Section01_概念

背景 二次型

• 例子

  1. 标准二次型 $f(x_{1},x_{2},x_{3}) = 2x^{2}_{1} + 3x_{2}^{2} - x^{2}_{3}$
     • 令 $\boldsymbol{A} = \left( \begin{smallmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & -1 \\ \end{smallmatrix} \right), \boldsymbol{X} = \left( \begin{smallmatrix} x_{1} \\ x_{2} \\ x_{3} \\ \end{smallmatrix} \right)$，则 $f = \boldsymbol{X}^{\intercal}\boldsymbol{AX}$，此时，$\boldsymbol{A}$ 为对角矩阵
  2. 非标准二次型 $f(x_{1},x_{2}) = x_{1}^{2} -4x_{1}x_{2}-x^{2}_{2}$
     • 令 $\boldsymbol{A} = \left( \begin{smallmatrix} 1 & -2 \\ -2 & -1 \end{smallmatrix} \right), \boldsymbol{X} = \left( \begin{smallmatrix} x_{1} \\ x_{2} \\ x_{3} \\ \end{smallmatrix} \right)$，则 $f = \boldsymbol{X}^{\intercal}\boldsymbol{AX}$，此时，$\boldsymbol{A}$ 为对称而非对角矩阵

• 判别

  $$f= \boldsymbol{X}^{\intercal}\boldsymbol{AX} \begin{cases} \text{标准二次型} \Leftrightarrow \boldsymbol{A}\text{为对角矩阵} \\ \text{非标准二次型} \Leftrightarrow \boldsymbol{A}\text{为对称而不对角矩阵} \\ \end{cases}$$

• 矩阵对角化

  将对称而不对角的 $\boldsymbol{A}$ 转化为对角 $\boldsymbol{A}$ 的过程

1. 特征值与特征向量 $\boldsymbol{A}$ 为 $n$ 阶方阵，若 $\exists\ \lambda\in\mathbb{C}, \boldsymbol{\alpha}\in \mathbb{C}_{n\times 1}$，使得 $\boldsymbol{A\alpha} = \lambda \boldsymbol{\alpha}$
   • $\lambda$ 为 $\boldsymbol{A}$ 特征值
   • $\boldsymbol{\alpha}$ 为 $\lambda$ 对应的特征向量
2. 特征方程 $\det(\lambda \boldsymbol{E} - \boldsymbol{A}) = 0$ 称为 $\boldsymbol{A}$ 的特征方程 $$\begin{array}{c} \det(\lambda \boldsymbol{E} - \boldsymbol{A}) = 0 \\ \Updownarrow \\ \left\vert \begin{matrix} \lambda - a_{11} & - a_{12} & \cdots & -a_{1n} \\ - a_{21} & \lambda - a_{22} & \cdots & -a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ - a_{n1} & - a_{n2} & \cdots & \lambda - a_{nn} \\ \end{matrix} \right\vert = 0 \\ \Updownarrow \\ \lambda^{n} - (a_{11} + a_{22} + \cdots + a_{nn})\lambda^{n-1} + \cdots = 0 \end{array}$$

   Notes

   1. $\lambda$ 不一定为实数
   2. 矩阵的迹 $a_{11} + a_{22} + \cdots + a_{nn} \triangleq \text{tr}(\boldsymbol{A})$
   3. $\det(\lambda \boldsymbol{E} - \boldsymbol{A}) = 0 \Rightarrow \lambda_{1},\cdots, \lambda_{n}$
      • $\sum\limits_{i=1}^{n} \lambda_{i} = \text{tr}(\boldsymbol{A})$
      • $\prod\limits_{i=1}^{n} \lambda_{i} = \det(\boldsymbol{A})$
   4. $\det(\boldsymbol{A}) = \prod\limits_{i=1}^{n}\lambda_{i}$
      • Case 1 $\boldsymbol{A}$ 可逆 $\Leftrightarrow$ $\det(\boldsymbol{A})\ne 0$ $\Leftrightarrow$ $\lambda_{i}\ne 0\quad (1\le i \le)$ $\Leftrightarrow$ $\boldsymbol{A}$ 满秩
      • Case 2 $\boldsymbol{A}$ 不可逆 $\Leftrightarrow$ $\det(\boldsymbol{A})= 0$ $\Leftrightarrow$ $\exists\ \lambda_{i} = 0\quad (1\le i \le)$ $\Leftrightarrow$ $\boldsymbol{A}$ 降秩
   5. 若 $\lambda_{0}$ 为特征值，其对应的特征向量，即 $(\lambda_{0}\boldsymbol{E} - \boldsymbol{A})\boldsymbol{X} = \boldsymbol{0}$ 的非零解

3. 相似矩阵 $\boldsymbol{A}, \boldsymbol{B}$ 为 $n$ 阶方阵，若存在可逆阵 $\boldsymbol{P}$，使得 $\boldsymbol{P}^{-1}\boldsymbol{A}\boldsymbol{P} = \boldsymbol{B}$，称 $\boldsymbol{A},\boldsymbol{B}$ 相似，记为 $\boldsymbol{A}\sim \boldsymbol{B}$
   1. $\begin{cases} \boldsymbol{A}\sim \boldsymbol{A},\quad (\boldsymbol{E}^{-1}\boldsymbol{A}\boldsymbol{E} = \boldsymbol{A}) \\ \boldsymbol{A} \sim \boldsymbol{B} \Rightarrow \boldsymbol{B} \sim \boldsymbol{A} \\ \boldsymbol{A} \sim \boldsymbol{B}, \boldsymbol{B} \sim \boldsymbol{C} \Rightarrow \boldsymbol{A}\sim \boldsymbol{C} \end{cases}$
   2. $\boldsymbol{A}\sim \boldsymbol{B} \nLeftarrow\Rightarrow r(\boldsymbol{A}) = r(\boldsymbol{B})$
   3. $\boldsymbol{A}\sim \boldsymbol{B} \nLeftarrow\Rightarrow \det(\lambda \boldsymbol{E} - \boldsymbol{A}) = \det(\lambda \boldsymbol{E} - \boldsymbol{B}) \Rightarrow \boldsymbol{A},\boldsymbol{B}\text{特征值相同}$

      Proof

      $\Rightarrow$ $$\begin{array}{ll} & \boldsymbol{A} \sim \boldsymbol{B} \Rightarrow \boldsymbol{B} = \boldsymbol{P}^{-1}\boldsymbol{AP} \\ & \det(\lambda \boldsymbol{E} - \boldsymbol{B}) = \det(\lambda \boldsymbol{E} - \boldsymbol{P}^{-1}\boldsymbol{AP}) \\ & = \det(\lambda \boldsymbol{P}^{-1}\boldsymbol{P} - \boldsymbol{P}^{-1}\boldsymbol{AP}) \\ & = \det(\boldsymbol{P}^{-1}(\lambda \boldsymbol{E} - \boldsymbol{A}) \boldsymbol{P}) \\ & = \det(\boldsymbol{P}^{-1})\det(\lambda\boldsymbol{E} - \boldsymbol{A})\det(\boldsymbol{P}) \\ \because & \det(\boldsymbol{P}^{-1})\det(\boldsymbol{P}) = 1 \\ \therefore & \det(\lambda \boldsymbol{E} - \boldsymbol{B}) = \det(\lambda \boldsymbol{E} - \boldsymbol{A}) = 0 \\ \end{array}$$ $\nLeftarrow$ 反例 $$\boldsymbol{A} = \left( \begin{matrix} 2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{matrix} \right), \boldsymbol{B} = \left( \begin{matrix} 0 & 1 & 2 \\ 0 & 0 & 1 \\ 0 & 0 & 2 \\ \end{matrix} \right)$$

   4. $\boldsymbol{A} \sim \boldsymbol{B}\Rightarrow \begin{cases} \text{tr}(\boldsymbol{A}) = \text{tr}(\boldsymbol{B}) \\ \det(\boldsymbol{A}) = \det(\boldsymbol{B}) \end{cases}$
   5. 若 $\boldsymbol{A} \sim \boldsymbol{B}\Rightarrow \begin{cases} \boldsymbol{A}^{\intercal}\sim \boldsymbol{B}^{\intercal} \\ f(\boldsymbol{A}) \sim f(\boldsymbol{B}) \end{cases}$
   6. 若 $\boldsymbol{A} \sim \boldsymbol{B}$，且 $\boldsymbol{A},\boldsymbol{B}$ 可逆 $\Rightarrow \begin{cases} \boldsymbol{A}^{-1}\sim \boldsymbol{B}^{-1} \\ \boldsymbol{A}^{*}\sim \boldsymbol{B}^{*} \end{cases}$

      Notes

      $$\begin{array}{ll} \boldsymbol{P}^{-1}\boldsymbol{AP} = \boldsymbol{B} \Rightarrow [(\boldsymbol{P^{-1}})^{\intercal}]^{-1}\boldsymbol{A}^{\intercal}(\boldsymbol{P^{-1}})^{\intercal} = \boldsymbol{B}^{\intercal} \\ \Rightarrow \boldsymbol{A}^{\intercal}\sim \boldsymbol{B}^{\intercal} \\\\ \boldsymbol{P}^{-1}\boldsymbol{AP} = \boldsymbol{B} \Rightarrow \boldsymbol{P}^{-1}\boldsymbol{A}^{-1}\boldsymbol{P} = \boldsymbol{B}^{-1} \\ \Rightarrow \boldsymbol{A}^{-1}\sim \boldsymbol{B}^{-1} \\\\ \boldsymbol{P}^{-1}\boldsymbol{A}^{-1}\boldsymbol{P} = \boldsymbol{B} \Rightarrow \boldsymbol{P}^{-1}\det(\boldsymbol{A})\boldsymbol{A}^{-1}\boldsymbol{P} = \det(\boldsymbol{B})\boldsymbol{B}^{-1} \\ \Rightarrow \boldsymbol{P}^{-1}\boldsymbol{A}^{*}\boldsymbol{P} = \boldsymbol{B}^{*}\Rightarrow \boldsymbol{A}^{*}\sim \boldsymbol{B}^{*} \\\\ \end{array}$$