Linear Algebra

Singular Value Decomposition

A = UEV^T (rotation, stretch, rotation)

U = Eigenvalue of AA^T : Unit Vector : Orthogonal

V = Eigenvalue of A^TA : Unit Vector : Orthogonal

U = [u1, u1, u3, .....,un]

E = [sing1,

sing2,

sing3,

...,

singn]

V ^T= [V1,

V2,

vn]

A in terms of Rank1 = u1sing1v1^T + u2sing2v2^T + ......

Rotation:

2x2:

{\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}{\text{ (rotation), }}\qquad {\begin{bmatrix}\cos \theta &\sin \theta \\\sin \theta &-\cos \theta \\\end{bmatrix}}{\text{ (reflection)}}

3x3:

{\begin{bmatrix}\cos \alpha \cos \gamma -\sin \alpha \sin \beta \sin \gamma &-\sin \alpha \cos \beta &-\cos \alpha \sin \gamma -\sin \alpha \sin \beta \cos \gamma \\\cos \alpha \sin \beta \sin \gamma +\sin \alpha \cos \gamma &\cos \alpha \cos \beta &\cos \alpha \sin \beta \cos \gamma -\sin \alpha \sin \gamma \\\cos \beta \sin \gamma &-\sin \beta &\cos \beta \cos \gamma \end{bmatrix}}

Ex:

[a11, a12

a21, a22]

cos (theta) = a11

sin (theta) = a21

[a11, a12, a13

a21, a22,a23

a31, a32,a33]

-sin(Beta) = a32

cos(alpha).cos(Beta) = a22

cos(Beta)sin(gama) = a31

SVD Example
SVD : YouTube