补充公式大全中的行列式
continuous-integration/drone/push Build is passing
Details
continuous-integration/drone/push Build is passing
Details
parent
40810e50d1
commit
50526116d3
|
|
@ -14,8 +14,242 @@ date: 2022-12-24T13:05:40+08:00
|
|||
|
||||
设在某个过程中有两个变量x和y，对变量x在允许的范围内的每一个确定的值，变量y按照某一确定的法则总有相应的值与之对应，则称y为x的函数，记为y=f(x)
|
||||
|
||||
---
|
||||
|
||||
# 线性代数
|
||||
|
||||
## 行列式
|
||||
|
||||
### 行列式定义和性质
|
||||
|
||||
#### 行列式定义
|
||||
|
||||
n阶行列式
|
||||
|
||||
$$
|
||||
\begin{vmatrix}
|
||||
a_{11} & a_{12} & \cdots & a_{1n} \\\\
|
||||
a_{21} & a_{22} & \cdots & a_{2n} \\\\
|
||||
\vdots & \vdots & \ddots & \vdots \\\\
|
||||
a_{n1} & a_{n2} & \cdots & a_{nn} \\\\
|
||||
\end{vmatrix}
|
||||
= \sum_{(j_1,j_2,\cdots j_n)}(-1)^{(j_1,j_2,\cdots j_n)}a_{1j_1}a_{2j_2} \cdots a_{nj}
|
||||
$$
|
||||
|
||||
$\sum_{(j_1,j_2,\cdots j_n)}$表示对所有n级排列求和
|
||||
|
||||
#### 行列式性质
|
||||
|
||||
* 行列互换，行列式的值不变，也即$D=D^T$
|
||||
* 任意两行（列）互换位置后，行列式改变符号
|
||||
* 如果行列式中有两行（列）对应元素相同，则行列式的值为0
|
||||
* 将行列式的某一行（列）乘以一个常数k后，行列式的值变为原来的k倍
|
||||
* 如果行列式的某一行（列）全为0，则行列式的值等于0
|
||||
* 行列式的某两行（列）元素对应成比例，则行列式的值等于0
|
||||
* 如果行列式某一行（列）的所有元素都可以写成两个元素的和，则该行列式可以写成两个行列式的和，这两个行列式的这一行（列）分别对应两个加数，其余行（列）与原行列式相等
|
||||
|
||||
$$
|
||||
\begin{vmatrix}
|
||||
a_{11} & a_{12} & \cdots & a_{1n} \\\\
|
||||
a_{21} & a_{22} & \cdots & a_{2n} \\\\
|
||||
\vdots & \vdots & \ddots & \vdots \\\\
|
||||
a_{i1}+b_{i1} & a_{i2}+b_{i2} & \ddots & a_{in}+b_{in} \\\\
|
||||
\vdots & \vdots & \ddots & \vdots \\\\
|
||||
a_{n1} & a_{n2} & \cdots & a_{nn} \\\\
|
||||
\end{vmatrix} =
|
||||
\begin{vmatrix}
|
||||
a_{11} & a_{12} & \cdots & a_{1n} \\\\
|
||||
a_{21} & a_{22} & \cdots & a_{2n} \\\\
|
||||
\vdots & \vdots & \ddots & \vdots \\\\
|
||||
a_{i1} & a_{i2} & \cdots & a_{in} \\\\
|
||||
\vdots & \vdots & \ddots & \vdots \\\\
|
||||
a_{n1} & a_{n2} & \cdots & a_{nn} \\\\
|
||||
\end{vmatrix} +
|
||||
\begin{vmatrix}
|
||||
a_{11} & a_{12} & \cdots & a_{1n} \\\\
|
||||
a_{21} & a_{22} & \cdots & a_{2n} \\\\
|
||||
\vdots & \vdots & \vdots & \vdots \\\\
|
||||
b_{i1} & b_{i2} & \cdots & b_{in} \\\\
|
||||
\vdots & \vdots & \vdots & \vdots \\\\
|
||||
a_{n1} & a_{n2} & \cdots & a_{nn} \\\\
|
||||
\end{vmatrix}
|
||||
$$
|
||||
|
||||
* 将行列式的某行（列）的k倍加到另一行（列）上，行列式的值不变
|
||||
|
||||
### 行列式展开定理
|
||||
|
||||
#### 余子式及代数余子式
|
||||
|
||||
在n阶行列式$D=|a_{ij}|$中，划掉$a_{ij}$所在第i行和第j列的所有元素后，余下$(n-1)^2$个元素按照原有次序构成一个(n-1)阶行列式，称为元素$a_{ij}$在D中的余子式，记作$M_{ij}$
|
||||
|
||||
$$
|
||||
M_{ij} =
|
||||
\begin{vmatrix}
|
||||
a_{11} & a_{12} & \cdots & a_{1(j-1)} & a_{1(j+1)} & \cdots & a_{1n} \\\\
|
||||
a_{21} & a_{22} & \cdots & a_{2(j-1)} & a_{2(j+1)} & \cdots & a_{2n} \\\\
|
||||
\vdots & \vdots & \vdots& \vdots& \vdots& \vdots & \vdots \\\\
|
||||
a_{(i-1)1} & a_{(i-1)2} & \cdots & a_{(i-1)(j-1)} & a_{(i-1)(j+1)} & \cdots & a_{(i-1)n} \\\\
|
||||
a_{(i+1)1} & a_{(i+1)2} & \cdots & a_{(i+1)(j+1)} & a_{(i+1)(j+1)} & \cdots & a_{(i+1)n} \\\\
|
||||
\vdots & \vdots & \vdots& \vdots& \vdots& \vdots & \vdots \\\\
|
||||
a_{n1} & a_{n2} & \cdots & a_{n(j-1)} & a_{n(j+1)} & \cdots & a_{nn} \\\\
|
||||
\end{vmatrix}
|
||||
$$
|
||||
|
||||
记作$A_{ij} = (-1)^{i+j}M_{ij}$，称作元素$a_{ij}$的代数余子式
|
||||
|
||||
#### 行列式按一行（列）展开
|
||||
|
||||
n阶行列式D等于其任一行（列）各元素与其代数余子式乘积之和
|
||||
|
||||
$D = a_{i1}A_{i1} + a_{i2}A_{i2} + \cdots +a_{in}A_{in}=\sum_{j=1}^na_{ij}A_{ij}(i=1,2,\cdots,n)=a_{1j}A_{1j} + a_{2j}A_{2j} + \cdots +a_{nj}A_{nj} = \sum_{i=1}^na_{ij}A_{ij}(j = 1,2,\cdots,n)$
|
||||
|
||||
推论：
|
||||
|
||||
行列式D的某一行（列）各元素与另一行（列）对应元素的代数余子式的乘积之和为零
|
||||
|
||||
$$
|
||||
\sum_{j=1}^na_{ij}A_{kj}= a_{i1}A_{k1} + a_{i2}A_{k2} + \cdots +a_{in}A_{kn} = 0 (i\not ={k})
|
||||
$$
|
||||
|
||||
$$
|
||||
\sum_{j=1}^na_{ji}A_{jk}= a_{1i}A_{1k} + a_{2i}A_{2k} + \cdots +a_{ni}A_{nk} = 0 (i\not ={k})
|
||||
$$
|
||||
|
||||
### 特殊行列式
|
||||
|
||||
#### 上三角、下三角、对角形行列式
|
||||
|
||||
$
|
||||
\begin{vmatrix}
|
||||
a_{11} & 0 & \cdots & 0 \\\\
|
||||
0 & a_{22} & \cdots & 0 \\\\
|
||||
\vdots & \vdots & \ddots & \vdots \\\\
|
||||
0 & 0 & \cdots & a_{nn} \\\\
|
||||
\end{vmatrix} =
|
||||
\begin{vmatrix}
|
||||
a_{11} & a_{12} & \cdots & a_{1n} \\\\
|
||||
0 & a_{22} & \cdots & a_{2n} \\\\
|
||||
\vdots & \vdots & \ddots & \vdots \\\\
|
||||
0 & 0 & \cdots & a_{nn} \\\\
|
||||
\end{vmatrix} =
|
||||
\begin{vmatrix}
|
||||
a_{11} & 0 & \cdots & 0 \\\\
|
||||
a_{21} & a_{22} & \cdots & 0 \\\\
|
||||
\vdots & \vdots & \ddots & \vdots \\\\
|
||||
a_{n1} & a_{n2} & \cdots & a_{nn} \\\\
|
||||
\end{vmatrix} =
|
||||
a_{11}a_{22}\cdots a_{nn}
|
||||
$
|
||||
|
||||
#### 次对角线行列式
|
||||
|
||||
$
|
||||
\begin{vmatrix}
|
||||
0 & \cdots & 0 & a_{1n} \\\\
|
||||
0 & \cdots & 2_{2(n-1)} & 0\\\\
|
||||
\vdots & \cdots & \vdots & \vdots \\\\
|
||||
a_{nn} & \cdots & 0 & 0 \\\\
|
||||
\end{vmatrix} =
|
||||
\begin{vmatrix}
|
||||
a_{11} & \cdots & a_{1(n-1)} & a_{1n} \\\\
|
||||
a_{21} & \cdots & a_{2(n-1)} & 0\\\\
|
||||
\vdots & \cdots & \vdots & \vdots \\\\
|
||||
a_{n1} & \cdots & 0 & 0 \\\\
|
||||
\end{vmatrix} =
|
||||
\begin{vmatrix}
|
||||
0 & \cdots & 0 & a_{1n} \\\\
|
||||
0 & \cdots & a_{2(n-1)} & a_{2n}\\\\
|
||||
\vdots & \cdots & \vdots & \vdots \\\\
|
||||
a_{n1} & \cdots & a_{n(n-1)} & a_{nn} \\\\
|
||||
\end{vmatrix} =
|
||||
(-1)^{\frac{n(n-1)}{2}}a_{1n}\cdots a_{n1}
|
||||
$
|
||||
|
||||
#### 范德蒙德行列式
|
||||
|
||||
$\begin{vmatrix}
|
||||
1 & 1 & 1 & \cdots & 1 \\\\
|
||||
a_1 & a_2 & a_3 & \cdots & a_n \\\\
|
||||
a_1^2 & a_2^2 & a_3^2 & \cdots & a_n^2 \\\\
|
||||
\vdots & \vdots & \vdots & \cdots & \vdots\\\\
|
||||
a_1^{n-1} & a_2^{n-1} & a_3^{n-1} & \cdots & a_n^{n-1}
|
||||
\end{vmatrix} =
|
||||
\prod_{1\leq i < j \leq n}(a_j-a_i)
|
||||
$
|
||||
|
||||
#### 拉普拉斯展开式
|
||||
|
||||
$
|
||||
\begin{vmatrix}
|
||||
a_{11} & a_{12} & \cdots & a_{1k} \\\\
|
||||
a_{21} & a_{22} & \cdots & a_{2k} & & O_{k\times l} \\\\
|
||||
\vdots & \vdots & & \vdots \\\\
|
||||
a_{k1} & a_{k2} & \cdots & a_{kk} \\\\
|
||||
c_{11} & c_{12} & \cdots & c_{1k} & b_{11} & b_{12} & \cdots & b_{1l} \\\\
|
||||
c_{21} & c_{22} & \cdots & c_{2k} & b_{21} & b_{22} & \cdots & b_{2l} \\\\
|
||||
\vdots & \vdots & & \vdots & \vdots & \vdots & &\vdots \\\\
|
||||
c_{l1} & c_{l2} & \cdots & c_{lk} & b_{l1} & b_{l2} &\cdots & b_{ll}
|
||||
\end{vmatrix} =
|
||||
\begin{vmatrix}
|
||||
a_{11} & a_{12} & \cdots & a_{1k} \\\\
|
||||
a_{21} & a_{22} & \cdots & a_{2k}\\\\
|
||||
\vdots & \cdots & \vdots & \vdots \\\\
|
||||
a_{k1} & a_{k2} & \cdots & a_{kk} \\\\
|
||||
\end{vmatrix}
|
||||
\begin{vmatrix}
|
||||
b_{11} & b_{12} & \cdots & b_{1l} \\\\
|
||||
b_{21} & b_{22} & \cdots & b_{2l} \\\\
|
||||
\vdots & \cdots & \vdots & \vdots \\\\
|
||||
b_{l1} & b_{l2} &\cdots & b_{ll} \\\\
|
||||
\end{vmatrix}
|
||||
$
|
||||
|
||||
---
|
||||
|
||||
$
|
||||
\begin{vmatrix}
|
||||
c_{11} & c_{12} & \cdots & c_{1k} & b_{11} & b_{12} & \cdots & b_{1l} \\\\
|
||||
c_{21} & c_{22} & \cdots & c_{2k} & b_{21} & b_{22} & \cdots & b_{2l} \\\\
|
||||
\vdots & \vdots & & \vdots & \vdots & \vdots & &\vdots \\\\
|
||||
c_{l1} & c_{l2} & \cdots & c_{lk} & b_{l1} & b_{l2} &\cdots & b_{ll} \\\\
|
||||
a_{11} & a_{12} & \cdots & a_{1k} \\\\
|
||||
a_{21} & a_{22} & \cdots & a_{2k} & & O_{k\times l} \\\\
|
||||
\vdots & \vdots & & \vdots \\\\
|
||||
a_{k1} & a_{k2} & \cdots & a_{kk}
|
||||
\end{vmatrix} = (-1)^{kd}
|
||||
\begin{vmatrix}
|
||||
a_{11} & a_{12} & \cdots & a_{1k} \\\\
|
||||
a_{21} & a_{22} & \cdots & a_{2k}\\\\
|
||||
\vdots & \cdots & \vdots & \vdots \\\\
|
||||
a_{k1} & a_{k2} & \cdots & a_{kk} \\\\
|
||||
\end{vmatrix}
|
||||
\begin{vmatrix}
|
||||
b_{11} & b_{12} & \cdots & b_{1l} \\\\
|
||||
b_{21} & b_{22} & \cdots & b_{2l} \\\\
|
||||
\vdots & \cdots & \vdots & \vdots \\\\
|
||||
b_{l1} & b_{l2} &\cdots & b_{ll} \\\\
|
||||
\end{vmatrix}
|
||||
$
|
||||
|
||||
### 行列式有关的重要公式
|
||||
|
||||
设A，B均为n阶方阵，k为常数，E为n阶单位矩阵,A*为A的伴随矩阵
|
||||
|
||||
* $|kA| = k^n \cdot {|A|}$
|
||||
* 若A是可逆矩阵，则有$|A^{-1} = \frac{1}{|A|}$
|
||||
* $|A \cdot B| = |A| \cdot |B|$
|
||||
* |A*| = |A|^{n-1}
|
||||
* $A \cdot A* = A* \cdot A = |A| \cdot E$
|
||||
* $|A| = \prod_{i=1}^n\lambda_i(\lambda_1,\lambda_2,\cdots,\lambda_n是A的全部特征值)$
|
||||
* $|A|\not ={0}\iff A$为可逆矩阵$\iff$A为满秩矩阵，即$r(A) = n$
|
||||
|
||||
---
|
||||
## 矩阵
|
||||
|
||||
### 矩阵定义
|
||||
|
||||
|
||||
## 向量
|
||||
|
||||
### n维向量定义及其运算
|
||||
|
|
|
|||
Loading…
Reference in New Issue