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

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.