Cholesky decomposition for symmetric matrices, Riemannian geometry, and random matrices

TL;DR

This paper introduces a new geometric and algebraic framework for symmetric matrices using Cholesky decompositions, revealing their Riemannian and Lie group structures, and extends these concepts to probabilistic models.

Contribution

It defines cones of symmetric matrices with sign patterns, establishes smooth Cholesky factorizations as diffeomorphisms, and uncovers their Riemannian and Lie group structures, extending to complex and probabilistic settings.

Findings

Cholesky-type factorizations form smooth diffeomorphisms

LPM cones are isometric Riemannian manifolds and abelian Lie groups

The framework extends to complex matrices and probabilistic models

Abstract

For each $n \geq 1$ and sign pattern $\epsilon \in \{ \pm 1 \}^n$, we introduce a cone of real symmetric matrices $LPM_n(\epsilon)$: those with leading principal $k \times k$ minors of signs $\epsilon_k$. These cones are pairwise disjoint and their union $LPM_n$ is an open dense cone in all symmetric matrices; they subsume positive and negative definite matrices, and symmetric (P-,) N-, PN-, almost P-, and almost N- matrices. We show that each $LPM_n$ matrix $A$ admits an uncountable family of Cholesky-type factorizations - yielding a unique lower triangular matrix $L$ with positive diagonals - with additional attractive properties: (i) each such factorization is algorithmic; and (ii) each such Cholesky map $A \mapsto L$ is a smooth diffeomorphism from $LPM_n(\epsilon)$ onto an open Euclidean ball. We then show that (iii) the (diffeomorphic) balls $LPM_n(\epsilon)$ are isometric Riemannian manifolds as well as isomorphic abelian Lie groups, each equipped with a translation-invariant Riemannian metric (and hence Riemannian means/barycentres). Moreover, (iv) this abelian metric group structure on each $LPM_n(\epsilon)$ - and hence the log-Cholesky metric on Cholesky space - yields an isometric isomorphism onto a finite-dimensional Euclidean space. The complex version of this also holds. In the latter part, we show that the abelian group $PD_n$ of positive definite matrices, with its bi-invariant log-Cholesky metric, is precisely the identity-component of a larger group with an alternate metric: the open dense cone $LPM_n$. This also holds for Hermitian matrices over several subfields $\mathbb{F} \subseteq \mathbb{C}$. As a result, (v) the groups $LPM_n^{\mathbb{F}}$ and $LPM_\infty^{\mathbb{F}}$ admit a rich probability theory, and the cones $LPM_n(\epsilon), TPM_n(\epsilon)$ admit Wishart densities with signed Bartlett decompositions.