Counting Cholesky factorizations of the zero matrix over $\mathbb{F}_2$

TL;DR

This paper studies the number of Cholesky factorizations of the zero matrix over the finite field with two elements, revealing equinumerosity with upper-triangular square roots and providing asymptotic counts.

Contribution

It proves the equinumerosity of Cholesky roots and upper-triangular square roots of the zero matrix over , and derives asymptotic formulas for their counts.

Findings

Number of Cholesky roots equals the number of upper-triangular square roots for each fixed rank.

Asymptotic formulas for the count of such factorizations.

Established equinumerosity over  for zero matrix factorizations.

Abstract

A square, upper-triangular matrix $U$ is a Cholesky root of a matrix $M$ provided $U^*U=M$, where $\cdot^*$ represents the conjugate transpose when working over the complex field and $U^*=U^T$ over the reals and finite fields. In this paper, we investigate the number of such factorizations over the finite field with two elements, $\mathbb{F}_2$, and prove the equinumerosity, for each fixed rank, of the Cholesky roots of and the upper-triangular square roots of the zero matrix. We then provide asymptotics for this count and finish with a few directions for future inquiry.