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arxiv: 2607.01099 · v1 · pith:L67XV2JEnew · submitted 2026-07-01 · 🧮 math.FA

On Circular Numerical Ranges of Companion Matrices with Repeated Eigenvalues

Hsin-Yi Lee , Wei-Qiang Huang This is my paper

Pith reviewed 2026-07-02 04:58 UTC · model grok-4.3

classification 🧮 math.FA
keywords companion matrixnumerical rangecircular numerical rangeJordan blockrepeated eigenvaluesLaurent polynomialChebyshev polynomials
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The pith

For n>3, a companion matrix with a single repeated eigenvalue has a circular numerical range only if it is the Jordan block.

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

The paper shows that companion matrices whose spectrum consists of one repeated value a can have a circular numerical range only when the matrix is the Jordan block, provided the dimension exceeds three. The argument proceeds by translating the circularity condition into the vanishing of coefficients in a Laurent polynomial that encodes the numerical range boundary. A decomposition of the matrix into a tridiagonal Toeplitz block plus a rank-two update, together with Chebyshev polynomials of the second kind, supplies an explicit formula for those coefficients and isolates the requirement that a must be zero. The same technique is used to treat the case in which the spectrum consists of zero together with a second value a of prescribed multiplicity. Readers care because the numerical range governs the behavior of powers and resolvents of the matrix.

Core claim

We prove that if an n×n (n > 3) companion matrix A with the spectrum σ(A) = { a } has a circular numerical range, then A is the Jordan block. This problem can be described by examining zeros of the Laurent polynomial arising from geometric properties of the numerical range. The difficulty is that the relevant Laurent coefficients involve both the repeated eigenvalue a and the radius parameter λ, so direct coefficient comparison does not isolate a. We address this by decomposing the relevant matrix into a tridiagonal Toeplitz part plus a rank-two update and using Chebyshev polynomials of the second kind. This reduction yields an explicit Laurent-coefficient formula whose vanishing under the c

What carries the argument

Decomposition of the companion matrix into a tridiagonal Toeplitz part plus a rank-two update, combined with Chebyshev polynomials of the second kind, to produce an explicit formula for the Laurent coefficients that must vanish for the numerical range to be circular.

If this is right

  • The only companion matrix with repeated spectrum that can have a circular numerical range is the Jordan block.
  • The eigenvalue a is forced to zero by the circularity condition once the coefficient formula is obtained.
  • The same structural conclusion extends to companion matrices whose spectrum consists of zero and one other value a.
  • Direct comparison of Laurent coefficients fails because they mix a and the radius; the decomposition is required to separate them.

Where Pith is reading between the lines

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

  • The result indicates that circular numerical ranges within the companion-matrix class are highly restrictive and select the most non-normal representative.
  • The reduction technique may apply to other low-rank perturbations of Toeplitz matrices whose numerical ranges are under study.
  • Classification of all matrices with circular numerical ranges inside additional structured families becomes feasible once the Laurent-coefficient method is available.

Load-bearing premise

The explicit formula for the Laurent coefficients derived from the tridiagonal Toeplitz decomposition plus rank-two update and Chebyshev polynomials correctly encodes the circularity condition and forces both a=0 and the Jordan structure.

What would settle it

Exhibit a companion matrix of size at least 4 whose spectrum is a single repeated value a, whose numerical range is circular, yet which is not similar to the Jordan block.

read the original abstract

We prove that if an $n\times n\ (n > 3)$ companion matrix $A$ with the spectrum $\sigma(A) = \{ a \}$ has a circular numerical range, then $A$ is the Jordan block. This problem can be described by examining zeros of the Laurent polynomial arising from geometric properties of the numerical range. The difficulty is that the relevant Laurent coefficients involve both the repeated eigenvalue $a$ and the radius parameter $\lambda$, so direct coefficient comparison does not isolate $a$. We address this by decomposing the relevant matrix into a tridiagonal Toeplitz part plus a rank-two update and using Chebyshev polynomials of the second kind. This reduction yields an explicit Laurent-coefficient formula whose vanishing under the circularity condition gives $a=0$. Furthermore, we extend this result when the spectrum is $\sigma(A) = \{0, a\}$ with algebraic multiplicities $n-m$ and $m$, respectively.

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.

Referee Report

0 major / 2 minor

Summary. The paper proves that for n>3, an n×n companion matrix A with spectrum σ(A)={a} has circular numerical range only if A is the Jordan block. The argument decomposes A into a tridiagonal Toeplitz matrix plus rank-two update, invokes the generating function and recurrence for Chebyshev polynomials of the second kind to obtain explicit Laurent coefficients, and shows that the circularity condition forces these coefficients to vanish only when a=0. The same reduction is extended to the two-eigenvalue case σ(A)={0,a} with algebraic multiplicities n-m and m.

Significance. If the derivation holds, the result gives a complete algebraic characterization of circular numerical ranges for this structured class of non-normal matrices, extending prior work on numerical ranges via explicit closed-form coefficient formulas rather than perturbation or numerical methods. The explicit Laurent-coefficient isolation via Chebyshev polynomials is a technical strength that makes the proof falsifiable by direct substitution.

minor comments (2)
  1. [Abstract] The abstract refers to 'the reduction step' without a section number; adding an explicit pointer (e.g., §3.2) would improve readability.
  2. In the two-eigenvalue extension, the multiplicities n-m and m appear in the Laurent polynomial; a short remark on how the rank-two update changes with m would clarify the transition from the single-eigenvalue case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation to accept the manuscript. The report accurately summarizes the main results and technical approach.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper's central derivation decomposes the companion matrix (with fixed characteristic polynomial (x-a)^n) into a tridiagonal Toeplitz matrix plus rank-two update, then applies the explicit generating functions and recurrence relations for Chebyshev polynomials of the second kind to produce closed-form Laurent coefficients. These coefficients are set to zero under the circular numerical range condition, yielding an algebraic system that isolates a=0 (hence the Jordan block) without any self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations. The technique relies on standard properties of numerical ranges, Laurent polynomials, and orthogonal polynomials, all independent of the target conclusion. The same reduction is applied to the two-eigenvalue case. No step reduces by construction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof relies on standard mathematical tools and assumptions about matrix decompositions and polynomial properties without introducing new free parameters or entities.

axioms (2)
  • domain assumption Properties of the numerical range being circular imply zeros of the associated Laurent polynomial.
    Stated in the abstract as the description of the problem.
  • standard math Chebyshev polynomials of the second kind can be used to find explicit formulas for the Laurent coefficients after the Toeplitz-plus-rank-two decomposition.
    Used in the reduction to isolate coefficients.

pith-pipeline@v0.9.1-grok · 5691 in / 1353 out tokens · 37232 ms · 2026-07-02T04:58:47.092036+00:00 · methodology

discussion (0)

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Reference graph

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