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
2.4605048700187635 0 0	0 1.699628148275319 0	0 0 0.23912327825655444
0.4152614854539272 -0.566497504206459 -0.711854145923831	0.9017526469087841 0.15312282248454143 0.4042221728546923	-0.12000026038137995 -0.8097122815928015 0.5744266346072238

Spreadsheet

	A	B	C
1	1	2	3
2	4	5	6
3	7	8	9

=SVD(A1:C3)

0.21483723836830051 -0.7766909903215562 0.40824829046386296

0.5723677939720628 -0.0756864701045544 -0.816496580927726

0.6650644100663488 0.6253180501124471 0.40824829046386313