The factorability of symmetric matrices and some implications for statistical linear models

Abstract

General conditions where a symmetric matrix is factorable by Cholesky decomposition are described. While numerical stability is a remaining issue whenever the Cholesky decomposition is used to factor indefinite matrices, the existence of such factors is demonstrated for matrix structures that are commonly found in statistics. Kalman filtering, for example, is rediscovered in the Cholesky decomposition of an indefinite matrix. Moreover, the Cholesky decomposition uniquely defines the likelihood function in linear statistical models, and this includes situations when the variance matrix is singular or when the Cholesky decomposition does not run to completion. Alternative methods of likelihood evaluation (which may involve, for example, the Bunch–Parlett factorization) are available only when the Cholesky decomposition exists. Suggestions are made for computing an adaptive-precision Cholesky decomposition when numerical stability is an issue.