特征值分解与奇异值分解

1.特征值分解(谱分解)

A=QΛQH{Q=[u1⃗u2⃗⋯uM⃗]Λ=diag{λ1,λ2,⋯ ,λM} \begin{align} &A=Q\Lambda Q^H\\ &\left\{ \begin{aligned} Q&=\begin{bmatrix} \vec{u_1} &\vec{u_2} &\cdots&\vec{u_M} \end{bmatrix}\\ \Lambda&=diag\{\lambda_1,\lambda_2,\cdots,\lambda_M\} \end{aligned} \right. \end{align} ​A=QΛQH{QΛ​=[u1​​​u2​​​⋯​uM​​​]=diag{λ1​,λ2​,⋯,λM​}​​​

2.奇异值分解

AL×M=YL×L[ΣK×K0K×(M−K)0(L−K)×K0(L−K)×(M−K)]XM×MH{X=[x1⃗x2⃗⋯xM⃗]Y=[y1⃗y2⃗⋯yM⃗]Σ=diag{σ1,σ2,⋯ ,σM} \begin{align} &A_{L\times M}=Y_{L\times L} \begin{bmatrix} \Sigma_{K\times K } & 0_{K\times (M-K)}\\ 0_{(L-K)\times K}& 0_{(L-K)\times (M-K)} \end{bmatrix} X_{M\times M}^H \\ &\left\{ \begin{aligned} X&=\begin{bmatrix} \vec{x_1} &\vec{x_2} &\cdots&\vec{x_M} \end{bmatrix}\\ Y&=\begin{bmatrix} \vec{y_1} &\vec{y_2} &\cdots&\vec{y_M} \end{bmatrix}\\ \Sigma&=diag\{\sigma_1,\sigma_2,\cdots,\sigma_M\} \end{aligned} \right. \end{align} ​AL×M​=YL×L​[ΣK×K​0(L−K)×K​​0K×(M−K)​0(L−K)×(M−K)​​]XM×MH​⎩⎨⎧​XYΣ​=[x1​​​x2​​​⋯​xM​​​]=[y1​​​y2​​​⋯​yM​​​]=diag{σ1​,σ2​,⋯,σM​}​​​

3.特征值分解与奇异值分解关系

$$\begin{array}{r}\{\begin{array}{rl}{A}^{H}A& =X\left[\begin{array}{cc}{\Sigma }^2& 0_{K\times (M-K)}\\ 0_{(M-K)\times K}& 0_{(M-K)\times (M-K)}\end{array}\right]X^H\\ A{A}^H& =Y\left[\begin{array}{cc}{\Sigma }^2& 0_{K\times (L-K)}\\ 0_{(L-K)\times K}& 0_{(L-K)\times (L-K)}\end{array}\right]Y^H\end{array}\end{array}$$

(1)特征值与奇异值

$A^{H}A$与$A{A}^H$的非零特征值等于$A_{L\times M}$的非零奇异值的平方:
$$\Lambda ={\Sigma }^2$$

(2)特征向量与奇异向量

左奇异向量矩阵$Y$是$A{A}^H$的特征向量矩阵
左奇异向量矩阵$X$是$A^{H}A$的特征向量矩阵