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

Joshua Cooper; Hays Whitlatch

Paper

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

Abstract

A square, upper-triangular matrix

is a Cholesky root of a matrix

provided

, where

represents the conjugate transpose when working over the complex field and

over the reals and finite fields. In this paper, we investigate the number of such factorizations over the finite field with two elements,

, 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.