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

Esther Garc\'ia, Miguel G\'omez Lozano, Peter Danchev

Pith reviewed 2026-05-10 08:33 UTC · model grok-4.3

classification 🧮 math.RA

keywords matrixfinitecharacteristicindexmathbbalgebraappldiagonalizable

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The pith

Every matrix over finite fields of char 2 with more than three elements is the sum of a diagonalizable matrix and a square-zero matrix.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Matrices are grids of numbers drawn from a finite field, a small arithmetic system where addition and multiplication follow special rules. In characteristic 2 fields the number 1 added to itself equals 0, which changes how matrices behave. A diagonalizable matrix can be turned into a diagonal grid by a suitable change of basis, like stretching along independent axes. A square-zero matrix becomes the zero grid when multiplied by itself. The sum of two matrices is formed by adding their entries one by one. The authors show that for any field with more than three elements this split always exists: any given matrix equals some diagonalizable matrix plus some square-zero matrix. For the smallest field with only two elements they prove a related but different split that uses a potent matrix (one that satisfies a power relation with index at most 4) instead of a diagonalizable one. The result combines with their earlier work to cover all finite fields of characteristic 2 except the two smallest ones.

Core claim

we completely resolve this question in the affirmative for any finite field of characteristic 2 having strictly more than three elements

Load-bearing premise

The finite field must have strictly more than three elements; the proof technique does not apply to the two smallest fields, which require separate arguments as noted in the abstract.

read the original 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]).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that every square matrix over a finite field of characteristic 2 with more than three elements can be expressed as the sum of a diagonalizable matrix and a square-zero matrix (nilpotency index at most 2). Combined with results from the authors' prior paper [7], this settles the decomposition question affirmatively for all finite fields except F_2 and F_3. For F_2 the paper establishes an alternative decomposition of every matrix as the sum of a potent matrix (index at most 4) and a square-zero matrix, substantiating examples from Šter and Shitov.

Significance. If the proofs hold, the work completely resolves the stated decomposition problem for matrices over finite fields of characteristic 2 (except the two smallest fields, where an alternative decomposition is supplied). It is a substantive contribution to linear algebra over finite fields, as it furnishes an affirmative answer for all but two cases and directly builds on the authors' earlier results in Linear Algebra and its Applications while confirming concrete examples from the literature.

minor comments (3)

The abstract refers to a 2026 publication for [7]; if this is still a preprint, consider updating the citation or adding a note on its status.
Clarify the precise definition of 'index of potency' for the potent matrix in the F_2 result (e.g., the smallest k such that A^k = A) to avoid ambiguity with nilpotency indices.
The title and abstract use 'square-zero' interchangeably with 'nilpotent of index at most 2'; a brief parenthetical in the introduction would help readers unfamiliar with the terminology.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive recommendation to accept.

Circularity Check

0 steps flagged

Minor self-citation for small fields; main theorem for >3 elements is independent

full rationale

The paper proves its main affirmative result for finite fields of characteristic 2 with more than three elements via a self-contained main theorem. It invokes the authors' own prior paper [7] solely to combine results and thereby settle the decomposition question for all fields except F2 and F3. This citation is not load-bearing for the central claim, which rests on the current paper's own arguments. The additional direct proof for F2 (using a potent matrix of index at most 4 plus nilpotent of index at most 2) is likewise presented without reduction to prior self-work. No self-definitional equations, fitted inputs renamed as predictions, or ansatzes imported via citation are present.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters or invented entities appear; the work rests entirely on standard facts from linear algebra over finite fields.

axioms (1)

standard math Standard properties of diagonalizable matrices, nilpotent matrices of index at most 2, and arithmetic in finite fields of characteristic 2
Invoked throughout the existence argument for the decomposition.

pith-pipeline@v0.9.0 · 5509 in / 1237 out tokens · 34755 ms · 2026-05-10T08:33:35.923927+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references

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Danchev, E

P. Danchev, E. Garc´ ıa and M. G. Lozano. Matrices over finite fields of odd characteristic as sums of diagonalizable and square-zero matricesLinear Algebra & Appl.,730:35–50, 2026

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Y. Shitov. The ringM 8k+4(Z2) is nil-clean of index four.Indag. Math. (N.S.),30:1077–1078, 2019

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J. ˇSter. On expressing matrices overZ 2 as the sum of an idempotent and a nilpotent.Linear Algebra & Appl.,544:339–349, 2018

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J. ˇSter. Nil-clean index ofM n(F2).Linear Algebra & Appl.,632:294–307, 2022. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria Email address:danchev@math.bas.bg Departamento de Matem´atica Aplicada, Ciencia e Ingenier ´ıa de Materiales y Tec- nolog´ıa Electr´onica, Universidad Rey Juan Carlos, 28933 M ´ostoles ...

2022