# Cholesky Decomposition

*Quant & Pricing — Finicade finance glossary*

Cholesky decomposition factors a covariance matrix into a triangular matrix and its transpose — the step that turns independent draws into correlated ones.

Cholesky decomposition factors a covariance matrix into a triangular matrix times its transpose, and that factor is what turns independent random draws into correlated ones. It's the standard first step of any multi-asset Monte Carlo. It only works on a positive semi-definite matrix, so a correlation matrix assembled from mismatched historical windows will fail outright — a useful early warning that the inputs are inconsistent.

**Also known as:** Cholesky factorisation, correlated random numbers

**Related terms:** [Monte Carlo Simulation](https://finicade.com/glossary/monte-carlo), [Correlation](https://finicade.com/glossary/correlation), [Covariance](https://finicade.com/glossary/covariance), [Portfolio Variance](https://finicade.com/glossary/portfolio-variance), [Copula](https://finicade.com/glossary/copula)

Source: https://finicade.com/glossary/cholesky-decomposition