Home Statistical Dictionary Cholesky Decomposition

Cholesky Decomposition

The factorization of a positive definite matrix into a lower triangular matrix times its own transpose -- the standard numerical test for positive definiteness, and the fast, stable way statistical software simulates correlated random variables from a target covariance matrix.

Linear Algebra for StatisticsAdvancedFree

In Plain English

The Cholesky decomposition breaks a positive definite matrix down into a single triangular 'square root' matrix, one that, multiplied by its own transpose, reconstructs the original. It's the computational engine behind two very practical statistical tasks, checking that a claimed covariance matrix is actually legitimate (the decomposition simply fails if it isn't), and generating simulated data with a specific target correlation structure by transforming independent random draws through this triangular matrix.

Definition

The Cholesky decomposition of a symmetric positive definite matrix $A$ expresses it as $A = L L^{⊤}$ , where $L$ is a unique lower triangular matrix with strictly positive diagonal entries. The decomposition exists if and only if $A$ is positive definite, making the successful computation of $L$ the standard practical test for positive definiteness, and is computed roughly twice as fast as a general matrix factorization since it exploits the matrix's symmetry. In statistics, the Cholesky decomposition is the standard method for simulating correlated multivariate normal data, generating a vector $z$ of independent standard normal draws and computing $x = μ + Lz$ produces a vector with exactly the target mean $μ$ and covariance matrix $Σ = L L^{⊤}$ , and it is also used as a numerically efficient way to solve the linear systems that arise in Gaussian process regression and Bayesian computation.

Formula

A = L L^{⊤}

Notation

L

—the unique lower triangular Cholesky factor

Properties

A matrix's Cholesky decomposition exists if and only if the matrix is symmetric positive definite -- attempting the decomposition and having it fail (encountering a negative number where a square root is required) is the standard, computationally cheap way statistical software checks whether a claimed covariance matrix is actually valid.
Generating correlated random data via $x = μ + Lz$ , with $z$ a vector of independent standard normal draws and $L$ the Cholesky factor of the target covariance matrix, is the standard simulation technique across Monte Carlo studies, Gaussian process sampling, and copula-based simulation whenever correlated multivariate normal data is needed.
The Cholesky decomposition is computationally cheaper than a general-purpose matrix factorization like LU decomposition because it exploits the input matrix's symmetry, needing to compute only the lower triangular factor rather than two separate factors, which is why it is the preferred method whenever the matrix being factored is already known to be symmetric positive definite.

At a Glance

RequiresA to be symmetric positive definite

FactorizationA = L L', L lower triangular with positive diagonal entries

Doubles asthe standard numerical test for positive definiteness

Main statistical usesimulating correlated multivariate normal data

Last updated September 8, 2026← Back to the Dictionary