Department Colloquium

Location: CW 113

Speaker: Prateek Kumar Vishwakarma, Laval University

Title:  Cholesky decomposition for almost all symmetric matrices

Abstract:

For an \(n \times n\) real symmetric matrix, the \(k\)-th leading principal minor is the determinant of its top-left \(k \times k\) submatrix, for \(k=1,\dots,n\). We study the open dense cone \(LPM_n\) of such matrices with all Leading Principal Minors nonzero. For this cone, we develop generalized Cholesky factorizations, which show that \(LPM_n\) decomposes into \(2^n\) disjoint, smoothly diffeomorphic "copies" of the positive definite cone \(\mathrm{PD}_n\), indexed by the sign patterns \(\epsilon \in \{\pm 1\}^n\) of the leading principal minors. In fact, we show that each of these "copies" admits uncountably many factorizations, each of which is algorithmic, and generalizes the one by Cholesky.

These novel factorizations have several notable applications. \textit{First,} they provide a framework for studying natural Riemannian geodesics on each non-convex "copy" of \(\mathrm{PD}_n\). \textit{Second,} each matrix cone in \(LPM_n\) with a given inertia is packed precisely by binomially many "copies" of \(\mathrm{PD}_n\). \textit{Third,} since these Cholesky factorizations have well-behaved Jacobians, probability densities on \(\mathrm{PD}_n\) extend naturally: first to each "copy" \(\boldsymbol{\longrightarrow}\) then to larger inertia cones \(\boldsymbol{\longrightarrow}\) ultimately to the full \(LPM_n\). This includes the Wishart distribution via the Bartlett decomposition, enabling probabilistic and statistical analysis of hyperbolic spaces of matrices with negative eigenvalues. This talk is based on recent joint work with Apoorva Khare (arXiv:2508.02715).