# Manuals/calci/SVD

**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

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 |

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 |

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## See Also

## References