行列式

¶ 定义

$S_n$ 表示 $\{1,2,\cdots,n\}$ 上所有排列的集合

定义函数 $N(\sigma)$ 表示 $\sigma$ 这个序列中逆序对的个数

定义函数 $\mathrm{sgn}(\sigma) = (-1)^{N(\sigma)}$

具体来说

$$\begin{aligned} S_3 = \{ &\left(\begin{array}{cccc} 1 & 2 &3 \end{array}\right), \\ &\left(\begin{array}{cccc} 1 & 3 &2 \end{array}\right), \\ &\left(\begin{array}{cccc} 2 & 1 &3 \end{array}\right), \\ &\left(\begin{array}{cccc} 2 & 3 & 1 \end{array}\right), \\ &\left(\begin{array}{cccc} 3 & 1 &2 \end{array}\right), \\ &\left(\begin{array}{cccc} 3 & 2 & 1 \end{array}\right)\} \end{aligned}$$

假如

$$\sigma=\left(\begin{array}{cccc} 3 & 1 &2 \end{array}\right)\in S_4$$

则 $\mathrm{sgn}(\sigma)=2$

¶ 一般的

$$\begin{vmatrix} a_{1,1} & \cdots & a_{1,n} \\ \vdots & \ddots & \vdots \\ a_{n,1} & \cdots & a_{n,n} \end{vmatrix} = \sum_{\sigma\in S_n} (\mathrm{sgn}(\sigma) \prod_{i=1}^n a_{i,\sigma(i)})$$

¶ 转置行列式

行列式

$$D=\begin{vmatrix} a_{1,1} & \cdots & a_{1,n} \\ \vdots & \ddots & \vdots \\ a_{n,1} & \cdots & a_{n,n} \end{vmatrix}=(a_{i,j})$$

定义其转置行列式

$$D^T=\begin{vmatrix} a_{1,1} & \cdots & a_{n,1} \\ \vdots & \ddots & \vdots \\ a_{1,n} & \cdots & a_{n,n} \end{vmatrix}=(b_{i,j})\\ a_{i,j}=b_{j,i}$$

¶ 二阶行列式

$$\begin{vmatrix} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \end{vmatrix} = a_{1,1} a_{2,2} - a_{1,2}a_{2,1}$$

¶ 三阶行列式

$$\begin{vmatrix} a_{1,1} & a_{1,2} & a_{1,3} \\ a_{2,1} & a_{2,2} & a_{2,3} \\ a_{3,1} & a_{3,2} & a_{3,3} \end{vmatrix} = a_{1,1}a_{2,2}a_{3,3} + a_{1,2}a_{2,3}a_{3,1}+ a_{1,3}a_{2,1}a_{3,2} - a_{1,3}a_{2,2}a_{3,1} - a_{1,1}a_{2,3}a_{3,2} - a_{1,2}a_{2,1}a_{3,3}$$

¶ 性质

对于行列式 $D$

• $D=D^T$
• 一行有公因子 $k$ ，可以提取 $k$

$$\begin{vmatrix} \vdots & \vdots & \vdots & \vdots \\ {\color{blue}k}a_{i,1} & {\color{blue}k}a_{i,2} & \dots & {\color{blue}k}a_{i,n} \\ \vdots & \vdots & \vdots & \vdots \\ \end{vmatrix}= {\color{blue}k}\begin{vmatrix} \vdots & \vdots & \vdots & \vdots \\ a_{i,1} & a_{i,2} & \dots & a_{i,n} \\ \vdots & \vdots & \vdots & \vdots \\ \end{vmatrix}$$

• 有一行或一列都为 $0$ ，则 $D=0$

$$\begin{vmatrix} \vdots & \vdots & \vdots & \vdots \\ {\color{blue}0} & {\color{blue}0} & \dots & {\color{blue}0} \\ \vdots & \vdots & \vdots & \vdots \\ \end{vmatrix} =\begin{vmatrix} {\color{blue}0} & \vdots & \vdots & \vdots \\ \vdots & \vdots & \vdots & \vdots \\ {\color{blue}0} & \vdots & \vdots & \vdots \\ \end{vmatrix}=0$$