Manuals/calci/SVD

SVD(Matrix)

• $Matrix$ is the set of values.

Description

• The singular value decomposition of a matrix A is the factorization of A into the product of three matrices $A=USV^{T}$
• Where the columns of U and V are orthonormal and the matrix S is diagonal with positive real entries
• Singular value decomposition is defined for all matrices (rectangular or square).
• The rank of a matrix is equal to the number of non-zero singular values.

Suppose A is a m × n matrix whose entries come from the field K, which is either the field of real numbers or the field of complex numbers.

• Then there exists a factorization, called a singular value decomposition of A, of the form

where
U is an m × m unitary matrix,
S is a diagonal m × n matrix with non-negative real numbers on the diagonal,
V is an n × n unitary matrix over K, and
$V^{T}$ is the conjugate transpose of V.

Example

Spreadsheet

	A	B	C
1	1	0	1
2	-1	-2	0
3	0	1	-1

=SVD(A1:C3)

0.12000026038175768 || -0.8097122815927454 || -0.5744266346072238

-0.9017526469088556 || 0.15312282248412068 || -0.40422217285469236

0.41526148545366265 || 0.5664975042066532 || -0.7117854145923829