行列行列の分割小行列、積

学習環境

線形代数演習〈理工系の数学入門コース/演習新装版〉 (浅野功義(著)、大関清太(著)、岩波書店)の第2章(行列)、2-4(行列の分割)、問題1の解答を求めてみる。

行列Aを

\begin{array}{l} A_{11} = [\begin{matrix} 1 & - 1 \\ 0 & 2 \end{matrix}] \\ A_{12} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}] \\ A_{21} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}] \\ A_{22} = [\begin{matrix} 1 & - 1 \\ 0 & 2 \end{matrix}] \\ A = [\begin{matrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix}] \end{array}

と分割。

行列Bを

\begin{array}{l} B_{11} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}] \\ B_{12} = [\begin{matrix} 2 & 3 \\ 0 & 1 \end{matrix}] \\ B_{21} = [\begin{matrix} 2 & 3 \\ 0 & 1 \end{matrix}] \\ B_{22} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}] \\ B = [\begin{matrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{matrix}] \end{array}

と分割。

A^{2} = [\begin{matrix} A_{11} A_{11} + A_{12} A_{21} & A_{11} A_{12} + A_{12} A_{22} \\ A_{21} A_{11} + A_{22} A_{21} & A_{21} A_{12} + A_{22} A_{22} \end{matrix}]

= [\begin{matrix} A_{11}^{2} & O \\ O & A_{22}^{2} \end{matrix}]

A_{11}^{2} = [\begin{matrix} 1 & - 1 \\ 0 & 2 \end{matrix}] [\begin{matrix} 1 & - 1 \\ 0 & 2 \end{matrix}] = [\begin{matrix} 1 & - 3 \\ 0 & 4 \end{matrix}]

A_{22}^{2} = [\begin{matrix} 1 & - 3 \\ 0 & 4 \end{matrix}]

A^{2} = [\begin{matrix} 1 & - 3 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 1 & - 3 \\ 0 & 0 & 0 & 4 \end{matrix}]

B^{2} = [\begin{matrix} B_{11} B_{11} + B_{12} B_{21} & B_{11} B_{12} + B_{12} B_{22} \\ B_{21} B_{11} + B_{22} B_{21} & B_{21} B_{12} + B_{22} B_{22} \end{matrix}]

= [\begin{matrix} B_{12} B_{21} & O \\ O & B_{21} B_{12} \end{matrix}]

[\begin{matrix} 2 & 3 \\ 0 & 1 \end{matrix}] [\begin{matrix} 2 & 3 \\ 0 & 1 \end{matrix}] = [\begin{matrix} 4 & 9 \\ 0 & 1 \end{matrix}]

B^{2} = [\begin{matrix} 4 & 9 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 4 & 9 \\ 0 & 0 & 0 & 1 \end{matrix}]

A B = [\begin{matrix} 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{matrix}]

= [\begin{matrix} O & A_{11} B_{12} \\ A_{22} B_{21} & O \end{matrix}]

A_{11} B_{12} = [\begin{matrix} 1 & - 1 \\ 0 & 2 \end{matrix}] [\begin{matrix} 2 & 3 \\ 0 & 1 \end{matrix}] = [\begin{matrix} 2 & 2 \\ 0 & 2 \end{matrix}]

A B = [\begin{matrix} 0 & 0 & 2 & 2 \\ 0 & 0 & 0 & 2 \\ 2 & 2 & 0 & 0 \\ 0 & 2 & 0 & 0 \end{matrix}]

B A = [\begin{matrix} B_{11} A_{11} + B_{12} A_{21} & B_{11} A_{12} + B_{12} A_{22} \\ B_{21} A_{11} + B_{22} A_{21} & B_{21} A_{12} + B_{22} A_{22} \end{matrix}]

= [\begin{matrix} O & B_{12} A_{22} \\ B_{21} A_{11} & O \end{matrix}]

B_{12} A_{22} = B_{21} A_{11} = [\begin{matrix} 2 & 3 \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & - 1 \\ 0 & 2 \end{matrix}] = [\begin{matrix} 2 & 4 \\ 0 & 2 \end{matrix}]

B A = [\begin{matrix} 0 & 0 & 2 & 4 \\ 0 & 0 & 0 & 2 \\ 2 & 4 & 0 & 0 \\ 0 & 2 & 0 & 0 \end{matrix}]

コード(Wolfram Language, Jupyter)

a = {{1, -1, 0, 0}, {0, 2, 0, 0}, {0, 0, 1, -1}, {0, 0, 0, 2}}

{{1, -1, 0, 0}, {0, 2, 0, 0}, {0, 0, 1, -1}, {0, 0, 0, 2}}

b = {{0, 0, 2, 3}, {0, 0, 0, 1}, {2, 3, 0, 0}, {0, 1, 0, 0}}

{{0, 0, 2, 3}, {0, 0, 0, 1}, {2, 3, 0, 0}, {0, 1, 0, 0}}

a . a

{{1, -3, 0, 0}, {0, 4, 0, 0}, {0, 0, 1, -3}, {0, 0, 0, 4}}

b . b

{{4, 9, 0, 0}, {0, 1, 0, 0}, {0, 0, 4, 9}, {0, 0, 0, 1}}

a . b

{{0, 0, 2, 2}, {0, 0, 0, 2}, {2, 2, 0, 0}, {0, 2, 0, 0}}

b . a

{{0, 0, 2, 4}, {0, 0, 0, 2}, {2, 4, 0, 0}, {0, 2, 0, 0}}