Cholesky Factorization

Cholesky decomposition

Cholesky decomposition or factorization is a decomposition of a symmetric and positive definite real matrix into the product of a lower triangular matrix and its conjugate transpose. This is useful for efficient numerical solutions. Cholesky decomposition is roughly twice as efficient as the LU decomposition used for solving systems of linear equations.

A square matrix $𝐴$ defined over the real numbers is defined as symmetric when $𝐴 = 𝐴^𝑇$.

A symmetric $(𝑛 \times 𝑛)$ real matrix, $𝑀$, is said to be positive definite if the scalar $𝑧^𝑇𝑀𝑧$ is strictly positive for every nonzero column vector $𝑧$ of $𝑛$ real numbers.

The Cholesky decomposition of a square symmetric positive definite matrix $𝐴$ is a decomposition of the form:

$$𝐴 = 𝐿𝐿^𝑇$$

In this equation:

• $𝐿$ is a lower triangular matrix with real and positive diagonal entries.
• $𝐿^𝑇$ denotes the transpose of $𝐿$.

Every real-valued symmetric positive-definite matrix has a unique Cholesky decomposition.