The group invertibility of matrices over B\'ezout domains

TL;DR

This paper investigates the conditions under which matrices over Bézout domains are similar when their products are group invertible, extending previous results to a broader algebraic setting.

Contribution

It generalizes the main result of Cao and Li by establishing similarity of matrices with group invertible products over Bézout domains.

Findings

$AB$ is similar to $CA$ when both are group invertible

$(AB)^{\#}$ is similar to $(CA)^{\#}$

Generalizes previous results to Bézout domains

Abstract

Let $R$ be a B\'ezout domain, and let $A,B,C\in R^{n\times n}$ with $ABA=ACA$. If $AB$ and $CA$ are group invertible, we prove that $AB$ is similar to $CA$. Moreover, we have $(AB)^{\#}$ is similar to $(CA)^{\#}$. This generalize the main result of Cao and Li(Group inverses for matrices over a B\'ezout domain, {\it Electronic J. Linear Algebra}, {\bf 18}(2009), 600--612).