行列の計算則   積について(1)

${\displaystyle A}$ が ${\displaystyle l\times m}$ 行列， ${\displaystyle B}$ が ${\displaystyle m\times n}$行列ならば

${\displaystyle \left(\alpha A\right)B=\alpha \left(AB\right)=A\left(\alpha B\right)}$

■証明

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

とし，

${\displaystyle \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)}$${\displaystyle =\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)}$

${\displaystyle \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)}$${\displaystyle =\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)

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

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

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

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

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

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

${\displaystyle \left(1\right)}$，${\displaystyle \left(2\right)}$，${\displaystyle \left(3\right)}$より

${\displaystyle c_{ij}=d_{ij}=e_{ij}}$

となり

${\displaystyle \left(\alpha A\right)B=\alpha \left(AB\right)=A\left(\alpha B\right)}$

が成り立つ．

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最終更新日： 2022年8月27日