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

Abstract

For each $n \geq 1$ and sign pattern $ε\in \{ \pm 1 \}^n$, we introduce a cone of real symmetric matrices $LPM_n(ε)$: those with leading principal $k \times k$ minors of signs $ε_k$. 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 $LPM_n(ε)$ onto an open Euclidean ball. We then show that (iii) the (diffeomorphic) balls $LPM_n(ε)$ 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 $LPM_n(ε)$ - 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 $LPM_n(ε), TPM_n(ε)$ admit Wishart densities with signed Bartlett decompositions.