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.