﻿ Cholesky – Algosim documentation
Algosim documentation: Cholesky

# Cholesky

Performs a Cholesky decomposition of a matrix.

## Syntax

• `Cholesky(A)`

• `A` is a positive-definite matrix

## Description

If `A` is a positive-definite Hermitian matrix, then `Cholesky(A)` returns the Cholesky factor `R` of `A`. This is an upper triangular matrix such that `A = R* ⋅ R`.

## Examples

`A ≔ ❨❨141, 101, 30, 72, 91❩, ❨101, 127, −7, 50, 81❩, ❨30, −7, 42, 4, −11❩, ❨72, 50, 4, 64, 63❩, ❨91, 81, −11, 63, 96❩❩`
```⎛141  101   30   72   91⎞
⎜101  127   −7   50   81⎟
⎜ 30   −7   42    4  −11⎟
⎜ 72   50    4   64   63⎟
⎝ 91   81  −11   63   96⎠
```
`R ≔ Cholesky(A)`
```⎛   11.874342087    8.50573440277     2.5264557632    6.06349383168    7.66358248171⎞
⎜              0    7.39273171903   −3.85369884705  −0.212975141659    2.13934489144⎟
⎜              0                0    4.55697558397   −2.66402370834   −4.85350662485⎟
⎜              0                0                0    4.48237234326   0.905253992078⎟
⎝              0                0                0                0    2.88386817207⎠
```
`R* ⋅ R`
```⎛141  101   30   72   91⎞
⎜101  127   −7   50   81⎟
⎜ 30   −7   42    4  −11⎟
⎜ 72   50    4   64   63⎟
⎝ 91   81  −11   63   96⎠
```