Cholesky

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matrix_cholesky

Cholesky factor L of a symmetric positive-definite matrix (A = L·Lᵀ)

Signature

Inputs

aSignal|Arrayrequired— A symmetric positive-definite 2-D matrix.

Outputs

lMatrix— The lower-triangular Cholesky factor $L$ with $A = L L^\top$.

Description

Cholesky factorizes a symmetric positive-definite matrix as $A = L L^{⊤}$ , returning the lower-triangular factor $L$ via the MKL LAPACK backend. It is roughly twice as efficient as a general LU factorization and is the standard tool for sampling correlated Gaussians, solving SPD systems, and whitening data.

The input must be genuinely positive-definite; a non-SPD matrix (indefinite or merely positive-semidefinite) produces a graceful backend error rather than a panic — which itself doubles as a positive-definiteness test. The output is a dimensionless lower-triangular matrix.

Mathematics

A = L L^{⊤}, L lower-triangular, L_{ii} > 0

A ≻ 0 ⟺ Cholesky succeeds

Examples

Correlating white noise

Given a target covariance $Σ$ , wire it into a to obtain $L$ with $Σ = L L^{⊤}$ . Multiplying $L$ by a vector of independent unit-variance samples (via tensor_matmul) produces samples with exactly the covariance $Σ$ .

Applications

Sampling correlated multivariate Gaussian noise ($x = L z$).
Efficiently solving SPD linear systems and normal equations.
Testing positive-definiteness of a covariance or Gram matrix.
Whitening / decorrelating data via $L^{-1}$.

Neat

Cholesky failure is a feature: a backend error on a non-SPD input is a cheap, exact positive-definiteness check.

At ~2× the speed of LU on SPD matrices, it is the preferred factorization wherever positive-definiteness is guaranteed.

Known issues

Indefinite or only semidefinite matrices error out — for those use matrix_eigh or matrix_svd instead.

Only the (assumed symmetric) triangle is used; a non-symmetric input is not a meaningful Cholesky target.