【线性代数】矩阵的乘法与求逆

矩阵乘法的几种种表示方法

1、一般形式

$$AB=C\\ C_{ij}=\sum\limits_{k=1}^{K}a_{ik}b_{kj}$$

2、矩阵与列向量相乘

$$\left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \\ \end{array} \right ] \left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \\ \end{array} \right ]= \left[ \begin{array}{cc} 7 & 10 \\ 15 & 22 \\ \end{array} \right ]$$ $$\left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \\ \end{array} \right ] \left[ \begin{array}{c} 1 \\ 3 \\ \end{array} \right ]= \left[ \begin{array}{cc} 7 \\ 15 \\ \end{array} \right ]$$ $$\left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \\ \end{array} \right ] \left[ \begin{array}{c} 2 \\ 4 \\ \end{array} \right ]= \left[ \begin{array}{cc} 10 \\ 22 \\ \end{array} \right ]$$

3、矩阵与行向量相乘

$$\left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \\ \end{array} \right ] \left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \\ \end{array} \right ]= \left[ \begin{array}{cc} 7 & 10 \\ 15 & 22 \\ \end{array} \right ]$$ $$\left[ \begin{array}{cc} 1 & 2 \\ \end{array} \right ] \left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \\ \end{array} \right ]= \left[ \begin{array}{cc} 7 & 10 \\ \end{array} \right ]$$ $$\left[ \begin{array}{cc} 3 & 4 \\ \end{array} \right ] \left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \\ \end{array} \right ]= \left[ \begin{array}{cc} 15 & 22 \\ \end{array} \right ]$$

4、矩阵分块相乘

$$A_1=B_1=\left[ \begin{array}{cc} 1 & 1 \\ 1 & 1 \\ \end{array} \right ], A_2=B_2=\left[ \begin{array}{cc} 2 & 2 \\ 2 & 2 \\ \end{array} \right ]$$ $$A_3=B_3=\left[ \begin{array}{cc} 3 & 3 \\ 3 & 3 \\ \end{array} \right ], A_4=B_4=\left[ \begin{array}{cc} 4 & 4 \\ 4 & 4 \\ \end{array} \right ]$$ $$A=\left[ \begin{array}{cc} A_1 & A_2 \\ A_3 & A_4 \\ \end{array} \right ], B=\left[ \begin{array}{cc} B_1 & B_2 \\ B_3 & B_4 \\ \end{array} \right ]$$ $$AB=\left[ \begin{array}{cc} A_1B_1+A_2B_3 & A_1B_2+A_2B_4 \\ A_3B_1+A_4B_3 & A_3B_2+A_4B_4 \\ \end{array} \right ]$$

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矩阵的逆

对于方阵，左逆=右逆

$$A^{-1}A=I$$

原矩阵乘以其逆矩阵得到单位矩阵

判断是否可逆的几种方法：

• 行列式为$0$

• 单位矩阵的各列是矩阵各列的线性组合

• 下式成立时，矩阵$A$不可逆：

  $$AX=0,$$

  其中$X$不是零向量。

  证明：

  $$A^{-1}AX=A^{-1}0$$

  $X=0$与假设矛盾，说明$A$不可逆。

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矩阵求逆(高斯-若尔当消元法)

假设矩阵为$A$:

$$A=\left[ \begin{array}{cc} 1 & 3 \\ 2 & 7 \\ \end{array} \right ]$$

消元过程如下：

$$\begin{aligned} \underbrace{\left[\begin{array}{cc|cc}1&3&1&0\\2&7&0&1\end{array}\right]}_{[A | I]} {\longrightarrow}\left[\begin{array}{cccc}1&3&1&0\\0&1&-2&1\end{array}\right] {\longrightarrow}\underbrace{\left[\begin{array}{cc|cc}1&0&7&-3\\0&1&-2&1\end{array}\right]}_{[I|A^{-1}]} \end{aligned}$$

通过消元，我们将矩阵$A$变成了单位矩阵$I$，则与此同时，矩阵$I$变成了$A$的逆矩阵。证明如下：

$$E[A|I]=[EA|EI]=[I|E]=[I|A^{-1}]\\ EA=I\rightarrow E=A^{-1}\\ EI=E=A^{-1}$$