行列の計算則   積について(1)

$A$$l×m$ 行列， $B$$m×n$行列ならば

$\left(\alpha A\right)B=\alpha \left(AB\right)=A\left(\alpha B\right)$

■証明

$A=\left(\begin{array}{cccc}{a}_{11}& {a}_{12}& \cdots & {a}_{1m}\\ {a}_{21}& {a}_{22}& \cdots & {a}_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{l1}& {a}_{l2}& \cdots & {a}_{lm}\end{array}\right)$$B=\left(\begin{array}{cccc}{b}_{11}& {b}_{12}& \cdots & {b}_{1n}\\ {b}_{21}& {b}_{22}& \cdots & {b}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {b}_{m1}& {b}_{m2}& \cdots & {b}_{mn}\end{array}\right)$

とし，

$\alpha A=\left(\begin{array}{cccc}\alpha {a}_{11}& \alpha {a}_{12}& \cdots & \alpha {a}_{1m}\\ \alpha {a}_{21}& \alpha {a}_{22}& \cdots & \alpha {a}_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ \alpha {a}_{l1}& \alpha {a}_{l2}& \cdots & \alpha {a}_{lm}\end{array}\right)$$=\left(\begin{array}{cccc}{{a}^{\prime }}_{11}& {{a}^{\prime }}_{12}& \cdots & {{a}^{\prime }}_{1m}\\ {{a}^{\prime }}_{21}& {{a}^{\prime }}_{22}& \cdots & {{a}^{\prime }}_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ {{a}^{\prime }}_{l1}& {{a}^{\prime }}_{l2}& \cdots & {{a}^{\prime }}_{lm}\end{array}\right)$

$\alpha B=\left(\begin{array}{cccc}\alpha {b}_{11}& \alpha {b}_{12}& \cdots & \alpha {b}_{1n}\\ \alpha {b}_{21}& \alpha {b}_{22}& \cdots & \alpha {b}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ \alpha {b}_{m1}& \alpha {b}_{m2}& \cdots & \alpha {b}_{mn}\end{array}\right)$$=\left(\begin{array}{cccc}{{b}^{\prime }}_{11}& {{b}^{\prime }}_{12}& \cdots & {{b}^{\prime }}_{1n}\\ {{b}^{\prime }}_{21}& {{b}^{\prime }}_{22}& \cdots & {{b}^{\prime }}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {{b}^{\prime }}_{m1}& {{b}^{\prime }}_{m2}& \cdots & {{b}^{\prime }}_{mn}\end{array}\right)$

とする．(行列のスカラー倍を参照)

$\left(\alpha A\right)B=\left(\begin{array}{cccc}{{a}^{\prime }}_{11}& {{a}^{\prime }}_{12}& \cdots & {{a}^{\prime }}_{1m}\\ {{a}^{\prime }}_{21}& {{a}^{\prime }}_{22}& \cdots & {{a}^{\prime }}_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ {{a}^{\prime }}_{l1}& {{a}^{\prime }}_{l2}& \cdots & {{a}^{\prime }}_{lm}\end{array}\right)\left(\begin{array}{cccc}{b}_{11}& {b}_{12}& \cdots & {b}_{1n}\\ {b}_{21}& {b}_{22}& \cdots & {b}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {b}_{m1}& {b}_{m2}& \cdots & {b}_{mn}\end{array}\right)$

$\left(\alpha A\right)B$$\left(i,j\right)$成分${c}_{ij}$は，行列の積の定義より

${c}_{ij}=\sum _{k=1}^{m}{{a}^{\prime }}_{ik}{b}_{kj}=\sum _{k=1}^{m}\left(\alpha {a}_{ik}\right){b}_{kj}$$=\alpha \sum _{k=1}^{m}{a}_{ik}{b}_{kj}$……(1)

$\alpha \left(AB\right)$$\left(i,j\right)$成分${d}_{ij}$は，行列の積の定義より

${d}_{ij}=\alpha \sum _{k=1}^{m}{a}_{ik}{b}_{kj}$……(2)

$A\left(\alpha B\right)=\left(\begin{array}{cccc}{a}_{11}& {a}_{12}& \cdots & {a}_{1m}\\ {a}_{21}& {a}_{22}& \cdots & {a}_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{l1}& {a}_{l2}& \cdots & {a}_{lm}\end{array}\right)\left(\begin{array}{cccc}{{b}^{\prime }}_{11}& {{b}^{\prime }}_{12}& \cdots & {{b}^{\prime }}_{1n}\\ {{b}^{\prime }}_{21}& {{b}^{\prime }}_{22}& \cdots & {{b}^{\prime }}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {{b}^{\prime }}_{m1}& {{b}^{\prime }}_{m2}& \cdots & {{b}^{\prime }}_{mn}\end{array}\right)$

$A\left(\alpha B\right)$$\left(i,j\right)$成分${e}_{ij}$は，行列の積の定義より

${e}_{ij}=\sum _{k=1}^{m}{a}_{ik}{{b}^{\prime }}_{kj}=\sum _{k=1}^{m}{a}_{ik}\left(\alpha {b}_{kj}\right)$$=\alpha \sum _{k=1}^{m}{a}_{ik}{b}_{kj}$……(3)

$\left(1\right)$$\left(2\right)$$\left(3\right)$より

${c}_{ij}={d}_{ij}={e}_{ij}$

となり

$\left(\alpha A\right)B=\alpha \left(AB\right)=A\left(\alpha B\right)$

が成り立つ．

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