1.特征值分解(谱分解)
A = Q Λ Q H { Q = [ u 1 ⃗ u 2 ⃗ ⋯ u M ⃗ ] Λ = d i a g { λ 1 , λ 2 , ⋯ , λ M } \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=[u1u2⋯uM]=diag{λ1,λ2,⋯,λM}
2.奇异值分解
A L × M = Y L × L [ Σ K × K 0 K × ( M − K ) 0 ( L − K ) × K 0 ( L − K ) × ( M − K ) ] X M × M H { X = [ x 1 ⃗ x 2 ⃗ ⋯ x M ⃗ ] Y = [ y 1 ⃗ y 2 ⃗ ⋯ y M ⃗ ] Σ = d i a g { σ 1 , σ 2 , ⋯ , σ M } \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×K0(L−K)×K0K×(M−K)0(L−K)×(M−K)]XM×MH⎩ ⎨ ⎧XYΣ=[x1x2⋯xM]=[y1y2⋯yM]=diag{σ1,σ2,⋯,σM}
3.特征值分解与奇异值分解关系
{ A H A = X [ Σ 2 0 K × ( M − K ) 0 ( M − K ) × K 0 ( M − K ) × ( M − K ) ] X H A A H = Y [ Σ 2 0 K × ( L − K ) 0 ( L − K ) × K 0 ( L − K ) × ( L − K ) ] Y H \begin{align} \left\{ \begin{aligned} A^HA&=X \begin{bmatrix} \Sigma^2 & 0_{K\times (M-K)}\\ 0_{(M-K)\times K}& 0_{(M-K)\times (M-K)} \end{bmatrix} X^H \\ AA^H&=Y \begin{bmatrix} \Sigma^2 & 0_{K\times (L-K)}\\ 0_{(L-K)\times K}& 0_{(L-K)\times (L-K)} \end{bmatrix} Y^H \\ \end{aligned} \right. \end{align} ⎩ ⎨ ⎧AHAAAH=X[Σ20(M−K)×K0K×(M−K)0(M−K)×(M−K)]XH=Y[Σ20(L−K)×K0K×(L−K)0(L−K)×(L−K)]YH
(1)特征值与奇异值
A
H
A
A^HA
AHA与
A
A
H
AA^H
AAH的非零特征值等于
A
L
×
M
A_{L\times M}
AL×M的非零奇异值的平方:
Λ
=
Σ
2
\Lambda = \Sigma ^2
Λ=Σ2
(2)特征向量与奇异向量
左奇异向量矩阵
Y
Y
Y是
A
A
H
AA^H
AAH的特征向量矩阵
左奇异向量矩阵
X
X
X是
A
H
A
A^HA
AHA的特征向量矩阵
