特征值分解与奇异值分解

1.特征值分解(谱分解)

$$\begin{array}{rl}& A=Q\Lambda Q^H\\ & \{\begin{array}{rl}Q& =\left[\begin{array}{cccc}{u}_1& u_2& \cdots & u_M\end{array}\right]\\ \Lambda & =diag\{{\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_M\}\end{array}\end{array}$$

2.奇异值分解

$$\begin{array}{rl}& A_{L\times M}=Y_{L\times L}\left[\begin{array}{cc}{\Sigma }_{K\times K}& 0_{K\times (M-K)}\\ 0_{(L-K)\times K}& 0_{(L-K)\times (M-K)}\end{array}\right]X_{M\times M}^H\\ & \{\begin{array}{rl}X& =\left[\begin{array}{cccc}{x}_1& x_2& \cdots & x_M\end{array}\right]\\ Y& =\left[\begin{array}{cccc}{y}_1& y_2& \cdots & y_M\end{array}\right]\\ \Sigma & =diag\{{\sigma }_1,{\sigma }_2,\cdots ,{\sigma }_M\}\end{array}\end{array}$$

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