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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
.
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