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

TL;DR

This paper proves that nonderogatory matrices over fields of positive odd characteristic can be decomposed into a sum of a p-potent and a nilpotent matrix with specific algebraic properties.

Contribution

It establishes a new decomposition result for nonderogatory matrices over fields of positive odd characteristic, linking p-potent and nilpotent matrices.

Findings

Every nonderogatory matrix over such fields can be expressed as a sum of a p-potent and a nilpotent matrix.

The decomposition satisfies E^p=E and V^3=0.

Provides a constructive proof for the decomposition.

Abstract

We prove that every nonderogatory matrix $A$ over a field of positive odd characteristic $p,$ that is sum of a $p$-potent matrix and a nilpotent matrix, has a decomposition $A=E+V,$ such that $E^p=E,$ and $V^3=0.$