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

Authors

  • V.M. Prokip Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova str., 79060, Lviv, Ukraine https://orcid.org/0000-0001-5539-7904
https://doi.org/10.15330/cmp.10.2.352-359

Keywords:

matrix, similarity, rank
Published online: 2018-12-31

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.

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How to Cite
(1)
Prokip, V. On the Similarity of Matrices $AB$ and $BA$ over a Field. Carpathian Math. Publ. 2018, 10, 352-359.