# Cholesky Factorization The Cholesky factorization of a symmetric positive definite matrix $A$ is the factorization $A=R^T R$, where $R$ is upper triangular with positive diagonal elements. It is a generalization of the property that a positive real number has a unique positive square root. The Cholesky factorization always exists and the requirement that the diagonal of $R$ be positive ensures that it is unique. For example: $G_4=\left[\begin{array}{llll}1 & 1 & 1 & 1 \\ 1 & 2 & 1 & 2 \\ 1 & 1 & 3 & 1 \\ 1 & 2 & 1 & 4\end{array}\right]=\left[\begin{array}{cccc}1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 0 & \sqrt{2} & 0 \\ 1 & 1 & 0 & \sqrt{2}\end{array}\right]\left[\begin{array}{cccc}1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & \sqrt{2} & 0 \\ 0 & 0 & 0 & \sqrt{2}\end{array}\right]$. For positive semidefinite matrix, there always exists a Cholesky factorization but it may not be unique. --- ## References