# Manuals/calci/CHOLESKYFACTORIZATION

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**CHOLESKYFACTORIZATION(Matrix)**

- is the array of numeric elements

## Description

- This function gives the value of Cholesky factorization.
- It is called Cholesky Decomposition or Cholesky Factorization.
- In , is the set of values.
- The Cholesky Factorization is only defined for symmetric or Hermitian positive definite matrices.
- Every positive definite matrix A can be factored as =

is lower triangular with positive diagonal elements is is the conjugate transpose value of

- Every Hermitian positive-definite matrix has a unique Cholesky decomposition.
- Here , is set of values to find the factorization value.
- Partition matrices in = is

## Algorithm

- Determine and = =
- Compute from =
- This is a Cholesky Factorization of order

*If the matrix A is Hermitian and positive semi-definite, then it still has a decomposition of the form A = LL^T if the diagonal entries of L are allowed to be zero. *Also A can be written as LL^T for some invertible L, lower triangular or otherwise, then A is Hermitian and positive definite.

## Examples

A | B | C | |
---|---|---|---|

1 | 25 | 15 | -5 |

2 | 15 | 18 | 0 |

3 | -5 | 0 | 11 |

=CHOLESKYFACTORIZATION(A1:C3)

5 | 0 | 0 |

3 | 3 | 0 |

-1 | 1 | 3 |

A | B | |
---|---|---|

1 | 8 | 14 |

2 | 10 | 32 |

=CHOLESKYFACTORIZATION(A1:B2)

2.8284271247461903 | 0 |

3.5355339059327373 | 4.415880433163924 |

## Related Videos

## See Also

## References