Section01_概念

1. 二次型 $f = \boldsymbol{X}^{\intercal}\boldsymbol{AX} \begin{cases} \text{标准二次型} \Leftrightarrow \boldsymbol{A} \text{为对角阵}\\ \text{非标准二次型} \Leftrightarrow \boldsymbol{A} \text{为对称但不对角矩阵} \end{cases}$
2. 标准化 $$f = \boldsymbol{X}^{\intercal}\boldsymbol{AX} \xlongequal[\boldsymbol{P} \text{可逆}]{\boldsymbol{X} = \boldsymbol{PY}} \boldsymbol{Y}^{\intercal}(\boldsymbol{P}^{\intercal}\boldsymbol{AP})\boldsymbol{Y}$$
   • 若 $\boldsymbol{P}^{\intercal}\boldsymbol{AP} = \left( \begin{matrix} \ell_{1} \\ & \ddots \\ & & \ell_{n} \end{matrix} \right)$，则 $f = \ell_{1}y_{1}^{2} + \cdots + \ell_{n}y_{n}^{2}$

     Notes

     1. $\boldsymbol{X} = \boldsymbol{PY}$，$\boldsymbol{P}$ 可逆
     2. $\boldsymbol{P}^{\intercal}\boldsymbol{AP} = \left( \begin{smallmatrix} \ell_{1} \\ & \ddots \\ & & \ell_{n} \end{smallmatrix} \right)$
     3. 矩阵合同 $\boldsymbol{A}, \boldsymbol{B}$ 为 $n$ 阶方阵，若 $\exists$ 可逆阵 $\boldsymbol{P}$，使得 $\boldsymbol{P}^{\intercal}\boldsymbol{AP} = \boldsymbol{B}$，称 $\boldsymbol{A}$ 与 $\boldsymbol{B}$ 合同，记为 $\boldsymbol{A}\cong \boldsymbol{B}$