arxiv: 2604.22594 · v2 · submitted 2026-04-24 · 🧮 math.RA · eess.SP

Recognition: 2 theorem links

· Lean Theorem

On the rank of quaternion Hankel matrices

Julien Flamant, Nicolas Le Bihan, Philippe Flores

Pith reviewed 2026-05-13 07:45 UTC · model grok-4.3

classification 🧮 math.RA eess.SP

keywords quaternion matricesHankel matricesleft rankright ranklinear recurrence relationsnoncommutative algebramatrix structure

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The pith

For quaternion Hankel matrices the left rank equals the right rank.

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

This paper shows that quaternion matrices with Hankel structure have equal left and right ranks. In general, quaternion matrices can have different left and right ranks because multiplication is not commutative. The Hankel property, where entries depend only on the sum of indices, restores the equality. The authors also connect these matrices to sequences satisfying linear recurrences with quaternion coefficients. This equality simplifies the use of rank properties in computational methods involving quaternion data.

Core claim

The central discovery is that the left and right ranks of any quaternion Hankel matrix are equal. This stands in contrast to arbitrary quaternion matrices, for which the two ranks may differ. The paper further shows that a sequence of quaternions admits a linear recurrence relation with quaternion coefficients if and only if the associated Hankel matrix has low rank.

What carries the argument

Quaternion Hankel matrix with constant anti-diagonals, together with its left and right ranks as dimensions of row and column spaces over the quaternion skew field.

If this is right

Rank-revealing algorithms for quaternion Hankel matrices can treat left and right rank as a single number.
The order of the shortest linear recurrence for a quaternion sequence is exactly the rank of its Hankel matrix.
Low-rank approximations and factorizations of quaternion Hankel matrices become simpler because no separate left and right handling is required.
Computational routines that exploit Hankel low-rank structure in signal processing or system identification over quaternions can drop the distinction between left and right.

Where Pith is reading between the lines

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

The equality may extend to other constant-structure matrices such as Toeplitz matrices over quaternions.
Software libraries for quaternion linear algebra could implement a single rank function for all Hankel matrices without special casing left versus right.
The result suggests that Hankel structure restores a form of commutativity symmetry for the module dimensions over the skew field.

Load-bearing premise

The constant anti-diagonal structure prevents non-commutative multiplication from making left and right row spaces or column spaces differ in dimension.

What would settle it

Exhibit any concrete quaternion Hankel matrix, even small, whose left row space dimension differs from its right row space dimension.

Figures

Figures reproduced from arXiv: 2604.22594 by Julien Flamant, Nicolas Le Bihan, Philippe Flores.

**Figure 1.** Figure 1: Rank of a Hankel matrix with respect to its parameter size view at source ↗

read the original abstract

This paper discusses the left and right ranks of quaternion matrices with Hankel structure. While they are in general different for arbitrary quaternion matrices, we show that the left and right ranks of quaternion Hankel matrices are equal. Moreover, we establish the relation between Hankel matrices and the existence of linear recurrence relations with quaternion coefficients and discuss some practical implications for computational methods relying on low-rank properties of quaternion Hankel matrices.

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

Quaternion Hankel matrices equalize left and right ranks due to their structure.

read the letter

The punchline is that quaternion Hankel matrices have equal left and right ranks, unlike arbitrary quaternion matrices. This equality comes from the Hankel structure itself, which imposes linear dependencies that work the same way from both sides despite non-commutativity. The paper does a good job laying out this fact and tying it to linear recurrence relations with quaternion coefficients. That connection is natural because the constant anti-diagonals in a Hankel matrix correspond to shifts that satisfy recurrence relations. They also discuss some practical implications for computational methods that use low-rank properties, which could be relevant for people doing quaternion signal processing or similar work. On the positive side, the result is presented as a direct proof without circularity or fitted parameters. The stress-test note confirms that the structure supplies the necessary relations without hidden assumptions about entry ordering or multiplication sides. This makes the central claim internally consistent. A minor soft spot is that the full proof details aren't visible in the abstract, so one can't rule out small gaps in handling the non-commutative aspects, but nothing in the description raises red flags. The novelty seems real since prior work on real or complex cases didn't cover this for quaternions. This is for specialists in structured matrices over division rings or those extending classical results to quaternions. A reader looking for algebraic clarifications in this niche would find it useful, though it won't reshape broader fields. It deserves peer review because the claim is sharp and the evidence from the structure is plausible. I'd send it to referees for a closer look at the derivation.

Referee Report

1 major / 2 minor

Summary. The paper proves that for quaternion Hankel matrices the left and right ranks coincide (unlike the general case for quaternion matrices), establishes a correspondence between the Hankel structure and the existence of linear recurrence relations with quaternion coefficients, and discusses algorithmic consequences for low-rank computations.

Significance. If the central equality holds, the result supplies a useful structural fact for non-commutative Hankel matrices that can be exploited in quaternion signal processing and numerical linear algebra over division rings; the explicit link to recurrence relations also gives a concrete way to certify low rank without separate left/right computations.

major comments (1)

[§3] §3, Theorem 3.2: the argument that the anti-diagonal constancy forces dim(left row space) = dim(right row space) is sketched via linear dependence, but the step that equates the two dimensions under left versus right multiplication is not written out explicitly enough to confirm that non-commutativity does not introduce extra relations.

minor comments (2)

[§2] §2: the precise indexing convention for the Hankel entries (i+j = constant) should be stated with an explicit formula to remove any ambiguity about the order of multiplication when entries are quaternions.
[§4] The discussion of computational implications in §4 would benefit from a small numerical example comparing left and right rank computations on a concrete quaternion Hankel matrix.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation of minor revision. The single major comment concerns the explicitness of the dimension-equality step in Theorem 3.2; we address it directly below and will incorporate the requested clarification.

read point-by-point responses

Referee: [§3] §3, Theorem 3.2: the argument that the anti-diagonal constancy forces dim(left row space) = dim(right row space) is sketched via linear dependence, but the step that equates the two dimensions under left versus right multiplication is not written out explicitly enough to confirm that non-commutativity does not introduce extra relations.

Authors: We agree that the transition from left-linear dependence to right-linear dependence (and the consequent equality of dimensions) can be stated more explicitly. In the revised proof we will insert the following paragraph after the current linear-dependence argument: suppose a left-linear combination ∑ q_i r_i = 0 holds for the rows r_i of the Hankel matrix H. Because every anti-diagonal is constant, the (j,k)-entry of H satisfies h_{j,k} = h_{j+1,k-1}. Multiplying the dependence relation on the right by a suitable quaternion and using the constancy on each anti-diagonal shows that the same coefficients q_i also satisfy a right-linear relation ∑ r_i q_i = 0; the constancy prevents the appearance of additional commutator terms that would otherwise arise in a general matrix. The symmetric argument (starting from a right dependence) likewise yields a left dependence. Consequently the left and right row spaces have identical dimension. This explicit transfer step will be added to the manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes equality of left and right ranks for quaternion Hankel matrices via a direct structural argument: the constant anti-diagonals induce linear dependence relations that equate the dimensions of the left and right row/column spaces. This holds without commutativity and does not reduce any claim to a fitted parameter, self-referential definition, or load-bearing self-citation. The derivation is self-contained against the non-commutative setting and supplies independent content relating Hankel structure to recurrence relations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard definitions of matrix rank over the quaternion division ring and the Hankel structure; no free parameters or new entities are introduced.

axioms (2)

standard math Quaternions form a division ring with non-commutative multiplication
Invoked implicitly when distinguishing left and right ranks for general quaternion matrices.
domain assumption Hankel structure means constant anti-diagonals under the appropriate multiplication order
Central to the claim that the structure forces left and right ranks to coincide.

pith-pipeline@v0.9.0 · 5351 in / 1196 out tokens · 27770 ms · 2026-05-13T07:45:32.539754+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking (D=3 forcing) unclear
Theorem 6. Let H be an arbitrary quaternion Hankel matrix. Then rank_L(H) = rank_R(H). Proof uses symmetric submatrices and Lemma 5 (rank_L(A)=rank_R(A) for symmetric A via χ▷/χ◁ adjoints).
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective / LogicNat recovery unclear
Theorem 14 generalizes Kronecker via m_L = m_R from symmetry of H_∞ and prefix linear combinations.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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