二次型

Quadratic Form
在线性代数和优化理论中非常重要

含有 $n$ 个变量 $x_{1}, x_{2}, \dots, x_{n}$ 的二次齐次函数称为二次型
若二次型只含有平方项，则称为标准二次型

\begin{aligned} f (x_{1}, x_{2}, \dots, x_{n}) & = a_{11} x_{1}^{2} + \dots + a_{n n} x_{n}^{2} + 2 a_{12} x_{1} x_{2} + \dots + 2 a_{n - 1, n} x_{n - 1} x_{n} \end{aligned}

将 $a_{j i} = a_{i j}$ 即写为 $2 a_{i j} x_{i} x_{j} = a_{i j} x_{i} x_{j} + a_{j i} x_{j} x_{i}$ ，并写为矩阵的形式：

\begin{aligned} f & = a_{11} x_{1}^{2} + \dots + a_{n n} x_{n}^{2} + a_{12} x_{1} x_{2} + a_{21} x_{2} x_{1} + \dots + a_{n - 1, n} x_{n - 1} x_{n} + a_{n, n - 1} x_{n} x_{n - 1} \\ = x_{1} (a_{11} x_{1} + a_{12} x_{2} + \dots + a_{1 n} x_{n}) + \dots + x_{n} (a_{n 1} x_{1} + a_{n 2} x_{2} + \dots + a_{n n} x_{n}) \\ = (x_{1}, x_{2}, \dots, x_{n}) (\begin{array}{c} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{n 1} & a_{n 2} & \dots & a_{n n} \end{array}) (\begin{array}{c} x_{1} \\ x_{2} \\ ⋮ \\ x_{n} \end{array}) \\ = x^{T} A x \end{aligned}

$A$ 为对称矩阵，称为二次型的矩阵
给定一个二次型就能唯一确定一个对称矩阵
给定一个对称矩阵就能唯一确定一个二次型

寻找可逆的线性变换 $x = C y$ 使得原二次型变为标准型

{\begin{cases} x_{1} = c_{11} y_{1} + c_{12} y_{2} + \dots + c_{1 n} y_{n} \\ x_{2} = c_{21} y_{1} + c_{22} y_{2} + \dots + c_{2 n} y_{n} \\ \dots \dots \\ x_{n} = c_{n 1} y_{1} + c_{n 2} y_{2} + \dots + c_{n n} y_{n} \end{cases}

\begin{aligned} f & = x^{T} A x = (C y)^{T} A C y = y^{T} (C^{T} A C) y \\ = k_{1} y_{1}^{2} + k_{2} y_{2}^{2} + \dots + k_{n} y_{n}^{2} \\ = (y_{1} + y_{2} + \dots + y_{n}) (\begin{array}{c} k_{1} \\ k_{2} \\ ⋱ \\ k_{n} \end{array}) (\begin{array}{c} y_{1} \\ y_{2} \\ ⋮ \\ y_{n} \end{array}) \end{aligned}