Manuals/calci/CHOLESKY

CHOLESKY (Matrix)

$Matrix$ is the array of numeric elements.

Description

This function gives the value of Cholesky factorization.
It is called Cholesky Decomposition or Cholesky Factorization.
The Cholesky Factorization is only defined for symmetric or Hermitian positive definite matrices.
Every positive definite matrix A can be factored as $A$ = $LL^{T}$

 $L$  is lower triangular with positive diagonal elements
 $L^{T}$  is is the conjugate transpose value of  $L$

Every Hermitian positive-definite matrix has a unique Cholesky decomposition.
Here $CHOLESKY(Matrix)$ , $Matrix$ is set of values to find the factorization value.
Partition matrices in $A$ = $LL^{T}$ is

${\begin{bmatrix}a_{11}&A_{21}^{T}\\A_{21}&A_{22}\end{bmatrix}}={\begin{bmatrix}l_{11}&0\\L_{21}&L_{22}\end{bmatrix}}{\begin{bmatrix}l_{11}&L_{21}^{T}\\0&L_{22}^{T}\end{bmatrix}}={\begin{bmatrix}l_{11}^{2}&L_{11}L_{21}^{T}\\L_{11}L_{21}&L_{21}L_{21}^{T}+L_{22}L_{22}^{T}\end{bmatrix}}$