Matrices over finite fields of odd characteristic as sums of diagonalizable and square-zero matrices

TL;DR

This paper proves that over finite fields of odd characteristic, every matrix can be decomposed into a diagonalizable matrix plus a square-zero matrix, settling a question about such decompositions for large enough fields.

Contribution

It establishes the existence of such decompositions over finite fields of odd characteristic, with specific results for fields of size 3 and larger fields, answering a previously posed open question.

Findings

Every matrix over finite fields with size at least 5 admits a diagonalizable plus square-zero decomposition.

Counterexamples exist for matrices over the field with 3 elements, especially for certain sizes and structures.

The results fully settle the question posed by Breaz regarding such decompositions over finite fields.

Abstract

Let $\mathbb{F}$ be a finite field of odd characteristic. When $|\mathbb{F}|\ge 5$, we prove that every matrix $A$ admits a decomposition into $D+M$ where $D$ is diagonalizable and $M^2=0$. For $\mathbb{F}=\mathbb{F}_3$, we show that such decomposition is possible for non-derogatory matrices of order at least 5, and more generally, for matrices whose first invariant factor is not a non-zero trace irreducible polynomial of degree 3; we also establish that matrices consisting of direct sums of companion matrices, all of them associated to the same irreducible polynomial of non-zero trace and degree 3 over $\mathbb{F}_3$, never admit such decomposition. These results completely settle the question posed by Breaz in Lin. Algebra & Appl. (2018) asking if it is true that for big enough positive integers $n\ge 3$ all matrices $A$ over a field of odd cardinality $q$ admit decompositions of the form $E+M$ with $E^q=D$ and $M^2=0$: the answer is {\it yes} for $q\ge 5$, but there are counterexamples for $q=3$ and each order $n=3k$, $k\ge 1$.