Singular Value Decomposition

V angle β

singular value

σ

1

singular value

σ

2

U angle ω

	b = ΣU T V x = Ax

	singular value decomposition

b = Uz	z = Σy	y = T V x

The singular value decomposition is a factorization of a matrix

A

into

A=UΣ

T

V

. A vector

x

is first rotated by an angle β via

T

V

x

, then

T

V

x

is scaled by a diagonal matrix

Σ

to form

Σ

T

V

x

. Finally the vector

Σ

T

V

x

is rotated by an angle ω to form

b=UΣ

T

V

x=Ax

.