Matrices over finite fields of odd characteristic as sums of diagonalizable and square-zero matrices
Pith reviewed 2026-05-19 06:40 UTC · model grok-4.3
The pith
Every matrix over a finite field of odd characteristic with at least five elements can be written as 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.
Core claim
When the finite field F has odd characteristic and size at least five, every matrix A over F admits a decomposition A = D + M with D diagonalizable over F and M^2 = 0. When F is the field with three elements, such a decomposition exists for all matrices except those that are direct sums of companion matrices associated to the same irreducible polynomial of degree three with nonzero trace.
What carries the argument
The rational canonical form decomposition into companion matrices of invariant factors, which allows explicit construction of the diagonalizable summand D by choosing appropriate scalars from the field to satisfy the square-zero condition on M = A - D.
Load-bearing premise
The field must contain enough distinct elements to select diagonal entries that make the difference matrix square-zero while keeping the diagonalizable part's minimal polynomial split into distinct linear factors.
What would settle it
Explicitly checking all possible 3 by 3 matrices over the field with three elements to confirm that only the direct sum of two companion matrices of the same nonzero-trace irreducible cubic fails the decomposition, or locating a matrix over a five-element field that cannot be decomposed this way.
read the original 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 $n=3k$, $k\ge 1$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proves that every matrix over a finite field F of odd characteristic with |F| >= 5 can be decomposed as A = D + M with D diagonalizable over F and M^2 = 0. For F = F_3 it gives a complete characterization: the decomposition exists for all non-derogatory matrices of size at least 5 and, more generally, whenever the first invariant factor is not a degree-3 irreducible of nonzero trace; it also exhibits explicit counterexamples consisting of direct sums of companion matrices of the same such irreducible.
Significance. The results furnish a definitive answer to the question posed by Breaz (Linear Algebra Appl. 2018) on the existence of decompositions E + M with E^q = D and M^2 = 0 for all sufficiently large n over fields of odd cardinality q. The positive statement for q >= 5 together with the precise invariant-factor obstruction for q = 3 constitute a substantial contribution to the additive theory of matrices over finite fields, obtained via standard rational-canonical-form reductions and explicit field-element constructions.
minor comments (1)
- Abstract: the phrase 'non-zero trace irreducible polynomial of degree 3' would be clearer if accompanied by an explicit example (e.g., x^3 + x + 1 over F_3) or a reference to the precise minimal polynomial used in the counterexample construction.
Simulated Author's Rebuttal
We thank the referee for their positive assessment of the manuscript, accurate summary of the results, and recommendation to accept. We are pleased that the contribution to the question posed by Breaz is recognized.
Circularity Check
No significant circularity
full rationale
The derivation proceeds via explicit block-by-block constructions on the rational canonical form, using the cardinality of the finite field to select suitable eigenvalues for the diagonalizable summand D such that A-D is square-zero. These steps rely on standard invariant-factor decompositions and direct verification of trace/minimal-polynomial conditions; no parameter is fitted to data and then renamed as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and the exceptional case over F3 is settled by exhibiting concrete counterexamples rather than by definition. The central claim is therefore self-contained against external matrix theory.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Every matrix over a field admits a rational canonical form with invariant factors that are monic polynomials.
- domain assumption Finite fields of odd characteristic admit the necessary trace and polynomial properties for the diagonalizable part.
Reference graph
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discussion (0)
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