$m$-nil-clean nonderogatory matrices

TL;DR

This paper proves that certain nonderogatory matrices over fields of positive odd characteristic can be decomposed into sums of idempotents and a nilpotent with specific nilpotency bounds, extending understanding of matrix decompositions.

Contribution

It establishes a decomposition result for nonderogatory matrices as sums of idempotents and a nilpotent in fields of positive odd characteristic, with explicit nilpotency constraints.

Findings

Matrices decompose into sums of idempotents and nilpotents under given conditions

Nilpotent component satisfies a specific power-zero property

Results extend matrix decomposition theory in positive characteristic fields

Abstract

We prove that if $\mathbb{F}$ is a field of positive odd characteristic $p,$ and $m,$ and $n$ are positive integers such that $m\geq2,$ and $n\leq p,$ every $n\times n$ nonderogatory matrix $A\in \mathbb{M}_n(\mathbb{F})$ which is sum of $m$ idempotents and a nilpotent, has a decomposition $A=E_1+E_2+\dots+E_m+V,$ such that $E_i^2=E_i,$ for every $i\in \{1,\dots,m\},$ and $V^{[\frac{p-2}{m}]+2}=0.$