On the nonsingular solutions of the matrix equation XAX=AXA and the centralizer of the matrix A

Recieved: 04/12/2023

Accepted: 12/15/2023

Published: 04/02/2024

Keywords: equation of the Young–Baxter type, similarity of matrices, centralizer of a matrix, diagonalizable matrix, Jordan block

To cite this article

Ikramov Kh.D., Chugunov V. N On the nonsingular solutions of the matrix equation XAX=AXA and the centralizer of the matrix A. // Moscow University Journal. Series 15. Computational Mathematics and Cybernetics. 2024. N 2, p.25-30 https://doi.org/10.55959/MSU/0137-0782-15-2024-47-2-25-30.

This work is licensed under a Сreative Commons Atribiution - NonCommercial 4.0 International (CC BY-NC 4.0)

Ikramov Kh.D. (Lomonosov Moscow State University (MSU))
Chugunov V. N ()

Abstract

The matrix equation XAX = AXA is called the equation of the Young–Baxter type. We examine this equation for matrices of order 2 under the assumption that A is a nonsingular matrix, and we are only interested in the nonsingular solutions. Using a unified rule, one can associate with every such solution a matrix commuting with A. In other words, this matrix is an element of the centralizer MA of A. No obvious reasons exist for two distinct solutions X1 and X2 to generate the same element of MA. Nevertheless, all the solutions (and there are infinitely many of them) generate one and the same matrix in the centralizer. We give an explanation of this surprising fact.