Integer matrix factorisations, superalgebras and the quadratic form obstruction

TL;DR

This paper investigates obstructions to factorizing integer matrices into specific products, using superalgebra techniques, and explores related quadratic form issues, providing new formulas and symmetry insights.

Contribution

It introduces a novel analysis of quadratic form obstructions in integer matrix factorizations via superalgebra methods and derives a determinant formula related to these decompositions.

Findings

Identified quadratic form obstructions to matrix factorization.

Derived a formula for the determinant in terms of adjugates.

Discovered a co-Latin symmetry space associated with the problem.

Abstract

We identify and analyse obstructions to factorisation of integer matrices into products $N^T N$ or $N^2$ of matrices with rational or integer entries. The obstructions arise as quadratic forms with integer coefficients and raise the question of the discrete range of such forms. They are obtained by considering matrix decompositions over a superalgebra. We further obtain a formula for the determinant of a square matrix in terms of adjugates of these matrix decompositions, as well as identifying a $\it co-Latin$ symmetry space.