Statistical Methods
Reconstruct a covariance matrix from its Cholesky factor
Define the task
A Cholesky factor can be stored as a lower or upper triangular matrix. Save the software convention alongside the factor. R returns an upper triangular factor by default, so reconstruction uses its transpose multiplied by the factor. Verify symmetry and positive definiteness of the intended covariance matrix before interpreting the factor.
Carry out the check
Compute the reconstruction and compare every entry with the input at a tolerance suited to its magnitude. An upper triangular factor multiplied by its transpose in the opposite order generally gives a different matrix. For simulated normal vectors, the factor orientation must also agree with whether samples are stored as rows or columns.
Worked hypothetical example
For a hypothetical upper factor with rows (2,1) and (0,3), transpose-times-factor gives rows (4,2) and (2,10). Factor-times-transpose instead gives rows (5,3) and (3,9). Both results are symmetric, so symmetry alone cannot detect the ordering error. A failed decomposition can indicate a non-positive-definite matrix or numerical issues. Do not silently alter correlations just to force a factorization.
Method checked 12 November 2026 against R documentation: The Cholesky Decomposition.