On the similarity of AB and BA for normal and other matrices

TL;DR

This paper investigates conditions under which the products AB and BA are similar, revealing that similarity can fail for Hermitian and normal matrices but holds when A is positive semidefinite and B is normal.

Contribution

It clarifies the conditions for similarity of AB and BA matrices, extending known results to cases involving positive semidefinite and normal matrices.

Findings

Similarity can fail when A is Hermitian and B is normal.

Similarity holds when A is positive semidefinite and B is normal.

Provides a nuanced understanding of matrix product similarity conditions.

Abstract

It is well-known that $AB$ and $BA$ are similar when $A$ and $B$ are complex square Hermitian matrices. In this note we answer a question of F. Zhang by demonstrating that similarity can fail if $A$ is Hermitian and $B$ is normal. Perhaps surprisingly, similarity does hold when $A$ is positive semidefinite and $B$ is normal.