Section01_定义

基本概念

1. 矩阵 $\boldsymbol{A} = \left( \begin{matrix} a_{11} & a_{12} &\cdots & a_{1n} \\ a_{21} & a_{22} &\cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} &\cdots & a_{mn} \\ \end{matrix} \right)\triangleq (a_{ij})_{m\times n}$
   1. 若 $\forall\ a_{ij} = 0, \boldsymbol{A} = \boldsymbol{0}$
   2. 若 $m=n$，$\boldsymbol{A}$ 为方阵
2. 同型矩阵与矩阵相等 $\boldsymbol{A}_{m\times n},\boldsymbol{B}_{m\times n}$ 称谓同型矩阵，设 $\boldsymbol{A} = (a_{ij})_{m\times n}, \boldsymbol{B} = (b_{ij})_{m\times n}$，若 $\forall\ a_{ij}= b_{ij}$，则 $\boldsymbol{A} = \boldsymbol{B}$
3. 三则运算
   1. $\boldsymbol{A}_{m\times n} = \left( \begin{matrix} a_{11} & a_{12} &\cdots & a_{1n} \\ a_{21} & a_{22} &\cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} &\cdots & a_{mn} \\ \end{matrix} \right), \boldsymbol{B}_{m\times n} = \left( \begin{matrix} b_{11} & b_{12} &\cdots & b_{1n} \\ b_{21} & b_{22} &\cdots & b_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ b_{m1} & b_{m2} &\cdots & b_{mn} \\ \end{matrix} \right)$ $$\boldsymbol{A}\pm \boldsymbol{B} = \left( \begin{matrix} a_{11}\pm b_{11} & a_{12}\pm b_{12} & \cdots & a_{1n}\pm b_{1n} \\ a_{21}\pm b_{21} & a_{22}\pm b_{22} & \cdots & a_{2n}\pm b_{2n} \\ \vdots &\vdots &\ddots & \vdots \\ a_{m1}\pm b_{m1} & a_{m2}\pm b_{m2} & \cdots & a_{mn}\pm b_{mn} \\ \end{matrix} \right) = (a_{ij}\pm b_{ij})_{m\times n}$$
   2. $k\cdot \boldsymbol{A} = \left( \begin{matrix} ka_{11} & ka_{12} & \cdots & ka_{1n} \\ ka_{21} & ka_{22} & \cdots & ka_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ ka_{m1} & ka_{m2} & \cdots & ka_{mn} \end{matrix} \right) = (ka_{ij})_{m\times n}$
   3. $\boldsymbol{A}_{m\times n} = \left( \begin{matrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{matrix} \right), \boldsymbol{B}_{n\times s} = \left( \begin{matrix} b_{11} & b_{12} & \cdots & b_{1s} \\ b_{21} & b_{22} & \cdots & b_{2s} \\ \vdots & \vdots & \ddots & \vdots \\ b_{n1} & b_{n2} & \cdots & b_{ns} \end{matrix} \right)$ $$\begin{split} &\boldsymbol{AB} = \boldsymbol{C}_{m\times s} = (c_{ij})_{m\times s} \\ &c_{ij} = a_{i 1}b_{1j} + a_{i2}b_{2j} +\cdots + a_{in}b_{nj} \end{split}$$

      Notes

      1. $\boldsymbol{A} \ne \boldsymbol{0}, \boldsymbol{B}\ne \boldsymbol{0} \nRightarrow \boldsymbol{AB}\ne \boldsymbol{0}$

         如 $\boldsymbol{A} = \left( \begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix} \right) \ne \boldsymbol{0}, \boldsymbol{B} = \left( \begin{matrix} 1 & -1 \\ -1 & 1 \\ \end{matrix} \right)\ne \boldsymbol{0}$ 但 $\boldsymbol{AB} = \boldsymbol{0}$

      2. $\boldsymbol{A} \ne \boldsymbol{0} \nRightarrow \boldsymbol{A}^{k} \ne \boldsymbol{0}$

         如 $\boldsymbol{A} = \left( \begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix} \right)\ne \boldsymbol{0}$ 但 $\boldsymbol{A}^{2} = \boldsymbol{0}$

      3. $\boldsymbol{AB}$ 与 $\boldsymbol{BA}$ 不一定相等
      4. 对于 $f(x) = a_{n}x^{n}+\cdots + a_{1}x + a_{0}$，将 $\boldsymbol{A}_{n\times n}$ 代入，则 $f(\boldsymbol{A}) = a_{n}\boldsymbol{A}^{n} + \cdots + a_{1}\boldsymbol{A}+a_{0}\boldsymbol{E}$，$f(\boldsymbol{A})$ 称为 $\boldsymbol{A}$ 的矩阵多项式，==$f(\boldsymbol{A})$ 可像多项式一样处理==

         如 $$\begin{split} x^{2} - x - 2 & \Rightarrow (x+1)(x-2) \\ \boldsymbol{A}^{2} - \boldsymbol{A} - 2 \boldsymbol{E} & \Rightarrow (\boldsymbol{A}+1)(\boldsymbol{A}-2) \end{split}$$

      5. 线性方程组的矩阵形式 $$\begin{split} & \boldsymbol{A} = \left( \begin{matrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{matrix} \right) \\ & \boldsymbol{x} = \left( \begin{matrix} x_{1}\\ x_{2}\\ \vdots\\x_{n} \end{matrix} \right)\quad \boldsymbol{\beta} = \left( \begin{matrix} b_{1} \\ b_{2} \\\vdots \\b_{m} \end{matrix} \right) \\\\ & \boldsymbol{AX} = \boldsymbol{0} \Leftrightarrow \begin{cases} a_{11}x_{1} + a_{12}x_{2} + \cdots + a_{1n}x_{n} = 0 \\ \cdots \\ a_{m1}x_{1} + a_{m2}x_{2} + \cdots + a_{mn}x_{n} = 0 \\ \end{cases} \\ & \boldsymbol{AX} = \boldsymbol{\beta} \Leftrightarrow \begin{cases} a_{11}x_{1} + a_{12}x_{2} + \cdots + a_{1n}x_{n} = b_{1} \\ \cdots \\ a_{m1}x_{1} + a_{m2}x_{2} + \cdots + a_{mn}x_{n} = b_{m} \\ \end{cases} \end{split}$$

4. 伴随矩阵 对于 $\boldsymbol{A}$ 为 $n\times n$ 的方阵，则称 $\boldsymbol{A}^{*} = \left( \begin{matrix} A_{11} & A_{21} & \cdots & A_{n1} \\ A_{12} & A_{22} & \cdots & A_{n2} \\ \vdots & \vdots & \ddots & \vdots \\ A_{1n} & A_{2n} & \cdots & A_{nn} \\ \end{matrix} \right)$ 为 $\boldsymbol{A}$ 的伴随矩阵 (注意此处为各元素对应的代数余子式组成矩阵的转置)
   • 注 $\boldsymbol{AA}^{*} = \vert \boldsymbol{A} \vert \boldsymbol{E}$

Q1 矩阵研究什么

背景 对于方程 $ax=b$

1. $a\ne0$ 时，$\frac{1}{a}\cdot a = 1$，则 $\frac{1}{a}\cdot ax = \frac{1}{a}b\Rightarrow x = \frac{1}{a}b$
2. $a = 0$ $\begin{cases} b = 0, & \text{无数解} \\ b \ne 0, & \text{无解} \end{cases}$

相似问题 对于方程组 $\boldsymbol{AX} = \boldsymbol{\beta}$

1. 对于 $\boldsymbol{A}_{n\times n}$ 若 $\exists\ \boldsymbol{B}_{n\times n}$，使得 $\boldsymbol{BA} = \boldsymbol{E}$ (==逆矩阵==)，则 $\boldsymbol{AX}=\boldsymbol{\beta}\Rightarrow \boldsymbol{BAX} = \boldsymbol{B\beta}\Rightarrow \boldsymbol{X} = \boldsymbol{B\beta}$
2. $\begin{cases} \boldsymbol{A}_{n\times n}, \text{但不满秩} \\ \boldsymbol{A}_{m\times n}, \text{且} m \ne n \end{cases}$ (==矩阵的秩==)