On the similarity of matrices $AB$ and $BA$ over a field

Abstract

Let $A$ and $B$ be $n$-by-$n$ matrices over a field. The study of the relationship between the products of matrices $AB$ and $BA$ has a long history. It is well-known that $AB$ and $BA$ have equal characteristic polynomials (and, therefore, eigenvalues, traces, etc.).  One beautiful result was obtained by H. Flanders in 1951. He determined the relationship between the elementary divisors of $AB$ and $BA$, which can be seen as a criterion when two matrices $C$ and $D$ can be realized as $C = AB$ and $D = BA$. If one of the matrices ($A$ or $B$) is invertible, then the matrices $AB$ and $BA$ are similar. If both $A$ and $B$ are singular then matrices $AB$ and $BA$ are not always similar. We give conditions under which matrices $AB$ and $BA$ are similar. The rank of matrices plays an important role in this investigation.