Difference between revisions of "Manuals/calci/SVD"

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*Where the columns of U and V are orthonormal and the matrix S is diagonal with positive real entries
 
*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).
 
*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.  
 
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.  
Line 15: Line 17:
 
  V is an n × n unitary matrix over K, and
 
  V is an n × n unitary matrix over K, and
 
  <math>V^T</math> is the conjugate transpose of V.
 
  <math>V^T</math> is the conjugate transpose of V.
 +
 +
==Example==
 +
{| class="wikitable"
 +
|+Spreadsheet
 +
|-
 +
! !! A !! B !!C     
 +
|-
 +
! 1
 +
| 1 || 0 || 1 
 +
|-
 +
! 2
 +
| -1 || -2 || 0
 +
|-
 +
! 3
 +
|0 || 1 || -1
 +
|}
 +
=SVD(A1:C3)
 +
{| border="1" cellpadding="5" cellspacing="0"
 +
|-
 +
|
 +
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
 +
|}
 +
 +
{| class="wikitable"
 +
|+Spreadsheet
 +
|-
 +
! !! A !! B !!C     
 +
|-
 +
! 1
 +
| 1 || 2 || 3 
 +
|-
 +
! 2
 +
| 4 || 5 || 6
 +
|-
 +
! 3
 +
| 7|| 8 || 9
 +
|}
 +
=SVD(A1:C3)
 +
{| border="1" cellpadding="5" cellspacing="0"
 +
|-
 +
|
 +
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==
 +
 +
{{#ev:youtube|v=Ls2TgGFfZnU|280|center|Singular Value Decomposition}}
 +
 +
==See Also==
 +
*[[Manuals/calci/LUDECOMPOSITION  | LUDECOMPOSITION ]]
 +
*[[Manuals/calci/CHOLESKYFACTORIZATION | CHOLESKYFACTORIZATION ]]
 +
*[[Manuals/calci/QRDECOMPOSITION | QRDECOMPOSITION ]]
 +
 +
==References==
 +
*[https://en.wikipedia.org/wiki/Singular_value_decomposition  Decomposition]
 +
 +
 +
*[[Z_API_Functions | List of Main Z Functions]]
 +
 +
*[[ Z3 |  Z3 home ]]

Latest revision as of 13:34, 2 May 2019

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


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