# 【陆勤践行】奇异值分解 - 最清晰易懂的svd 科普

******线性变换的几何解释**

Mvi = λivi

2*2矩阵奇异值分解的几何实质是：对于任意2*2矩阵，总能找到某个正交网格到另一个正交网格的转换与矩阵变换相对应。

Mv1= σ1u1

Mv2= σ2u2

x= (v1·****x)v1+ (v2·****x)v2

Mx= (v1·****x)Mv1+ (v2·****x)Mv2

Mx= (v1·****x) σ1u1+ (v2·****x) σ2u2

v·x = vTx

Mx=u1σ1v1Tx+u2σ2v2Tx

M=u1σ1v1T+u2σ2v2T

M = UΣ_V_T

Mvi= σiui

Mvj= σjuj

Mvi·Mvj=viTMTMvj=vi·MTMvj= λjvi·****vj= 0

Mvi·Mvj= σiσjui·****uj= 0

M = u1σ1 v1T

σ1= 14.72

σ2= 5.22

σ3= 3.31

M=u1σ1v1T+u2σ2v2T+u3σ3v3T

σ1= 14.15 σ2= 4.67 σ3= 3.00 σ4= 0.21 σ5= 0.19 … σ15= 0.05

M≈u1σ1v1T+u2σ2v2T+u3σ3v3T

Noisy image Improved image

```-1.03	0.74	-0.02	0.51	-1.31	0.99	0.69	-0.12	-0.72	1.11
-2.23	1.61	-0.02	0.88	-2.39	2.02	1.62	-0.35	-1.67	2.46```

σ1= 6.04 σ2= 0.22

References:

• Gilbert Strang, ** Linear Algebra and Its Applications**. Brooks Cole.Strang’s book is something of a classic though some may find it to be a little too formal.
• William H. Press et al, Numercial Recipes in C: The Art of Scientific Computing. Cambridge University Press.Authoritative, yet highly readable. Older versions are available online.
• Dan Kalman, A Singularly Valuable Decomposition: The SVD of a Matrix, The College Mathematics Journal ** 27** (1996), 2-23.Kalman’s article, like this one, aims to improve the profile of the singular value decomposition. It also a description of how least-squares computations are facilitated by the decomposition.
• If You Liked This, You’re Sure to Love That, The New York Times, November 21, 2008.This article describes Netflix’s prize competition as well as some of the challenges associated with it.

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