Product of two matrices similar to companion matrices over sufficiently large fields
Pith reviewed 2026-06-26 15:13 UTC · model grok-4.3
The pith
A square matrix of size n over a field with at least 2n elements is the product of two companion-similar matrices precisely when its rank exceeds n-2.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
In a field containing at least 2n elements, a square matrix A of size n admits a factorization A = BC in which B and C are similar to companion matrices if and only if rank(A) > n-2.
What carries the argument
The iff condition based on rank(A) > n-2, which classifies exactly which matrices admit the desired factorization when the field is large enough.
Load-bearing premise
The field has at least 2n elements to allow selection of sufficiently many distinct values in the construction.
What would settle it
An n by n matrix of rank n-1 over a field with exactly 2n elements that admits no factorization into two companion-similar matrices would falsify the iff statement.
read the original abstract
In this note, we prove that a square matrix of size $n$ over a field containing at least $2n$ elements can be expressed as the product of two matrices similar to companion matrices, that is to say matrices with the same minimal and characteristic polynomial, if and only if the rank of $A$ is greater than $n-2$, using only elementary facts. We will also give some partial results valid over smaller fields.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves that an n x n matrix A over a field F with |F| >= 2n can be written as a product of two matrices each similar to a companion matrix if and only if rank(A) > n-2. The 'only if' direction uses the fact that matrices similar to companion matrices have rank at least n-1. The 'if' direction gives an explicit construction of suitable monic polynomials of degree n, relying on the field cardinality to ensure the existence of appropriate roots or coefficients. Partial results are stated for smaller fields. The proof uses only elementary facts from linear algebra.
Significance. If correct, the result gives a clean, rank-based characterization of matrices that factor into two companion-similar factors. The elementary nature of the argument and the explicit 2n bound on field size are strengths; the construction appears constructive and the necessity direction is standard from Jordan form considerations. This may be of interest for factorization questions in matrix semigroups over finite or large fields.
minor comments (1)
- The abstract and introduction could explicitly state the precise definition of 'similar to companion matrices' (same minimal and characteristic polynomial) to avoid any ambiguity for readers.
Simulated Author's Rebuttal
We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript.
Circularity Check
No circularity; direct elementary iff proof from matrix rank properties and polynomial construction
full rationale
The paper states an iff characterization proved using only elementary facts: the 'only if' direction follows immediately from the fact that any matrix similar to a companion matrix has rank at least n-1 (as it has a single Jordan block per eigenvalue), while the 'if' direction is an explicit construction of two such matrices whose product equals A, enabled by choosing suitable monic polynomials when the field has cardinality at least 2n. No self-definitional steps, fitted inputs renamed as predictions, self-citation chains, or imported uniqueness theorems appear. The derivation is self-contained against external linear-algebra benchmarks and does not reduce to its own inputs.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Standard field and matrix axioms (characteristic and minimal polynomials, similarity, rank)
Reference graph
Works this paper leans on
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discussion (0)
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