On the Number of Cholesky Roots of the Zero Matrix over F2

TL;DR

This paper investigates the number of Cholesky roots of the zero matrix over the finite field F2, establishing a bijection with upper-triangular square roots and contributing to understanding matrix factorizations in finite fields.

Contribution

It proves a rank-preserving bijection between Cholesky roots and upper-triangular roots of the zero matrix over F2, revealing new structural insights.

Findings

Number of Cholesky roots of zero matrix over F2 characterized

Existence of a rank-preserving bijection established

Structural relationship between different matrix factorizations over F2

Abstract

A square, upper-triangular matrix U is a Cholesky root of a matrix M provided U*U=M, where * represents the conjugate transpose. Over finite fields, as well as over the reals, it suffices for U^TU=M. In this paper, we investigate the number of such factorizations over the finite field with two elements, F2, and prove the existence of a rank-preserving bijection between the number of Cholesky roots of the zero matrix and the upper-triangular square roots the zero matrix.