# Manuals/calci/CHOLESKYFACTORIZATION

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

1. Determine and 2. =  = 3. Compute from
4. = • 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