3.【线性代数】——矩阵乘法和逆矩阵-1. 矩阵乘法

1.1 常规方法

$$\underset{A_{m\ast n}}{\underbrace{\left[\begin{array}{cccc}...& ...& ...& ...\\ a_{31}& a_{32}& a_{33}& a_{34}\\ ...& ...& ...& ...\end{array}\right]}}\underset{B_{n\ast p}}{\underbrace{\left[\begin{array}{cccc}...& ...& ...& b_{14}\\ ...& ...& ...& b_{24}\\ ...& ...& ...& b_{34}\\ ...& ...& ...& b_{44}\end{array}\right]}}=\underset{C_{m\ast p}}{\underbrace{\left[\begin{array}{cccc}...& ...& ...& ...\\ ...& ...& ...& C_{34}\\ ...& ...& ...& ...\end{array}\right]}}$$
$$C_{34}=A_{ro{w}_3}\ast B_{co{l}_4}=\sum _{i=1}^n{a}_{3i}\ast b_{i4}$$

1.2 列向量组合

已知
$$\begin{array}{rl}\left[\begin{array}{ccc}{A}_{11}& A_{12}& A_{13}\\ A_{21}& A_{22}& A_{23}\\ A_{31}& A_{32}& A_{33}\end{array}\right]\left[\begin{array}{c}{B}_{11}\\ B_{21}\\ B_{31}\end{array}\right]& =B_{11}\ast A_{col1}+B_{21}\ast A_{col2}+B_{31}\ast A_{col3}\\ & =\left[\begin{array}{c}{B}_{11}\ast A_{11}+B_{21}\ast A_{12}+B_{31}\ast A_{13}\\ B_{11}\ast A_{21}+B_{21}\ast A_{22}+B_{31}\ast A_{23}\\ B_{11}\ast A_{31}+B_{21}\ast A_{32}+B_{31}\ast A_{33}\end{array}\right]\end{array}$$
那么
$$\begin{array}{rl}\underset{A}{\underbrace{\left[\begin{array}{ccc}{A}_{11}& A_{12}& A_{13}\\ A_{21}& A_{22}& A_{23}\\ A_{31}& A_{32}& A_{33}\end{array}\right]}}\underset{B}{\underbrace{\left[\begin{array}{cc}{B}_{11}& B_{12}\\ B_{21}& B_{22}\\ B_{31}& B_{32}\end{array}\right]}}& =\underset{C}{\underbrace{\left[\begin{array}{cc}{B}_{11}\ast A_{col1}+B_{21}\ast A_{col2}+B_{31}\ast A_{col3}& B_{12}\ast A_{col1}+B_{22}\ast A_{col2}+B_{32}\ast A_{col3}\end{array}\right]}}\\ & =\left[\begin{array}{cc}{B}_{11}\ast A_{11}+B_{21}\ast A_{12}+B_{31}\ast A_{13}& B_{12}\ast A_{11}+B_{22}\ast A_{12}+B_{32}\ast A_{13}\\ B_{11}\ast A_{21}+B_{21}\ast A_{22}+B_{31}\ast A_{23}& B_{12}\ast A_{21}+B_{22}\ast A_{22}+B_{32}\ast A_{23}\\ B_{11}\ast A_{31}+B_{21}\ast A_{32}+B_{31}\ast A_{33}& B_{12}\ast A_{31}+B_{22}\ast A_{32}+B_{32}\ast A_{33}\end{array}\right]\end{array}$$
C矩阵是A矩阵的列向量组合

1.3 行向量组合

已知
$$\begin{array}{rl}\left[\begin{array}{ccc}{A}_{11}& A_{12}& A_{13}\end{array}\right]\left[\begin{array}{cc}{B}_{11}& B_{12}\\ B_{21}& B_{22}\\ B_{31}& B_{32}\end{array}\right]& =A_{11}\ast B_{row1}+A_{12}\ast B_{row2}+A_{13}\ast B_{row3}\\ & =\left[\begin{array}{cc}{A}_{11}\ast B_{11}& A_{11}\ast B_{12}\\ +& +\\ A_{12}\ast B_{21}& A_{12}\ast B_{22}\\ +& +\\ A_{13}\ast B_{31}& A_{13}\ast B_{32}\end{array}\right]\end{array}$$
那么
$$\begin{array}{rl}\underset{A}{\underbrace{\left[\begin{array}{ccc}{A}_{11}& A_{12}& A_{13}\\ A_{21}& A_{22}& A_{23}\\ A_{31}& A_{32}& A_{33}\end{array}\right]}}\underset{B}{\underbrace{\left[\begin{array}{cc}{B}_{11}& B_{12}\\ B_{21}& B_{22}\\ B_{31}& B_{32}\end{array}\right]}}& =\underset{C}{\underbrace{\left[\begin{array}{c}{A}_{11}\ast B_{row1}+A_{12}\ast B_{row2}+A_{13}\ast B_{row3}\\ A_{21}\ast B_{row1}+A_{22}\ast B_{row2}+A_{23}\ast B_{row3}\\ A_{31}\ast B_{row1}+A_{32}\ast B_{row2}+A_{33}\ast B_{row3}\end{array}\right]}}\end{array}$$
C矩阵是B矩阵的行向量组合

1.4 单行和单列的乘积和

$$\begin{array}{rl}\left[\begin{array}{cc}2& 7\\ 3& 8\\ 4& 9\end{array}\right]\left[\begin{array}{cc}1& 6\\ 1& 1\end{array}\right]& =\left[\begin{array}{c}2\\ 3\\ 4\end{array}\right]\left[\begin{array}{cc}1& 6\end{array}\right]+\left[\begin{array}{c}7\\ 8\\ 9\end{array}\right]\left[\begin{array}{cc}1& 1\end{array}\right]\\ & =\left[\begin{array}{cc}9& 19\\ 11& 26\\ 13& 33\end{array}\right]\end{array}$$

1.5 块乘法

$$\left[\begin{array}{ccc}{A}_1& |& A_2\\ ——& ——& ——\\ A_3& |& A_4\end{array}\right]\left[\begin{array}{ccc}{B}_1& |& B_2\\ ——& ——& ——\\ B_3& |& B_4\end{array}\right]=\left[\begin{array}{ccc}{A}_1\ast B_1+A_2\ast B_3& |& A_1\ast B_2+A_2\ast B_4\\ ————————& ——& ————————\\ A_3\ast B_1+A_4\ast B_3& |& A_3\ast B_2+A_4\ast B_4\end{array}\right]$$