Cholesky decomposition for symmetric matrices over finite fields

TL;DR

This paper extends Cholesky decomposition theory to symmetric matrices over finite fields, demonstrating its compatibility with Frobenius maps and applying it to define group operations and enumerate sub-cones.

Contribution

It develops a finite field Cholesky factorization for LPM cones, compatible with Frobenius maps, and uses it to define group structures and count sub-cones.

Findings

Extended Cholesky factorization to finite fields with asymptotic density 1

Proved compatibility with Frobenius maps in finite fields

Enumerated sub-cones of LPM matrices using the developed factorization

Abstract

Inspired by the seminal work of Andr\'e-Louis Cholesky -- whose contributions remain crucial in broader sciences even after more than a century -- Cooper, Hanna and Whitlatch (2024) developed a theory of positive matrices over finite fields, and Khare and Vishwakarma (2025) described a general Cholesky factorization for a dense sub-family of the cone of Hermitian matrices over real/complex fields, whose leading principal minors (LPM) are nonzero. Building on this, we develop a parallel theory within the finite field setting. Specifically $(i)$ we extend the general Cholesky factorization to the LPM cone over finite fields which has asymptotic density $1$. We show that $(ii)$ this factorization is compatible with the entrywise Frobenius map, recently studied in the context of positivity preservers by Guillot, Gupta, Vishwakarma, and Yip [J. Algebra, 2025]. We also $(iii)$ leverage the Cholesky-structures to define meaningful group operations on the matrix cone, and as an application $(iv)$ enumerate sub-cones of LPM matrices using our general Cholesky factorizations.