On reality of eigenvalues of banded block Toeplitz matrices

Dario Giandinoto

arxiv: 2411.16266 · v1 · submitted 2024-11-25 · 🧮 math.SP · math.CA

On reality of eigenvalues of banded block Toeplitz matrices

Dario Giandinoto This is my paper

Pith reviewed 2026-05-23 17:25 UTC · model grok-4.3

classification 🧮 math.SP math.CA

keywords Toeplitz matricesblock Toeplitz matricesasymptotic spectrumeigenvalue realitybanded matricesconjecturespectral theory

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The pith

A conjecture gives necessary and sufficient conditions for the asymptotic spectrum of real banded block Toeplitz matrices to lie entirely on the real line.

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

The paper formulates a conjecture that identifies the precise conditions under which the asymptotic eigenvalues of any real banded block Toeplitz matrix remain real numbers. This extends an earlier conjecture that covered only the non-block, scalar case. A sympathetic reader would care because these matrices arise in discretizations of differential equations and signal processing, where knowing the spectrum is real determines stability and oscillation behavior without needing to compute large eigenvalue problems directly. The authors supply a partial proof for some instances and numerical checks that support the conditions across several block sizes and bandwidths.

Core claim

The paper formulates and partially proves a general conjecture providing necessary and sufficient conditions for the reality of the asymptotic spectrum of an arbitrary real banded block Toeplitz matrix, and presents numerical experiments supporting it. This conjecture is a direct generalization of the already existing one in the case of banded Toeplitz matrices.

What carries the argument

The formulated conjecture on necessary and sufficient conditions that force the asymptotic spectrum of a real banded block Toeplitz matrix to be real.

If this is right

For any real banded block Toeplitz matrix meeting the conditions, the limiting eigenvalues as size grows can be asserted real without direct computation.
The result applies uniformly to systems of coupled equations represented by block entries rather than scalars.
Partial analytic proof covers selected bandwidths and block dimensions, with the remaining cases resting on the numerical evidence.
The conjecture supplies a structural test based on the matrix symbol or generating function that replaces full spectral analysis for large systems.

Where Pith is reading between the lines

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

If the conjecture holds, it could supply a quick pre-check before running iterative eigensolvers on very large block Toeplitz systems arising from multi-physics models.
The block extension may link to stability criteria for vector-valued recurrence relations or multi-component lattice models.
Further tests on non-constant block patterns or higher bandwidths could reveal whether extra algebraic constraints appear beyond those already stated.

Load-bearing premise

The stated conditions in the conjecture are both necessary and sufficient to guarantee that the asymptotic spectrum lies on the real line in the block setting.

What would settle it

A single explicit real banded block Toeplitz matrix whose asymptotic spectrum contains a non-real eigenvalue while satisfying every condition in the conjecture, or a matrix whose spectrum is entirely real yet violates one of the conditions.

Figures

Figures reproduced from arXiv: 2411.16266 by Dario Giandinoto.

**Figure 2.** Figure 2: Eigenvalues of matrix Tn(B1) with size n = 100 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 4.** Figure 4: Eigenvalues of matrix Tn(B2) with size n = 100 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 6.** Figure 6: ζ = 70 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗

**Figure 8.** Figure 8: ζ = 40 [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗

read the original abstract

We formulate and partially prove a general conjecture providing necessary and sufficient conditions for the reality of the asymptotic spectrum of an arbitrary real banded block Toeplitz matrix. Additionally we present numerical experiments supporting it. This conjecture is a direct generalization of the already existing one in the case of banded Toeplitz 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

Generalizes scalar banded Toeplitz reality conjecture to blocks with partial proof and numerics.

read the letter

The punchline is that this work takes the known conjecture for scalar banded Toeplitz matrices and extends it to the block case, stating necessary and sufficient conditions for the asymptotic spectrum to be real, with a partial proof and numerical backing. What is new is the formulation of the block version. The authors show how the conditions generalize and prove it in some cases while leaving others for future work. The numerical experiments are used to check the conjecture where the proof does not reach. This approach is straightforward and builds directly on the scalar literature. The generalization appears clean, and the decision to include numerics helps make the claim more concrete. The main limitation is that the proof is partial. This means the conjecture is not fully resolved, and readers will need to see which parts are covered and how the numerics fill the gaps. The abstract gives no specific data on the experiments, such as matrix sizes or error measures, so the support level is hard to gauge from the summary alone. The work is for people in operator theory and matrix analysis who deal with structured matrices and their spectra. It would be of interest to those who have looked at the scalar Toeplitz case and want to see the block extension. It deserves peer review. The conjecture is a genuine extension, the partial proof is honest, and having referees look at the details would help assess how far it goes.

Referee Report

1 major / 0 minor

Summary. The paper formulates a conjecture giving necessary and sufficient conditions for the asymptotic spectrum of an arbitrary real banded block Toeplitz matrix to be real. It provides a partial proof of this conjecture together with numerical experiments, presenting the result as a direct generalization of an existing conjecture for the scalar (non-block) banded Toeplitz case.

Significance. If the conjecture is correct, it would supply an explicit criterion for reality of the limiting spectrum in the block setting, extending the scalar theory to a broader class of matrices that arise in applications such as multivariate time series and discretized PDEs. The numerical support is noted but the partial character of the proof limits the immediate strength of the contribution.

major comments (1)

[Abstract and introduction] The manuscript states that only a partial proof is given; without the full details of which parts of the conjecture are proved and which remain open (including any dependence on the block size or bandwidth), it is impossible to assess whether the central claim is supported at the level required for publication.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for greater clarity regarding the scope of our partial proof. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract and introduction] The manuscript states that only a partial proof is given; without the full details of which parts of the conjecture are proved and which remain open (including any dependence on the block size or bandwidth), it is impossible to assess whether the central claim is supported at the level required for publication.

Authors: We agree that the abstract and introduction would benefit from an explicit summary of the proved versus open cases. The body of the paper already contains the detailed statements and proofs for the cases that are established (along with the numerical experiments supporting the general conjecture). In the revised version we will add a dedicated paragraph in the introduction that delineates the proved portions, notes their dependence on block size and bandwidth, and identifies the remaining open cases. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper formulates a new conjecture as a direct generalization of an existing scalar banded Toeplitz conjecture, provides a partial proof, and supplies numerical experiments. The abstract and described content show no self-definitional reductions, no fitted inputs renamed as predictions, and no load-bearing self-citations; the generalization rests on external prior work for the scalar case with independent support added for the block case.

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 conjecture is stated at the level of necessary and sufficient conditions without visible fitting or new postulated objects.

pith-pipeline@v0.9.0 · 5555 in / 1143 out tokens · 34843 ms · 2026-05-23T17:25:15.709297+00:00 · methodology

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/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Conjecture 1.1: Λ0(B)⊂R iff Γ(B) contains k Jordan curves having 0 in their interior. Thm 1.2 proves necessity via Newton polygon of f and paths 0↔∞ intersecting Nj⊂Γ(B).
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Γ(B) defined by Re/Im f(x+iy,α+iβ)=0 with β=0; algebraic curve g(x,y)=0.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · 1 internal anchor

[1]

Alexandersson, P

P. Alexandersson, P. Br¨ and´ en, and B. Shapiro. ”An inverse problem in P´ olya-Schur theory. I.” Non-degenerate and degenerate operators, submitted (2020). 12 DARIO GIANDINOTO

work page 2020
[2]

Ammari, S

H. Ammari, S. Barandun, and P. Liu, Perturbed block Toeplitz matrices and the non- Hermitian skin effect in dimer systems of subwavelength resonators, arXiv:2307.13551, 25 July 2023

work page arXiv 2023
[3]

Ammari, S

H. Ammari, S. Barandun, Y. De Bruijn, P. Liu, and C. Thalhammer, Spectra and pseudo- spectra of tridiagonal k-Toeplitz matrices and the topological origin of the non-Hermitian skin effect, arXiv:2401.12626, Jan. 23, 2024

work page arXiv 2024
[4]

Haoyan Chen, and Yi Zhang, Real spectra in one-dimensional single-band non-Hermitian Hamiltonians, arXiv:2303.15186, 19 sept. 2023

work page arXiv 2023
[5]

Bebiano, J

N. Bebiano, J. Da Providˆ encia and J. P. Da Providˆ encia, Classes of non-hermitian operators with real eigenvalues, Electronic Journal of Linear Algebra, Volume 21, pp. 98-109, October 2010

work page 2010
[6]

B¨ ottcher and S

A. B¨ ottcher and S. M. Grudsky. Spectral properties of banded Toeplitz matrices, Society for Industrial and Applied Mathematics, 2005

work page 2005
[7]

Delvaux, Equilibrium problem for the eigenvalues of banded block Toeplitz matrices, Math

S. Delvaux, Equilibrium problem for the eigenvalues of banded block Toeplitz matrices, Math. Nach. Volume 285, Issue 16, November 2012, Pages 1935-1962

work page 2012
[8]

E. J. Meier, Al. Fangzhao, B. Gadway, (2016-12-23), Observation of the topological soliton state in the Su–Schrieffer–Heeger model. Nature Communications. 7 (1): 13986. arXiv:1607.02811

work page internal anchor Pith review Pith/arXiv arXiv 2016
[9]

Han and W

R. Han and W. Schlag, Non-perturbative localization for quasi-periodic Jacobi block matrices, arXiv: 2309.03423

work page arXiv
[10]

W. P. Su, J. R. Schrieffer, A. J. Heeger, (1979-06-18). Solitons in Polyacetylene. Physical Review Letters. 42 (25): 1698–1701

work page 1979
[11]

Kawabata, and M

K. Kawabata, and M. Sato, Real spectra in non-Hermitian topological insulators, Phys. Rev. Res. 2, 033391 (2020). Pascal-like determinants are recursive, Advances in Applied Mathematics 33 (2004) 431–450

work page 2020
[12]

Shapiro, F

B. Shapiro, F. ˇStampach, Non-selfajoint Toeplitz matrices whose principal submatrices have real spectrum, Constr. Approx. 49 (2019), no. 2, 191–226, https://doi.org/10.1007/s00365- 017-9408-0

work page doi:10.1007/s00365- 2019
[13]

Shapiro, F

B. Shapiro, F. ˇStampach, Errata to ”Non-self-adjoint Toeplitz matrices whose princi- pal submatrices have real spectrum”, February 2023, Constructive Approximation DOI: 10.1007/s00365-022-09614-0

work page doi:10.1007/s00365-022-09614-0 2023
[14]

Schmidt, F

P. Schmidt, F. Spitzer, The Toeplitz matrices of an arbitrary Laurent polynomial, Mathe- matica Scandinavica 8 (1960), 15–38

work page 1960
[15]

Widom, Asymptotic behavior of block Toeplitz matrices and determinants, Adv

H. Widom, Asymptotic behavior of block Toeplitz matrices and determinants, Adv. Math., 13, 284-322 (1974)

work page 1974
[16]

Widom, Asymptotic behavior of block Toeplitz matrices and determinants

H. Widom, Asymptotic behavior of block Toeplitz matrices and determinants. II, Adv. Math., 21, l-29 (1976),

work page 1976
[17]

Yang, Y.-K

G. Yang, Y.-K. Li, Y. Fu , Zh. Wang, and Y. Zhang, Complex semiclassical theory for non- Hermitian quantum system, Phys. Rev. 109, 045110 (2024). Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden Email address: dario.giandinoto@math.su.se

work page 2024

[1] [1]

Alexandersson, P

P. Alexandersson, P. Br¨ and´ en, and B. Shapiro. ”An inverse problem in P´ olya-Schur theory. I.” Non-degenerate and degenerate operators, submitted (2020). 12 DARIO GIANDINOTO

work page 2020

[2] [2]

Ammari, S

H. Ammari, S. Barandun, and P. Liu, Perturbed block Toeplitz matrices and the non- Hermitian skin effect in dimer systems of subwavelength resonators, arXiv:2307.13551, 25 July 2023

work page arXiv 2023

[3] [3]

Ammari, S

H. Ammari, S. Barandun, Y. De Bruijn, P. Liu, and C. Thalhammer, Spectra and pseudo- spectra of tridiagonal k-Toeplitz matrices and the topological origin of the non-Hermitian skin effect, arXiv:2401.12626, Jan. 23, 2024

work page arXiv 2024

[4] [4]

Haoyan Chen, and Yi Zhang, Real spectra in one-dimensional single-band non-Hermitian Hamiltonians, arXiv:2303.15186, 19 sept. 2023

work page arXiv 2023

[5] [5]

Bebiano, J

N. Bebiano, J. Da Providˆ encia and J. P. Da Providˆ encia, Classes of non-hermitian operators with real eigenvalues, Electronic Journal of Linear Algebra, Volume 21, pp. 98-109, October 2010

work page 2010

[6] [6]

B¨ ottcher and S

A. B¨ ottcher and S. M. Grudsky. Spectral properties of banded Toeplitz matrices, Society for Industrial and Applied Mathematics, 2005

work page 2005

[7] [7]

Delvaux, Equilibrium problem for the eigenvalues of banded block Toeplitz matrices, Math

S. Delvaux, Equilibrium problem for the eigenvalues of banded block Toeplitz matrices, Math. Nach. Volume 285, Issue 16, November 2012, Pages 1935-1962

work page 2012

[8] [8]

E. J. Meier, Al. Fangzhao, B. Gadway, (2016-12-23), Observation of the topological soliton state in the Su–Schrieffer–Heeger model. Nature Communications. 7 (1): 13986. arXiv:1607.02811

work page internal anchor Pith review Pith/arXiv arXiv 2016

[9] [9]

Han and W

R. Han and W. Schlag, Non-perturbative localization for quasi-periodic Jacobi block matrices, arXiv: 2309.03423

work page arXiv

[10] [10]

W. P. Su, J. R. Schrieffer, A. J. Heeger, (1979-06-18). Solitons in Polyacetylene. Physical Review Letters. 42 (25): 1698–1701

work page 1979

[11] [11]

Kawabata, and M

K. Kawabata, and M. Sato, Real spectra in non-Hermitian topological insulators, Phys. Rev. Res. 2, 033391 (2020). Pascal-like determinants are recursive, Advances in Applied Mathematics 33 (2004) 431–450

work page 2020

[12] [12]

Shapiro, F

B. Shapiro, F. ˇStampach, Non-selfajoint Toeplitz matrices whose principal submatrices have real spectrum, Constr. Approx. 49 (2019), no. 2, 191–226, https://doi.org/10.1007/s00365- 017-9408-0

work page doi:10.1007/s00365- 2019

[13] [13]

Shapiro, F

B. Shapiro, F. ˇStampach, Errata to ”Non-self-adjoint Toeplitz matrices whose princi- pal submatrices have real spectrum”, February 2023, Constructive Approximation DOI: 10.1007/s00365-022-09614-0

work page doi:10.1007/s00365-022-09614-0 2023

[14] [14]

Schmidt, F

P. Schmidt, F. Spitzer, The Toeplitz matrices of an arbitrary Laurent polynomial, Mathe- matica Scandinavica 8 (1960), 15–38

work page 1960

[15] [15]

Widom, Asymptotic behavior of block Toeplitz matrices and determinants, Adv

H. Widom, Asymptotic behavior of block Toeplitz matrices and determinants, Adv. Math., 13, 284-322 (1974)

work page 1974

[16] [16]

Widom, Asymptotic behavior of block Toeplitz matrices and determinants

H. Widom, Asymptotic behavior of block Toeplitz matrices and determinants. II, Adv. Math., 21, l-29 (1976),

work page 1976

[17] [17]

Yang, Y.-K

G. Yang, Y.-K. Li, Y. Fu , Zh. Wang, and Y. Zhang, Complex semiclassical theory for non- Hermitian quantum system, Phys. Rev. 109, 045110 (2024). Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden Email address: dario.giandinoto@math.su.se

work page 2024