
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 be coprime integers such that . We characterize the matrices such that and are similar. If is invertible, we prove that is a polynomial in and . To achieve this, we study the matrix equation . We show that for such matrices, and commute. When is diagonalizable, is a root of and is a power of . We explicitly solve the previous equation when has distinct eigenvalues or when has a sole eigenvalue. In the second part, we completely solve the case of the more general matrix equation .
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