From 50526116d35e59f1a912c8a4a1e59d8a22aacd1e Mon Sep 17 00:00:00 2001 From: InkSoul Date: Tue, 15 Aug 2023 21:46:27 +0800 Subject: [PATCH] =?UTF-8?q?=E8=A1=A5=E5=85=85=E5=85=AC=E5=BC=8F=E5=A4=A7?= =?UTF-8?q?=E5=85=A8=E4=B8=AD=E7=9A=84=E8=A1=8C=E5=88=97=E5=BC=8F?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- content/mathematics/公式大全.md | 234 ++++++++++++++++++++++++++++++++ 1 file changed, 234 insertions(+) diff --git a/content/mathematics/公式大全.md b/content/mathematics/公式大全.md index 200090a..eb78864 100644 --- a/content/mathematics/公式大全.md +++ b/content/mathematics/公式大全.md @@ -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