Matrices over Finite Fields of Characteristic 2 as Sums of Diagonalizable and Square-Zero Matrices

TL;DR

This paper characterizes when matrices over finite fields of characteristic 2 can be decomposed into sums of diagonalizable and nilpotent matrices, providing complete solutions for most finite fields and new results for the field with two elements.

Contribution

It fully resolves the decomposition problem for finite fields of characteristic 2 with more than three elements and introduces new decompositions for matrices over the field with two elements.

Findings

Matrices over finite fields with more than three elements can be decomposed into diagonalizable and nilpotent sums.

Over _2, matrices are expressible as sums of potent and nilpotent matrices with bounded indices.

The results confirm and extend recent examples and conjectures in matrix decomposition theory.

Abstract

We investigate the problem asking when any square matrix whose entries lie in a finite field of characteristic 2 is decomposable into the sum of a diagonalizable matrix and a nilpotent matrix with index of nilpotency at most 2 and, as a result, we completely resolve this question in the affirmative for any finite field of characteristic 2 having strictly more than three elements. Our main theorem of that type, combined with results from our recent publication in Linear Algebra & Appl. (2026) (see [7]), totally settle this problem for all finite fields different from $\mathbb{F}_2$ and $\mathbb{F}_3$. However, in this paper we also prove that each matrix over $\mathbb{F}_2$ is expressible as the sum of a potent matrix with index of potency not exceeding 4 and a nilpotent matrix with index of nilpotency not exceeding 2, thus substantiating recent examples due to \v{S}ter in Linear Algebra & Appl. (2018) and Shitov in Indag. Math. (2019) (see, respectively, [9] and [8]).