# 奇异值分解Singular Value Decomposition (SVD)

## Singular Value Decomposition (SVD)

Posted by xuepro on July 30, 2019

### 初等变换

-1）对调矩阵的两行，记为$E_{i,j} = r_i \leftrightarrow r_j$

-2）将某一行的元素乘以非零常数，记为 $E_{s} =r_i\times s$

-3）将某一行的元素乘以非零常数后，加到另一行上，记为$E_{i,j(s)}r_i+r_j\times s$

• 对一个矩阵$A_{mn}$进行初等行变换相当于用相应的初等矩阵$T_{mm}$左乘矩阵A。

• 对一个矩阵$A_{mn}$进行初等列变换相当于用相应的初等矩阵$T_{nn}$右乘矩阵A。

$$A_1^TA_1 = A_1A_1^T = I$$ $$A_2^TA_2 = A_2A_2^T = I$$ 那么： $$(A_1A_2)^TA_1A_2 = A_2^TA_1^T A_1A_2 = I$$ $$A_1A_2(A_1A_2)^T = A_1A_2A_2^TA_1^T = I$$

$A_{mn} = U_{mm} D_{mn} V_{nn}^T$

$A ={E_1}^T \cdots {E_h}^TD{\hat{E_k}}^T \cdots {\hat{E_1}}$

$${U_{mm}}^TU_{mm} = U_{mm}{U_{mm}}^T = I$$ $${V_{nn}}^TV_{nn} = V_{nn}{V_{nn}}^T = I$$

$A_{mn}{A_{mn}}^T U_{mm} = U_{mm} D_{mn} {V_{nn}}^T V_{nn} {D_{mn}}^T {U_{mm}}^T U_{mm} = U_{mm} D_{mn}{D_{mn}}^T$

$AA^TU = U D_{mn}{D_{mn}}^T = US_{mm}$ $A^TAV = V {D_{mn}}^T D_{mn} = = V \hat{S}_{nn}$

$D^T = \left(\begin{array}{cc} d_{11}&0\\ 0&d_{22}\\ 0&0 \end{array}\right)$

$D D^T = \begin{bmatrix} {d_{11}}^2 & 0 \\ 0 &{d_{22}}^2 \\ \end{bmatrix}$ $D^T D = \begin{bmatrix} {d_{11}}^2 & 0& 0 \\ 0 &{d_{22}}^2& 0 \\ 0 &0 & 0 \end{bmatrix}$

$A^TAV = A^TA (V_1,V_2,\cdots, V_n) = (V_1 {d_{11}}^2,V_2{d_{22}}^2,\cdots, V_n{d_{nn}}^2 )$

$U = AVD^{-1}$ $A^T = VD^TU^T$

Singular Value Decomposition (SVD) Tutorial: Applications, Examples, Exercises