1.正定矩阵、半正定矩阵

推荐文章：如何理解正定矩阵和半正定矩阵

1.1 正定矩阵

1.1.1 判断正定矩阵

1.矩阵的所有特征值都为正数

下面以对称矩阵为例，对称矩阵的特征值为正数，所以对称矩阵是正定矩阵

$$\begin{gathered} {\lambda }_1>0、{\lambda }_2>0 \\ \{\begin{array}{l}{\lambda }_1{\lambda }_2=detS=ac-b^2>0\\ {\lambda }_1+{\lambda }_2=trS=a+c>0\end{array} \end{gathered}$$

2.矩阵消元后的每个主元都为正数

3.矩阵的所有顺序主子式的行列式都是正的

4.对于所有非零向量（不仅仅是特征向量）$\mathbf{x}$，都有 ${\mathbf{x}}^{T}S\mathbf{x}>0$【在许多应用中，${\mathbf{x}}^{T}S\mathbf{x}$ 的结果代表系统中的能量】

$$\begin{gathered} S\mathbf{x}=\lambda \mathbf{x} \\ {\mathbf{x}}^{T}S\mathbf{x}={\mathbf{x}}^T\lambda \mathbf{x} \\ {\mathbf{x}}^{T}S\mathbf{x}=\lambda {\mathbf{x}}^T\mathbf{x} \\ {\mathbf{x}}^{T}S\mathbf{x}=\lambda ||\mathbf{x}|{|}^2 \\ \lambda >0 \\ {\mathbf{x}}^{T}S\mathbf{x}=\lambda ||\mathbf{x}|{|}^2>0 \\ {\mathbf{x}}^{T}S\mathbf{x}>0 \end{gathered}$$

5.如果矩阵A的列是线性无关的，则$S=A^{T}A$是正定矩阵

$${\mathbf{x}}^{T}S\mathbf{x}={\mathbf{x}}^T(A^{T}A)\mathbf{x}=({\mathbf{x}}^T{A}^T)(A\mathbf{x})=(A\mathbf{x}{)}^T(A\mathbf{x})=||A\mathbf{x}|{|}^2>0$$

综上：矩阵正定的五个等价判定

1.2 半正定矩阵

1.2.1 判定半正定矩阵

半正定矩阵是正定矩阵的推广，相比正定矩阵，判定条件多了一个可以等于0的条件

1.所有矩阵特征值 $\ge 0$
2.消元后的所有主元 $\ge 0$
3.矩阵的所有顺序主子式的行列式都 $\ge 0$
4.对于所有非零向量（不仅仅是特征向量）$\mathbf{x}$，都有 ${\mathbf{x}}^{T}S\mathbf{x}\ge 0$
5.如果矩阵A的列是线性有关的，则$S=A^{T}A$是正定矩阵

1.3 椭圆 $a{x}^2+2bxy+c{y}^2=1$

例子：

1.3.1 与对称矩阵$S$有关的椭圆

$$\begin{gathered} {\mathbf{x}}^{T}S\mathbf{x}=1 \\ \left[\begin{array}{cc}x& y\end{array}\right]\left[\begin{array}{cc}5& 4\\ 4& 5\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=1 \\ S=\left[\begin{array}{cc}5& 4\\ 4& 5\end{array}\right] \end{gathered}$$
求解矩阵 $S$ 的特征值和特征向量
$$\begin{gathered} detS={\lambda }_1{\lambda }_2=9 \\ trS={\lambda }_1+{\lambda }_2=10 \\ {\lambda }_1=9、{\lambda }_2=1 \end{gathered}$$

$$\begin{gathered} S{\mathbf{x}}_1={\lambda }_1{\mathbf{x}}_1 \\ \left[\begin{array}{cc}5& 4\\ 4& 5\end{array}\right]\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]=9\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right] \\ {a}_1=a_2 \\ {\mathbf{x}}_1=\left[\begin{array}{c}1\\ 1\end{array}\right] \end{gathered}$$

$$\begin{gathered} S{\mathbf{x}}_2={\lambda }_2{\mathbf{x}}_2 \\ \left[\begin{array}{cc}5& 4\\ 4& 5\end{array}\right]\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]=\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right] \\ {a}_1=-a_2 \\ {\mathbf{x}}_2=\left[\begin{array}{c}1\\ -1\end{array}\right] \end{gathered}$$

$${\mathbf{q}}_1=\frac{{\mathbf{x}}_1}{||{\mathbf{x}}_1||}、{\mathbf{q}}_2=\frac{{\mathbf{x}}_2}{||{\mathbf{x}}_2||}$$

$$\begin{gathered} Q=[{\mathbf{q}}_1{\mathbf{q}}_2]=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right] \\ \Lambda =\left[\begin{array}{cc}{\lambda }_1& 0\\ 0& {\lambda }_2\end{array}\right]=\left[\begin{array}{cc}9& 0\\ 0& 1\end{array}\right] \end{gathered}$$

In xy system，axes are along the eigenvectors of $S$

1.3.2 与特征值矩阵$\Lambda$有关的椭圆

$$\begin{gathered} S=Q\Lambda Q^T（PrincipalAxisTheorem） \\ {\mathbf{x}}^{T}S\mathbf{x}=({\mathbf{x}}^{T}Q)\Lambda (Q^T\mathbf{x})=X^T\Lambda X \end{gathered}$$

$$\begin{gathered} {\mathbf{x}}^{T}S\mathbf{x}=\left[\begin{array}{cc}x& y\end{array}\right]\left[\begin{array}{cc}5& 4\\ 4& 5\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=1 \\ S=Q\Lambda Q^T=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]\left[\begin{array}{cc}9& 0\\ 0& 1\end{array}\right]\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right] \\ \left[\begin{array}{c}X\\ Y\end{array}\right]=Q^T\left[\begin{array}{c}x\\ y\end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}{c}x+y\\ x-y\end{array}\right] \\ X=\frac{x+y}{\sqrt{2}}、Y=\frac{x-y}{\sqrt{2}} \end{gathered}$$

$$\begin{gathered} {\lambda }_1{X}^2+{\lambda }_2{Y}^2=1 \\ 9(\frac{x+y}{\sqrt{2}}{)}^2+(\frac{x-y}{\sqrt{2}}{)}^2=1 \end{gathered}$$

In XY system，axes are along the eigenvectors of $\Lambda$

1.4 重要应用：检验最小值

矩阵正定说明其表示的二次曲面开口朝上
矩阵负定说明其表示的二次曲面开口朝下