Solutions of the matrix equation $p(X)=A$, with polynomial function $p(\lambda)$ over field extensions of $\mathbb{Q}$

TL;DR

This paper investigates solutions to matrix equations of the form p(X)=A over field extensions of the rationals, providing explicit constructions under certain conditions and introducing a new canonical form combining rational and Jordan forms.

Contribution

It offers explicit solutions for nonderogatory matrices to polynomial matrix equations over field extensions, and introduces a novel canonical form blending rational and Jordan forms.

Findings

Necessary conditions for solutions are identified.

Explicit solution constructions are provided under additional conditions.

A new canonical form combining rational and Jordan forms is introduced.

Abstract

Let $\mathbb{H}$ be a field with $\mathbb{Q}\subset\mathbb{H}\subset\mathbb{C}$, and let $p(\lambda)$ be a polynomial in $\mathbb{H}[\lambda]$, and let $A\in\mathbb{H}^{n\times n}$ be nonderogatory. In this paper we consider the problem of finding a solution $X\in\mathbb{H}^{n\times n}$ to $p(X)=A$. A necessary condition for this to be possible is already known from a paper by M.P. Drazin. Under an additional condition we provide an explicit construction of such solutions. The similarities and differences with the derogatory case will be discussed as well. One of the tools needed in the paper is a new canonical form, which may be of independent interest. It combines elements of the rational canonical form with elements of the Jordan canonical form.