Cholesky decomposition

Statement

Let $A$ be a positive definite $n \times n$ complex Hermitian matrix. Then there is exactly one lower triangular matrix $L$ 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.