Generalizing the notion of rank to noncommutative quadratic forms

TL;DR

This paper introduces a new concept of rank for noncommutative quadratic forms on two or three generators, extending classical notions and providing analogs for minors and determinants in this setting.

Contribution

It proposes a novel definition of rank for noncommutative quadratic forms and develops methods to express forms as sums of products, including noncommutative analogs of minors and determinants.

Findings

Defined a rank notion for noncommutative quadratic forms

Developed methods to rewrite forms as sums of products

Established noncommutative analogs of minors and determinants

Abstract

In 2010, Cassidy and Vancliff extended the notion of a quadratic form on n generators to the noncommutative setting. In this article, we suggest a notion of rank for such noncommutative quadratic forms, where n = 2 or 3. Since writing an arbitrary quadratic form as a sum of squares fails in this context, our methods entail rewriting an arbitrary quadratic form as a sum of products. In so doing, we find analogs for 2 x 2 minors and determinant of a 3 x 3 matrix in this noncommutative setting.