式展開: 余因子行列

余因子行列

余因子行列

i

行

j

列の要素を

a_{i j}

とする

n

次正方行列

A

において，

i

行と

j

列の要素を取り除いた残りの行列を

M_{i j}

とした時，

{\tilde{a}}_{i j} = (- 1)^{i + j} | M_{j i} | (添 え 字 の 順 序 に 注 意)

を要素とする行列

\tilde{A}

を余因子行列と呼ぶ．

2x2行列

\begin{array}{rcl} A_{2 \times 2} & = & [\begin{array}{c} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}] \\ {\tilde{A}}_{2 \times 2} & = & [\begin{array}{c} (- 1)^{1 + 1} | M_{11} | & (- 1)^{1 + 2} | M_{21} | \\ (- 1)^{2 + 1} | M_{12} | & (- 1)^{2 + 2} | M_{22} | \end{array}] \\ = & [\begin{array}{c} | M_{11} | & - | M_{21} | \\ - | M_{12} | & | M_{22} | \end{array}] \\ = & {[\begin{array}{c} | M_{11} | & - | M_{12} | \\ - | M_{21} | & | M_{22} | \end{array}]}^{T} \dots A^{T} : A の 転 置 行 列 \\ = & {[\begin{array}{c} a_{22} & - a_{21} \\ - a_{12} & a_{11} \end{array}]}^{T} \\ = & [\begin{array}{c} a_{22} & - a_{12} \\ - a_{21} & a_{11} \end{array}] \\ A_{2 \times 2} & = & [\begin{array}{c} a & b \\ c & d \end{array}] \\ {\tilde{A}}_{2 \times 2} & = & [\begin{array}{c} d & - b \\ - c & a \end{array}] \end{array}

3x3行列

\begin{array}{rcl} A_{3 \times 3} & = & [\begin{array}{c} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}] \\ {\tilde{A}}_{3 \times 3} & = & [\begin{array}{c} (- 1)^{1 + 1} | M_{11} | & (- 1)^{1 + 2} | M_{21} | & (- 1)^{1 + 3} | M_{31} | \\ (- 1)^{2 + 1} | M_{12} | & (- 1)^{2 + 2} | M_{22} | & (- 1)^{2 + 3} | M_{32} | \\ (- 1)^{3 + 1} | M_{13} | & (- 1)^{3 + 2} | M_{23} | & (- 1)^{3 + 3} | M_{33} | \end{array}] \\ = & [\begin{array}{c} | M_{11} | & - | M_{21} | & | M_{31} | \\ - | M_{12} | & | M_{22} | & - | M_{32} | \\ | M_{13} | & - | M_{23} | & | M_{33} | \end{array}] \\ = & {[\begin{array}{c} | M_{11} | & - | M_{12} | & | M_{13} | \\ - | M_{21} | & | M_{22} | & - | M_{23} | \\ | M_{31} | & - | M_{32} | & | M_{33} | \end{array}]}^{T} \dots A^{T} : A の 転 置 行 列 \\ = & {[\begin{array}{c} | \begin{matrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{matrix} | & - | \begin{matrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{matrix} | & | \begin{matrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{matrix} | \\ - | \begin{matrix} a_{12} & a_{13} \\ a_{32} & a_{33} \end{matrix} | & | \begin{matrix} a_{11} & a_{13} \\ a_{31} & a_{33} \end{matrix} | & - | \begin{matrix} a_{11} & a_{12} \\ a_{31} & a_{32} \end{matrix} | \\ | \begin{matrix} a_{12} & a_{13} \\ a_{22} & a_{23} \end{matrix} | & - | \begin{matrix} a_{11} & a_{13} \\ a_{21} & a_{23} \end{matrix} | & | \begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} | \end{array}]}^{T} \\ = & {[\begin{array}{c} a_{22} a_{33} - a_{23} a_{32} & - (a_{21} a_{33} - a_{23} a_{31}) & a_{21} a_{32} - a_{22} a_{31} \\ - (a_{12} a_{33} - a_{13} a_{32}) & a_{11} a_{33} - a_{13} a_{31} & - (a_{11} a_{32} - a_{12} a_{31}) \\ a_{12} a_{23} - a_{13} a_{22} & - (a_{11} a_{23} - a_{13} a_{21}) & a_{11} a_{22} - a_{12} a_{21} \end{array}]}^{T} \\ = & [\begin{array}{c} a_{22} a_{33} - a_{23} a_{32} & - (a_{12} a_{33} - a_{13} a_{32}) & a_{12} a_{23} - a_{13} a_{22} \\ - (a_{21} a_{33} - a_{23} a_{31}) & a_{11} a_{33} - a_{13} a_{31} & - (a_{11} a_{23} - a_{13} a_{21}) \\ a_{21} a_{32} - a_{22} a_{31} & - (a_{11} a_{32} - a_{12} a_{31}) & a_{11} a_{22} - a_{12} a_{21} \end{array}] \\ = & [\begin{array}{c} a_{22} a_{33} - a_{23} a_{32} & a_{13} a_{32} - a_{12} a_{33} & a_{12} a_{23} - a_{13} a_{22} \\ a_{23} a_{31} - a_{21} a_{33} & a_{11} a_{33} - a_{13} a_{31} & a_{13} a_{21} - a_{11} a_{23} \\ a_{21} a_{32} - a_{22} a_{31} & a_{12} a_{31} - a_{11} a_{32} & a_{11} a_{22} - a_{12} a_{21} \end{array}] \end{array}

余因子行列と元の行列との積

\begin{array}{r} \tilde{A} A = A \tilde{A} = | A | I \end{array}

余因子行列と元の行列との積 3x3の例

\begin{array}{rcl} {\tilde{A}}_{3 \times 3} A_{3 \times 3} & = & [\begin{array}{c} | M_{11} | & - | M_{21} | & | M_{31} | \\ - | M_{12} | & | M_{22} | & - | M_{32} | \\ | M_{13} | & - | M_{23} | & | M_{33} | \end{array}] [\begin{array}{c} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}] \\ = & [\begin{array}{c} | M_{11} | a_{11} - | M_{21} | a_{21} + | M_{31} | a_{31} & | M_{11} | a_{12} - | M_{21} | a_{22} + | M_{31} | a_{32} & | M_{11} | a_{13} - | M_{21} | a_{23} + | M_{31} | a_{33} \\ - | M_{12} | a_{11} + | M_{22} | a_{21} - | M_{32} | a_{31} & - | M_{12} | a_{12} + | M_{22} | a_{22} - | M_{32} | a_{32} & - | M_{12} | a_{13} + | M_{22} | a_{23} - | M_{32} | a_{33} \\ | M_{13} | a_{11} - | M_{23} | a_{21} + | M_{33} | a_{31} & | M_{13} | a_{12} - | M_{23} | a_{22} + | M_{33} | a_{32} & | M_{13} | a_{13} - | M_{23} | a_{23} + | M_{33} | a_{33} \end{array}] \\ = & [\begin{array}{c} | A_{3 \times 3} | & | M_{11} | a_{12} - | M_{21} | a_{22} + | M_{31} | a_{32} & | M_{11} | a_{13} - | M_{21} | a_{23} + | M_{31} | a_{33} \\ - | M_{12} | a_{11} + | M_{22} | a_{21} - | M_{32} | a_{31} & | A_{3 \times 3} | & - | M_{12} | a_{13} + | M_{22} | a_{23} - | M_{32} | a_{33} \\ | M_{13} | a_{11} - | M_{23} | a_{21} + | M_{33} | a_{31} & | M_{13} | a_{12} - | M_{23} | a_{22} + | M_{33} | a_{32} & | A_{3 \times 3} | \end{array}] \dots 計 算 は 下 記 “ 対 角 要 素 に つ い て ” に 記 載 \\ = & [\begin{array}{c} | A_{3 \times 3} | & 0 & 0 \\ 0 & | A_{3 \times 3} | & 0 \\ 0 & 0 & | A_{3 \times 3} | \end{array}] \dots 計 算 は 下 記 “ そ の 他 要 素 に つ い て ” に 記 載 \\ = & | A_{3 \times 3} | [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}] \\ = & | A_{3 \times 3} | I_{3} \dots I_{3} : 3 \times 3 の 単 位 行 列 \end{array}

対角要素について

\begin{array}{rcl} | A_{3 \times 3} | & = & | \begin{array}{c} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} | \\ = & a_{11} M_{11} - a_{21} M_{21} + a_{31} M_{31} \\ = & - a_{12} M_{12} + a_{22} M_{22} - a_{32} M_{32} \\ = & a_{13} M_{13} - a_{23} M_{23} + a_{33} M_{33} \end{array}

その他要素について

\begin{array}{rcl} | M_{11} | a_{12} - | M_{21} | a_{22} + | M_{31} | a_{32} & = & | \begin{array}{c} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} | a_{12} - | \begin{array}{c} a_{12} & a_{13} \\ a_{32} & a_{33} \end{array} | a_{22} + | \begin{array}{c} a_{12} & a_{13} \\ a_{22} & a_{23} \end{array} | a_{32} \\ = & a_{12} (a_{22} a_{33} - a_{23} a_{32}) - a_{22} (a_{12} a_{33} - a_{13} a_{32}) + a_{32} (a_{12} a_{23} - a_{13} a_{22}) \\ = & a_{12} a_{22} a_{33} - a_{12} a_{23} a_{32} - a_{12} a_{22} a_{33} + a_{13} a_{22} a_{32} + a_{12} a_{23} a_{32} - a_{13} a_{22} a_{32} \\ = & 0 \\ | M_{11} | a_{13} - | M_{21} | a_{23} + | M_{31} | a_{33} & = & | \begin{array}{c} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} | a_{13} - | \begin{array}{c} a_{12} & a_{13} \\ a_{32} & a_{33} \end{array} | a_{23} + | \begin{array}{c} a_{12} & a_{13} \\ a_{22} & a_{23} \end{array} | a_{33} \\ = & a_{13} (a_{22} a_{33} - a_{23} a_{32}) - a_{23} (a_{12} a_{33} - a_{13} a_{32}) + a_{33} (a_{12} a_{23} - a_{13} a_{22}) \\ = & a_{13} a_{22} a_{33} - a_{13} a_{23} a_{32} - a_{12} a_{23} a_{33} + a_{13} a_{23} a_{32} + a_{12} a_{23} a_{33} - a_{13} a_{22} a_{33} \\ = & 0 \\ - | M_{12} | a_{11} + | M_{22} | a_{21} - | M_{32} | a_{31} & = & - | \begin{array}{c} a_{21} & a_{23} \\ a_{31} & a_{33} \end{array} | a_{11} + | \begin{array}{c} a_{11} & a_{13} \\ a_{31} & a_{33} \end{array} | a_{21} - | \begin{array}{c} a_{11} & a_{13} \\ a_{21} & a_{23} \end{array} | a_{31} \\ = & - a_{11} (a_{21} a_{33} - a_{23} a_{31}) + a_{21} (a_{11} a_{33} - a_{13} a_{31}) - a_{31} (a_{11} a_{23} - a_{13} a_{21}) \\ = & - a_{11} a_{21} a_{33} + a_{11} a_{23} a_{31} + a_{11} a_{21} a_{33} - a_{13} a_{21} a_{31} - a_{11} a_{23} a_{31} + a_{13} a_{21} a_{31} \\ = & 0 \\ - | M_{12} | a_{13} + | M_{22} | a_{23} - | M_{32} | a_{33} & = & - | \begin{array}{c} a_{21} & a_{23} \\ a_{31} & a_{33} \end{array} | a_{13} + | \begin{array}{c} a_{11} & a_{13} \\ a_{31} & a_{33} \end{array} | a_{23} - | \begin{array}{c} a_{11} & a_{13} \\ a_{21} & a_{23} \end{array} | a_{33} \\ = & - a_{13} (a_{21} a_{33} - a_{23} a_{31}) + a_{23} (a_{11} a_{33} - a_{13} a_{31}) - a_{33} (a_{11} a_{23} - a_{13} a_{21}) \\ = & - a_{13} a_{21} a_{33} + a_{13} a_{23} a_{31} + a_{11} a_{23} a_{33} - a_{13} a_{23} a_{31} - a_{11} a_{23} a_{33} + a_{13} a_{21} a_{33} \\ = & 0 \\ | M_{13} | a_{11} - | M_{23} | a_{21} + | M_{33} | a_{31} & = & | \begin{array}{c} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array} | a_{11} - | \begin{array}{c} a_{11} & a_{12} \\ a_{31} & a_{32} \end{array} | a_{21} + | \begin{array}{c} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} | a_{31} \\ = & a_{11} (a_{21} a_{32} - a_{22} a_{31}) - a_{21} (a_{11} a_{32} - a_{12} a_{31}) + a_{31} (a_{11} a_{22} - a_{12} a_{21}) \\ = & a_{11} a_{21} a_{32} - a_{11} a_{22} a_{31} - a_{11} a_{21} a_{32} + a_{12} a_{21} a_{31} + a_{11} a_{22} a_{31} - a_{12} a_{21} a_{31} \\ = & 0 \\ | M_{13} | a_{12} - | M_{23} | a_{22} + | M_{33} | a_{32} & = & | \begin{array}{c} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array} | a_{12} - | \begin{array}{c} a_{11} & a_{12} \\ a_{31} & a_{32} \end{array} | a_{22} + | \begin{array}{c} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} | a_{32} \\ = & a_{12} (a_{21} a_{32} - a_{22} a_{31}) - a_{22} (a_{11} a_{32} - a_{12} a_{31}) + a_{32} (a_{11} a_{22} - a_{12} a_{21}) \\ = & a_{12} a_{21} a_{32} - a_{12} a_{22} a_{31} - a_{11} a_{22} a_{32} + a_{12} a_{22} a_{31} + a_{11} a_{22} a_{32} - a_{12} a_{21} a_{32} \\ = & 0 \end{array}

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