---
tag: 0WKG8CYY6A
kind: statement
role: theorem
name: Cholesky decomposition
msc: [15A23, 65F05]
uses: [0QQCG3JWJ4, 0EZSD6T9E0]
sources:
  - url: https://en.wikipedia.org/wiki/Cholesky_decomposition
    locator: Statement
same_as:
  wikidata: Q515375
  wikipedia: Cholesky decomposition
---

## Statement

Let $A$ be a [[positive definite|0KX66CZMBB]] $n \times n$ complex Hermitian matrix.
Then there is exactly one [[lower triangular matrix|00J3KZP8Y0]] $L$ with positive real diagonal entries such that
$$A = L L^*.$$
If $A$ is real, then $L$ is real.

## Remarks

The factor $L$ is the *Cholesky factor* of $A$.
For example, $\begin{pmatrix} 4 & 2 \\ 2 & 5 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 1 & 2 \end{pmatrix} \begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix}$.
It needs about half the work of an LU decomposition, and it solves systems $Ax = b$ with positive definite $A$.
A positive semidefinite matrix also has a decomposition $A = L L^*$, but $L$ need not be unique.

## Explanations

### L1 human

A positive definite matrix is the product of a lower triangular matrix and its conjugate transpose.
This is like writing a positive number as the square of its positive square root.
It is the fastest standard way to solve linear systems with such a matrix.