# 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 ```

## Related Videos

Singular Value Decomposition