行列式学习

tonyfang posted @ 2015年8月23日 21:55 in math with tags math , 1069 阅读

矩阵A的行列式定义:$det(A)=\sum\limits_{\sigma \in S_n}sgn(\sigma)\prod\limits_{i=1}^na_{i,\sigma(i)}$
这么一长串公式什么意思呢?

$S_n$表示集合$C={1,2,...,n}$上置换的全体,也就是集合$C$到自身双射的全体。

$a_{i,j}$表示矩阵中第$i$行第$j$列的元素。$sgn(\sigma)$表示置换$\sigma \in S_n$的符号差,也就是逆序对的个数。对于任意正整数$n$,$card(S_n)=n!$,所以是个有限次的求和。

主对角线以下(上)的元素都为0的矩阵叫做上(下)三角矩阵,它的行列式的值与对角行列式一样。

感谢图片提供:这里

行列式的性质:

1.在行列式中,如果有一行(列)的元素都是0,那么行列式的值为0.

2.行列式中,某一行(列)的元素有公因子$k$,则可以提出$k$后算,算完再乘进去。

3.在行列式中,某一行(列)的元素是两数之和,那么可以拆分成两个行列式相加。

4.行列式的两行(列)互换,改变行列式正负符号。

5.行列式中,如果有两行/列对应成比例或相同,那么行列式的值为0。

6.将一行(列)的$k$倍加进另一行里,行列式的值不变。

7.矩阵行列式=矩阵的转置的行列式。

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