Cholesky decomposition for symmetric matrices over finite fields

Abstract: Inspired by the seminal work of Andr\'e-Louis Cholesky -- whose contributions remain crucial even after more than a century in broader sciences -- 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 family of the dense 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 asymptomatic 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.