 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}$

$\rightarrow \begin{pmatrix} 0 & x_{21} & -x_{12} & 0 \\ x_{12} & -x_{11}+x_{22} & 0 & -x_{12} \\ 0 & 0 & 1+x_{12} & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \rightarrow \begin{pmatrix} x_{12} & -x_{11}+x_{22} & 0 & -x_{12} \\ 0 & x_{21} & -x_{12} & 0 \\ 0 & 0 & 1+x_{12} & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}$

$\rightarrow \begin{pmatrix} x_{12} & -x_{11}+x_{22} & 0 & -x_{12} \\ 0 & x_{21} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \rightarrow \begin{pmatrix} x_{12} & 0 & 0 & -x_{12} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \rightarrow \begin{pmatrix} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}$

これから

$\begin{pmatrix} a \\ b \\ c \\ d \end{pmatrix} = \alpha \begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix}$

$\begin{pmatrix} \alpha & 0 \\ 0 & \alpha \end{pmatrix} \quad (\alpha \text{ は任意})$