Companion matrices as sums of $p$-potent and nilpotent matrices

TL;DR

This paper characterizes when a companion matrix over a field of odd characteristic can be expressed as the sum of a p-potent and a nilpotent matrix, based on its trace being a multiple of the unity.

Contribution

It provides a necessary and sufficient condition for representing companion matrices as sums of p-potent and nilpotent matrices in fields of odd characteristic.

Findings

Companion matrices can be decomposed into p-potent and nilpotent matrices under specific trace conditions.

The trace of the companion matrix determines the possibility of such a decomposition.

The result applies to fields with odd characteristic, expanding understanding of matrix decompositions in algebra.

Abstract

We prove that, over a field $\mathbb{F}$ of odd characteristic $p$, a companion matrix $C$ is the sum of $E$ and $N$, with $E$ $p$-potent (i.e. $E^p = E$,) and $N$ nilpotent, if and only if the trace of $C$ is an integer multiple of unity of $\mathbb{F}$.