Manuals/calci/SVD

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SVD(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
  • 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
 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 0.8872306883463938 0.4082482904638627
0.5205873894647103 0.2496439529883539 -0.8164965809277261
0.82633754056112 -0.3879427823696853 0.4082482904638632
16.848103352614217  0  0
0  1.0683695145547085  0
0  0  0
0.4796711778777768 -0.7766909903215562 0.40824829046386296
0.5723677939720628 -0.0756864701045544 -0.816496580927726
0.6650644100663488 0.6253180501124471 0.40824829046386313

Related Videos

Singular Value Decomposition

See Also

References