MIT线性代数笔记03-矩阵乘法和逆矩阵

Linear Algebra-Lecture03 矩阵乘法和逆矩阵

Gilbert Strang

矩阵乘法

对于矩阵乘法
$$\mathbf{A}\mathbf{B}=\mathbf{C}$$
主要有5种方法可用于计算：
【前提条件】：$\mathbf{A},\mathbf{B}$两个矩阵行列要匹配，$\mathbf{A}$的列数要等于$\mathbf{B}$的行数。
[ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a m 1 a m 2 ⋯ a m n ] [ b 11 b 12 ⋯ b 1 p b 21 b 22 ⋯ b 2 p ⋮ ⋮ ⋱ ⋮ b n 1 b n 2 ⋯ b n p ] = [ c 11 c 12 ⋯ c 1 p c 21 c 22 ⋯ c 2 p ⋮ ⋮ ⋱ ⋮ c m 1 c m 2 ⋯ c m p ] \left[ \begin{matrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{matrix} \right] \left[ \begin{matrix} b_{11} & b_{12} & \cdots & b_{1p} \\ b_{21} & b_{22} & \cdots & b_{2p} \\ \vdots & \vdots & \ddots & \vdots \\ b_{n1} & b_{n2} & \cdots & b_{np} \\ \end{matrix} \right]= \left[ \begin{matrix} c_{11} & c_{12} & \cdots & c_{1p} \\ c_{21} & c_{22} & \cdots & c_{2p} \\ \vdots & \vdots & \ddots & \vdots \\ c_{m1} & c_{m2} & \cdots & c_{mp} \\ \end{matrix} \right] ⎣⎢⎢⎢⎡​a11​a21​⋮am1​​a12​a22​⋮am2​​⋯⋯⋱⋯​a1n​a2n​⋮amn​​⎦⎥⎥⎥⎤​⎣⎢⎢⎢⎡​b11​b21​⋮bn1​​b12​b22​⋮bn2​​⋯⋯⋱⋯​b1p​b