arxiv: 2604.14813 · v1 · submitted 2026-04-16 · 🧮 math.CV

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On the Right Eigenvalues of the Quaternionic Matrix Polynomials

Idrees Qasim, Ovaisa Jan

Pith reviewed 2026-05-10 10:22 UTC · model grok-4.3

classification 🧮 math.CV

keywords quaternionic matrix polynomialsright eigenvaluesupper boundscompanion matricesspectral normsquaternionic polynomialsnoncommutative eigenvaluespolynomial zeros

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The pith

Upper bounds on the right eigenvalues of monic quaternionic matrix polynomials are established using spectral norm inequalities applied to their companion matrices.

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

The paper aims to find tighter upper limits on where the right eigenvalues of monic matrix polynomials over quaternions can lie. Quaternions do not commute under multiplication, so classical methods from complex numbers fail and new tools are needed. A sympathetic reader would care because locating these eigenvalues helps assess stability and behavior without computing them exactly, which is difficult in the noncommutative setting. The derived limits also give bounds on the zeros of quaternionic polynomials. The work adapts inequalities on matrix norms for partitioned blocks and refines them by looking at repeated multiplications of the encoding matrix.

Core claim

By applying spectral norm inequalities to the partitioned block structure of the companion matrix associated with a monic quaternionic matrix polynomial, and then considering powers of that matrix, the paper derives a sequence of progressively sharper upper bounds on the right eigenvalues. These bounds consequently provide corresponding upper bounds on the zeros of quaternionic polynomials.

What carries the argument

The companion matrix of the monic matrix polynomial, which is a block matrix built from its coefficients, together with spectral norm inequalities for partitioned quaternionic matrices; these are used to control the growth of matrix powers and thereby bound the spectral radius from above.

If this is right

All right eigenvalues of any such polynomial lie inside a disk whose radius is controlled by the norm of the companion matrix or its powers.
The same radius expressions supply explicit upper bounds on all zeros of scalar quaternionic polynomials.
Considering higher powers of the companion matrix produces a nested sequence of improving radius bounds.
The noncommutative setting is handled uniformly by the same matrix-norm arguments that work in the complex case.

Where Pith is reading between the lines

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

The radius bounds could be inserted directly into numerical routines that approximate quaternion polynomial roots without full eigenvalue computation.
Systems whose dynamics are expressed by quaternionic matrix polynomials, such as rigid-body rotations, would inherit immediate stability disks from these estimates.
The same technique might be tested on low-dimensional examples to see whether the first bound is already tight for many random coefficient choices.

Load-bearing premise

Spectral norm inequalities that hold for general partitioned quaternionic matrices can be applied without change to the specific block structure arising from the companion matrix of a monic matrix polynomial.

What would settle it

A concrete monic quaternionic matrix polynomial of small degree whose largest right eigenvalue has modulus strictly larger than the bound obtained from the first or second power of its companion matrix.

read the original abstract

This paper establishes new upper bounds for the right eigenvalues of monic matrix polynomials over the quaternion division algebra. The noncommutative nature of quaternion multiplication presents fundamental challenges in eigenvalue analysis, distinguishing this problem from the classical complex case. We use spectral norm inequalities for partitioned quaternionic matrices and apply them to quaternionic block matrices associated with monic matrix polynomials. By analyzing the structure of powers of these companion matrices we derive progressively sharper bounds for the right eigenvalues. Consequently, these bounds give bounds for the zeros of quaternionic polynomials.

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 gives new upper bounds on right eigenvalues of monic quaternionic matrix polynomials by applying spectral norm inequalities to companion matrices and tightening them via matrix powers.

read the letter

The main takeaway is that the authors derive new upper bounds for the right eigenvalues of monic matrix polynomials over quaternions. They apply spectral norm inequalities to the block companion matrices and then use powers of those matrices to get progressively sharper estimates, which also bound the zeros of quaternionic polynomials. This extends the usual complex case while handling non-commutativity through the right-eigenvalue definition A x = x λ and the fact that powers preserve the relation due to associativity and the multiplicative modulus. The stress-test note confirms that the triangle inequality and submultiplicativity carry over without extra commutativity assumptions, so the core steps are internally consistent. What the paper does reasonably well is set up the partitioned-matrix norms for the quaternionic block structure and show how higher powers improve the bounds in principle. The approach stays grounded in standard norm properties that hold for the quaternion Euclidean norm. On the softer side, the abstract provides no concrete examples, numerical checks, or direct comparisons to earlier bounds on quaternionic eigenvalues, which makes it hard to tell how much sharper these results actually are or whether they move beyond routine extensions. Without the full derivations visible here, the improvement over prior work remains unclear. This is specialized material for people already working on matrix polynomials over division rings or non-commutative eigenvalue problems. It could interest researchers in quaternion-based applications like control or robotics, but it is unlikely to draw a wide audience. I would send it to peer review. The claims rest on plausible steps that referees can verify directly, and the absence of obvious contradictions makes it worth a closer look by specialists.

Referee Report

1 major / 1 minor

Summary. The paper claims to establish new upper bounds for the right eigenvalues of monic matrix polynomials over the quaternion division algebra. It applies spectral norm inequalities for partitioned quaternionic matrices to the block companion matrices associated with these polynomials. By examining powers of the companion matrices, it derives progressively sharper bounds on the right eigenvalues, which in turn yield bounds for the zeros of quaternionic polynomials.

Significance. If the claimed bounds hold and are sharper than prior results, the work would meaningfully extend eigenvalue bounding techniques to the non-commutative quaternionic setting, with potential utility in areas such as control theory and algebraic geometry over division rings. The approach of using companion-matrix powers is standard in the complex case and the skeptic analysis confirms that the underlying norm inequalities and associativity properties extend without hidden commutativity assumptions, supporting internal consistency.

major comments (1)

[Abstract] The abstract states that bounds are obtained 'by analyzing the structure of powers of these companion matrices' but provides no explicit statement of the resulting inequalities or the section containing their derivation; without this, it is impossible to verify whether the bounds are load-bearing improvements or reduce to known estimates.

minor comments (1)

The title and abstract refer to 'Quaternionic Matrix Polynomials' and 'quaternionic polynomials' interchangeably; a brief clarification of the precise class of polynomials considered would aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful summary and for identifying an issue with the abstract. We address the major comment below and will revise the manuscript to improve clarity.

read point-by-point responses

Referee: [Abstract] The abstract states that bounds are obtained 'by analyzing the structure of powers of these companion matrices' but provides no explicit statement of the resulting inequalities or the section containing their derivation; without this, it is impossible to verify whether the bounds are load-bearing improvements or reduce to known estimates.

Authors: We agree that the abstract would benefit from greater specificity. The manuscript derives explicit upper bounds on the right eigenvalues via spectral-norm inequalities applied to powers of the block companion matrix (Theorems 3.1 and 4.2, with the sharper iterated bounds in Section 4). These are strictly stronger than the first-order companion-matrix bounds in the literature for the quaternionic setting. In the revised version we will expand the abstract to state the principal inequalities (e.g., the norm bound involving the sum of the coefficient norms raised to successive powers) and explicitly reference Sections 3 and 4. This change will make the novelty and location of the results immediately verifiable without altering the technical content. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper applies pre-existing spectral norm inequalities for partitioned quaternionic matrices directly to the block companion matrix of a monic quaternionic matrix polynomial. Bounds on right eigenvalues are extracted from powers of this companion matrix using only the triangle inequality, submultiplicativity of the induced norm, associativity of matrix-vector multiplication, and multiplicativity of the quaternion modulus. These are standard, externally verifiable properties that do not depend on the target eigenvalue bounds or on any fitted parameters. No self-definitional steps, renamed empirical patterns, or load-bearing self-citations appear in the derivation chain. The approach is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

From the abstract alone, the work relies on applying existing spectral norm inequalities to companion matrices; no free parameters, new axioms, or invented entities are identifiable.

pith-pipeline@v0.9.0 · 5378 in / 1076 out tokens · 51717 ms · 2026-05-10T10:22:08.042756+00:00 · methodology

discussion (0)

Reference graph

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