Similar Powers of a Matrix

TL;DR

This paper characterizes matrices where different powers are similar, showing that invertible such matrices are polynomials in those powers, and explicitly solves related matrix equations, especially for 2x2 matrices.

Contribution

It provides a complete characterization of matrices with similar powers and explicit solutions for specific matrix equations, including the 2x2 case.

Findings

Matrices with coprime powers being similar are characterized.

Invertible matrices with similar powers are polynomials in those powers.

Explicit solutions are given for 2x2 matrix equations.

Abstract

Let $p,q$ be coprime integers such that $|p|+|q|>2$. We characterize the matrices $A\in\mathcal{M}_n(\mathbb{C})$ such that $A^p$ and $A^q$ are similar. If $A$ is invertible, we prove that $A$ is a polynomial in $A^p$ and $A^q$. To achieve this, we study the matrix equation $B^{-1}A^pB=A^q$. We show that for such matrices, $B^{-1}AB$ and $A$ commute. When $A$ is diagonalizable, $A$ is a root of $I_n$ and $B^{-1}AB$ is a power of $A$. We explicitly solve the previous equation when $A$ has $n$ distinct eigenvalues or when $A$ has a sole eigenvalue. In the second part, we completely solve the $2\times{2}$ case of the more general matrix equation $A^{r}B^{s}A^{r'}B^{s'}=\pm{I}_2$.