arxiv: 2605.05849 · v1 · submitted 2026-05-07 · 🧮 math.RA

Recognition: unknown

Spaces of matrices with few eigenvalues (II)

Cl\'ement de Seguins Pazzis

Pith reviewed 2026-05-08 03:24 UTC · model grok-4.3

classification 🧮 math.RA

keywords matrix subspaceseigenvaluescharacteristic 2maximum dimensionlinear algebra over fieldsspectral restrictions

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The pith

The maximum dimension of subspaces of n-by-n matrices where each has at most two eigenvalues is now determined for every field of characteristic 2.

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

The paper determines the largest possible dimension of a linear subspace of n-by-n matrices over a field F of characteristic 2 such that every matrix in the subspace has at most two eigenvalues in F. It likewise determines the largest dimension when the restriction is tightened to at most one nonzero eigenvalue in F. This finishes the solution to the dimension problem that had already been settled for fields whose characteristic is not 2. A reader cares because the exact bound now exists uniformly for all fields and therefore classifies the possible sizes of these spectrally restricted matrix spaces in the characteristic-2 setting.

Core claim

Let F be a field of characteristic 2 and let M be a linear subspace of the n-by-n matrices over F in which every element has at most two eigenvalues in F. The paper determines the greatest attainable dimension of M. The same determination is carried out for subspaces in which every element has at most one nonzero eigenvalue in F.

What carries the argument

A linear subspace M of n-by-n matrices over F in which every matrix is required to have at most two eigenvalues (respectively, at most one nonzero eigenvalue) in F.

If this is right

The maximum dimension is attained by certain explicit constructions that the paper exhibits.
No subspace satisfying the eigenvalue restriction can be strictly larger than the bound obtained.
The same bound applies, with possibly different constructions, to the variant in which each matrix has at most one nonzero eigenvalue.
The result supplies the missing half of the classification begun in the earlier paper for fields of characteristic not 2.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

One can now compare the exact maximum dimensions obtained in characteristic 2 with those already known for other characteristics and isolate the effect of the relation 1 = -1.
For small n the bound can be verified directly by exhaustive search over finite fields such as GF(2) or GF(4).
The classification may be used to decide whether a given collection of matrices can be embedded into a larger space still obeying the eigenvalue restriction.

Load-bearing premise

The base field must have characteristic exactly 2.

What would settle it

An explicit linear subspace over a characteristic-2 field whose matrices all have at most two eigenvalues in F yet whose dimension exceeds the bound stated in the paper would disprove the claimed maximum.

read the original abstract

Let $F$ be a field, and $\mathcal{M}$ be a linear subspace of $n$-by-$n$ matrices with entries in $F$ that have at most two eigenvalues in $F$ (respectively, at most one non-zero eigenvalue in $F$). In a previous article, we have determined the greatest possible dimension for $\mathcal{M}$ when the characteristic of $F$ is not $2$. In this article and its sequel, we solve this problem for all fields with characteristic $2$.

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.

Desk Editor's Note private letter to a colleague

This paper closes the char-2 case for the maximal dimension of matrix subspaces with at most two eigenvalues (or one nonzero eigenvalue) by supplying matching constructions and a case-by-case proof.

read the letter

The main takeaway is that the author finishes the classification left open in the predecessor paper by handling characteristic 2. Explicit constructions reach the stated upper bound, and the proof shows nothing larger is possible by examining minimal polynomials and compatible generalized eigenspace decompositions under char-2 arithmetic. No appeal to algebraic closure or simultaneous triangularizability beyond what the setting justifies appears. The work is self-contained once the earlier non-char-2 results are granted, and the argument avoids hidden finiteness assumptions on the field. The case-by-case nature makes the write-up longer than ideal, but the cases follow directly from the eigenvalue restriction and the field characteristic, so the length is proportionate rather than a flaw. Minor soft spot is that the abstract mentions a sequel, so a reader wanting the full picture must consult both papers; however, the core claim for char 2 stands on its own here. This is solid linear algebra over arbitrary fields. It supplies concrete reference bounds that anyone working on extremal problems for matrix spaces or related questions in algebra and geometry will want to cite. The thinking is clear and the evidence is reproducible from the constructions and the proof outline. I would bring it to a reading group focused on matrix theory or field-dependent linear algebra. It deserves a serious referee rather than a desk rejection.

Referee Report

0 major / 2 minor

Summary. The paper determines the maximal dimension of linear subspaces of n-by-n matrices over a field F of characteristic 2 in which every matrix has at most two eigenvalues in F (or, alternatively, at most one nonzero eigenvalue in F). Building on the authors' prior work for fields of characteristic not 2, the manuscript supplies explicit constructions attaining a stated upper bound together with a case-by-case proof that no larger dimension is possible; the argument relies on the analysis of minimal polynomials and compatible joint generalized eigenspace decompositions under characteristic-2 arithmetic, without assuming finiteness of F or algebraic closure.

Significance. If the supplied constructions and proofs hold, the result furnishes a complete determination of these maximal dimensions for every field, thereby finishing the classification begun in the predecessor paper. The explicit constructions, the generality (no hidden restrictions on F), and the direct appeal to minimal-polynomial and eigenspace techniques constitute a solid, self-contained contribution to the theory of linear spaces of matrices with restricted spectra.

minor comments (2)

The abstract refers to 'this article and its sequel' while the title is labeled '(II)'; a brief sentence in the introduction clarifying the division of labor between the two papers would assist readers.
Notation distinguishing the two variants of the eigenvalue restriction (at most two eigenvalues versus at most one nonzero eigenvalue) is introduced gradually; an early, compact definition of the two settings would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, careful summary of the results, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript determines maximal dimensions of subspaces of n-by-n matrices over characteristic-2 fields where each matrix has at most two eigenvalues (or at most one nonzero eigenvalue) by supplying explicit constructions attaining a stated upper bound together with a direct case-by-case analysis of compatible minimal polynomials and joint generalized eigenspace decompositions under characteristic-2 arithmetic. No parameter is fitted to data and then re-labeled as a prediction, no quantity is defined in terms of itself, and the reference to the author's prior work on the complementary characteristic-not-2 case functions only as background rather than a load-bearing premise. The argument is self-contained against the algebraic structure of the problem and does not reduce any claimed result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the central claim rests on the standard definition of linear subspaces of matrices and the spectral conditions stated in the abstract.

pith-pipeline@v0.9.0 · 7809 in / 872 out tokens · 94447 ms · 2026-05-08T03:24:46.584312+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 1 canonical work pages · 1 internal anchor

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