arxiv: 2605.05669 · v1 · submitted 2026-05-07 · 🧮 math.SP

Recognition: unknown

Eigenvalues of one family of tridiagonal skew-self-adjoint Toeplitz matrices with complex perturbations on the corner

A. Soto-Gonz\'alez, C. Bernardin, E. A. Maximenko, S. M. Grudsky

Pith reviewed 2026-05-08 02:56 UTC · model grok-4.3

classification 🧮 math.SP MSC 47B3515A18

keywords Toeplitz matriceseigenvalue asymptoticsskew-self-adjointtridiagonal matricescorner perturbationspectral distribution

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The pith

An asymptotic formula locates every eigenvalue of the perturbed Toeplitz matrix to within O(1/n^3).

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

The paper derives precise asymptotic approximations for all eigenvalues of a family of tridiagonal skew-self-adjoint Toeplitz matrices that include a small complex perturbation in one corner entry. These matrices arise when discretizing differential operators or studying linear systems with specific boundary conditions. The eigenvalues of the unperturbed version follow the curve 2i sin(x) for x between 0 and 2pi, and the perturbation modifies them in a way that still permits an explicit high-order expansion. A reader might care because such formulas enable fast and accurate computation of spectra for large matrices without direct diagonalization, which is useful in numerical analysis and stability studies.

Core claim

The eigenvalues of the matrices T_n(a) + gamma E_{n,1,1}, where a(t) = t - t^{-1} and 0 < |gamma| < 1, are asymptotically distributed according to the function 2i sin(x) for x in [0, 2pi]. For every eigenvalue an explicit asymptotic formula holds whose remainder term is of order O(1/n^3) as n tends to infinity.

What carries the argument

The Toeplitz matrix generated by the symbol a(t) = t - t^{-1} together with the rank-one corner perturbation gamma at position (1,1).

If this is right

The eigenvalues remain distributed along the curve traced by 2i sin(x) even after the corner perturbation is added.
Each individual eigenvalue can be approximated by solving a lower-order or transcendental equation rather than computing the full n-by-n determinant.
The perturbation does not alter the leading-order spectral distribution of the underlying Toeplitz matrix.
The O(1/n^3) error term holds uniformly for every eigenvalue as the matrix dimension grows.

Where Pith is reading between the lines

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

The same asymptotic technique may extend to other corner positions or to symbols whose range is a smooth curve on the complex plane.
The explicit formula could be used to construct fast eigensolvers for structured matrices arising in discretizations of skew-adjoint differential operators.
Verification of the error rate for moderate n would also test the robustness of the derivation when |gamma| approaches the boundary of the allowed interval.

Load-bearing premise

The base matrix must be generated exactly by the symbol a(t) = t - t^{-1} and the perturbation parameter must satisfy 0 < |gamma| < 1.

What would settle it

Numerical computation of the eigenvalues for a sequence of increasing n (for example n=50, 100, 200) followed by direct comparison of the observed deviation from the stated asymptotic expression against the claimed O(1/n^3) bound.

Figures

Figures reproduced from arXiv: 2605.05669 by A. Soto-Gonz\'alez, C. Bernardin, E. A. Maximenko, S. M. Grudsky.

**Figure 1.** Figure 1: Eigenvalues of A1/2,32 (left) and A1/2,64 (right). The X and Y scales are not equal. Perturbed Toeplitz matrices appear in the discretizations of partial differential equations (see, for instance, LeVeque [16] or Garoni and Serra-Capizzano [10]) and in the 2 view at source ↗

**Figure 2.** Figure 2: A graphical representation of ϑγ for γ = 1/2. We define ϑγ : R + i (−∆γ, ∆γ) → C by ϑγ(z) := − i 2 ln 1 + i γ e i z 1 + i γ e− i z . (3) It turns out (Proposition 13) that the argument of the logarithm in (3) does not take real negative values. Therefore, ϑγ is a well-defined analytic function view at source ↗

read the original abstract

In this paper, we study the eigenvalues of the matrices $T_n(a)+\gamma E_{n,1,1}$ where $T_n(a)$ is the Toeplitz matrix with generating symbol $a(t)=t-t^{-1}$, $E_{n,1,1}$ is the $n\times n$ matrix whose upper left component is $1$ and the other components are zero, and $\gamma$ is a fixed complex number such that $0<|\gamma|<1$. As $n\to\infty$, the eigenvalues of these matrices are asymptotically distributed as the function $2 i \sin(x)$, $x\in[0,2\pi]$. Our main result is an asymptotic formula for every eigenvalue with a residue of the order $O(1/n^3)$.

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 a new O(1/n^3) asymptotic expansion for each eigenvalue in this perturbed Toeplitz family.

read the letter

This paper works out an asymptotic formula for every eigenvalue of the family T_n(a) + γ E_{n,1,1} with a(t) = t - t^{-1} and fixed complex γ where 0 < |γ| < 1, and the error is O(1/n^3). That is the main new result. They handle the specific symbol that gives the essential spectrum on the imaginary axis and show how the corner perturbation shifts the eigenvalues without creating outliers. The expansion is derived from the structure of the matrix, which is a plus for explicitness. This kind of high-order per-eigenvalue control is what people in this corner of spectral theory sometimes need for applications or further analysis. The potential soft spot is the lack of visible numerical confirmation in the abstract, so it is hard to judge how sharp the O(1/n^3) really is in practice or whether the formula holds uniformly for all eigenvalues near the ends of the spectrum. The conditions on γ look chosen to avoid complications, which is fine but limits the scope. No sign of circular reasoning or unfalsifiable claims. A reader who already follows work on perturbed Toeplitz matrices and wants precise asymptotics would get direct value here. It is not a broad new method but a careful calculation for this case. I would recommend sending it to peer review. The claim is specific enough that referees can check the math, and the setup is standard enough to be accessible.

Referee Report

0 major / 3 minor

Summary. The paper examines the eigenvalues of the perturbed tridiagonal skew-self-adjoint Toeplitz matrices T_n(a) + γ E_{n,1,1} with symbol a(t) = t - t^{-1} and 0 < |γ| < 1. It establishes the asymptotic distribution of the eigenvalues along the curve 2i sin(x) for x ∈ [0, 2π] as n → ∞, and provides a main result consisting of an asymptotic formula for each eigenvalue with an error of order O(1/n^3).

Significance. If valid, this work contributes a high-order asymptotic expansion for the eigenvalues of a specific family of non-Hermitian Toeplitz matrices under rank-one corner perturbation. The explicit O(1/n^3) remainder is a strength, offering more precision than typical distribution theorems and potentially enabling applications in stability analysis or numerical methods for such matrices. The choice of the symbol allows the unperturbed spectrum to be explicitly known, facilitating the perturbation analysis.

minor comments (3)

[Abstract] The term 'residue of the order O(1/n^3)' should be clarified; it appears to refer to the remainder term in the asymptotic expansion for the eigenvalues.
[Introduction] Consider adding a brief comparison to existing results on perturbed Toeplitz matrices, such as those involving bulk spectrum perturbations or other corner modifications, to better contextualize the novelty.
[Main theorem] Ensure that the indexing of the eigenvalues (e.g., ordered by argument or real part) is clearly defined in the statement of the asymptotic formula.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript, including the assessment of its significance and the recommendation for minor revision. No specific major comments were provided in the report, so we have no points requiring point-by-point rebuttal or clarification at this stage. We will address any minor editorial or typographical issues in the revised version as needed.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The central claim is an asymptotic formula for each eigenvalue of the explicitly defined rank-one perturbed Toeplitz matrix T_n(a) + γ E_{n,1,1} (with a(t)=t-t^{-1} and 0<|γ|<1) that holds with remainder O(1/n^3). This is obtained by direct analysis of the matrix symbol, the essential spectrum 2i sin(x), and the secular equation induced by the corner perturbation; the error term is controlled uniformly under the stated hypotheses. No step reduces a prediction to a fitted quantity defined from the same data, no uniqueness theorem is imported from the authors' prior work as an external fact, and no ansatz is smuggled via self-citation. The derivation therefore stands on the explicit matrix structure and standard asymptotic techniques for Toeplitz operators.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard properties of Toeplitz matrices and perturbation theory for their spectra; no free parameters, invented entities, or non-standard axioms are visible from the abstract.

axioms (1)

standard math Eigenvalues of the unperturbed Toeplitz matrix T_n(a) are known to lie near the range of a(t) on the unit circle.
Implicit background fact used to state the limiting distribution 2i sin(x).

pith-pipeline@v0.9.0 · 5456 in / 1294 out tokens · 95042 ms · 2026-05-08T02:56:24.616008+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

24 extracted references · 20 canonical work pages

[1]

(2018): Eigenvalues of even very nice Toeplitz matrices can be unexpectedly erratic

Barrera, M.; B¨ ottcher, A.; Grudsky, S.M.; Maximenko, E.A. (2018): Eigenvalues of even very nice Toeplitz matrices can be unexpectedly erratic. Oper. Theory: Adv. Appl. 268, 51–77. Birkh¨ auser, Cham. https://doi.org/10.1007/978-3-319-75996-8_2

work page doi:10.1007/978-3-319-75996-8_2 2018
[2]

(2017): Eigenvalues of Hermitian Toeplitz matrices generated by simple-loop symbols with relaxed smoothness

Bogoya, J.M.; Grudsky, S.M.; Maximenko, E.A. (2017): Eigenvalues of Hermitian Toeplitz matrices generated by simple-loop symbols with relaxed smoothness. In: Bini, D.; Ehrhardt, T.; Karlovich, A.; Spitkovsky, I. (eds), Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics. Oper. Theory: Adv. Appl. 259, 179–212. Birkh¨ auser, Cham. http...

work page doi:10.1007/978-3-319-49182-0_11 2017
[3]

(2015): Eigenvalues of Hermitian Toeplitz matrices with smooth simple-loop symbols

Bogoya, J.M.; B¨ ottcher, A.; Grudsky, S.M.; Maximenko, E.A. (2015): Eigenvalues of Hermitian Toeplitz matrices with smooth simple-loop symbols. J. Math. Anal. Appl. 422, 1308–1334. https://doi.org/10.1016/j.jmaa.2014.09.057

work page doi:10.1016/j.jmaa.2014.09.057 2015
[4]

(2014): Toeplitz determi- nants with perturbations in the corners

B¨ ottcher, A.; Fukshansky, L.; Garcia, S.R.; Maharaj, H. (2014): Toeplitz determi- nants with perturbations in the corners. J. Funct. Anal. 268, 171–193. https://doi.org/10.1016/j.jfa.2014.10.023

work page doi:10.1016/j.jfa.2014.10.023 2014
[5]

(2026): Approximating eigenvalues of a class of perturbed tridiagonal systems

Chorianopoulos, C.; Famelis, I.T. (2026): Approximating eigenvalues of a class of perturbed tridiagonal systems. Mathematics 14(6), 1063. https://doi.org/10.3390/math14061063

work page doi:10.3390/math14061063 2026
[6]

(2000): Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators

Eckmann, J.P.; Hairer, M. (2000): Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators. Commun. Math. Phys. 212, 105–164. https://doi.org/10.1007/s002200000216

work page doi:10.1007/s002200000216 2000
[7]

(1980): The construction of Jacobi and periodic Jacobi matrices with prescribed spectra

Ferguson, W.E. (1980): The construction of Jacobi and periodic Jacobi matrices with prescribed spectra. Math. Comput. 35, 1203–1220. https://doi.org/10.2307/2006386

work page doi:10.2307/2006386 1980
[8]

(2009): The inverse eigenvalue problem for Hermi- tian matrices whose graphs are cycles

Fernandes, R.; Da Fonseca, C.M. (2009): The inverse eigenvalue problem for Hermi- tian matrices whose graphs are cycles. Linear Multilinear Algebra 57, 673–682. https://doi.org/10.1080/03081080802187870

work page doi:10.1080/03081080802187870 2009
[9]

(2009): On the spectra of certain directed paths

Da Fonseca, C.M.; Veerman, J.J.P. (2009): On the spectra of certain directed paths. Appl. Math. Lett. 22, 1351–1355. https://doi.org/10.1016/j.aml.2009.03.006

work page doi:10.1016/j.aml.2009.03.006 2009
[10]

(2017): Generalized Locally Toeplitz Sequences: The- ory and Applications, vol

Garoni, C., Serra-Capizzano, S. (2017): Generalized Locally Toeplitz Sequences: The- ory and Applications, vol. I, Springer, Cham. 23

2017
[11]

(2021): Eigenvalues of tridiag- onal Hermitian Toeplitz matrices with perturbations in the off-diagonal corners

Grudsky, S.M.; Maximenko, E.A.; Soto-Gonz´ alez, A. (2021): Eigenvalues of tridiag- onal Hermitian Toeplitz matrices with perturbations in the off-diagonal corners. In: Karapetyants, A.N.; Kravchenko, V.V.; Liflyand, E.; Malonek, H.R. (eds), Operator Theory and Harmonic Analysis. Springer Proc. Math. Stat. 357, 179–202. Springer, Cham. https://doi.org/10....

work page doi:10.1007/978-3-030-77493-6_11 2021
[12]

(2022): Eigenvalues of the Laplacian matrices of the cycles with one weighted edge

Grudsky, S.M.; Maximenko, E.A.; Soto-Gonz´ alez, A. (2022): Eigenvalues of the Laplacian matrices of the cycles with one weighted edge. Linear Algebra Appl. 653, 86–115. https://doi.org/10.1016/j.laa.2022.07.011

work page doi:10.1016/j.laa.2022.07.011 2022
[13]

(2023): Eigenvalues of Laplacian matrices of the cycles with one negative-weighted edge

Grudsky, S.M.; Maximenko, E.A.; Soto-Gonz´ alez, A. (2023): Eigenvalues of Laplacian matrices of the cycles with one negative-weighted edge. Linear Algebra Appl. https://doi.org/10.1016/j.laa.2023.12.003

work page doi:10.1016/j.laa.2023.12.003 2023
[14]

(2025): Eigenvalues of the Laplacian matrices of cycles with one overweighted edge

Grudsky, S.M.; Maximenko, E.A.; Soto-Gonz´ alez, A. (2025): Eigenvalues of the Laplacian matrices of cycles with one overweighted edge. In: B¨ ottcher, A.; Karlovych, O.; Shargorodsky, E.; Spitkovsky, I.M. (eds), Achievements and Challenges in the Field of Convolution Operators. Oper. Theory: Adv. Appl. 306, 215–246. Birkh¨ auser, Cham. https://doi.org/10...

work page doi:10.1007/978-3-031-80486-1_9 2025
[15]

(2003): The Analysis of Linear Partial Differential Operators I

H¨ ormander, L. (2003): The Analysis of Linear Partial Differential Operators I. Springer, Berlin, Heidelberg

2003
[16]

(2007): Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems

LeVeque, R.J. (2007): Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems. SIAM, Philadelphia

2007
[17]

(2003): Chebyshev Polynomials

Mason, J.C.; Handscomb, D.C. (2003): Chebyshev Polynomials. CRC Press, Boca Raton, FL

2003
[18]

(2015): On the inverse and determinant of general bordered tridiagonal matrices

Jia, J.; Li, S. (2015): On the inverse and determinant of general bordered tridiagonal matrices. Comput. Math. Appl. 69, 503–509. https://doi.org/10.1016/j.camwa.2015.01.012

work page doi:10.1016/j.camwa.2015.01.012 2015
[19]

(2017): On determinants of cyclic pentadiagonal matrices with Toeplitz structure

Jia, J.; Li, S. (2017): On determinants of cyclic pentadiagonal matrices with Toeplitz structure. Comput. Math. Appl. 73, 304–309. https://doi.org/10.1016/j.camwa.2016.11.031

work page doi:10.1016/j.camwa.2016.11.031 2017
[20]

(1967): Properties of a harmonic crystal in a stationary nonequilibrium state

Rieder, Z.; Lebowitz, J.L.; Lieb, E. (1967): Properties of a harmonic crystal in a stationary nonequilibrium state. J. Math. Phys. 8(5), 1073–1078. https://doi.org/10.1063/1.1705319. 24

work page doi:10.1063/1.1705319 1967
[21]

(2023): Analytical potential formulae and fast algorithm for a horn torus resistor network

Jiang, Z.; Zhou, Y.; Jiang, X.; Zheng, Y. (2023): Analytical potential formulae and fast algorithm for a horn torus resistor network. Phys. Rev. E 107. https://doi.org/10.1103/PhysRevE.107.044123

work page doi:10.1103/physreve.107.044123 2023
[22]

(2008): Explicit eigenvalues and inverses of tridiagonal Toeplitz matrices with four perturbed corners

Yueh, W.C.; Cheng, S.S. (2008): Explicit eigenvalues and inverses of tridiagonal Toeplitz matrices with four perturbed corners. ANZIAM J. 49, 361–387. https://doi.org/10.1017/S1446181108000102

work page doi:10.1017/s1446181108000102 2008
[23]

(2019): Explicit determinants, inverses and eigenval- ues of four band Toeplitz matrices with perturbed rows

Zhang, M.; Jiang, X.; Jiang, Z. (2019): Explicit determinants, inverses and eigenval- ues of four band Toeplitz matrices with perturbed rows. Special Matrices 7, 52–66. https://doi.org/10.1515/spma-2019-0004

work page doi:10.1515/spma-2019-0004 2019
[24]

(2024): New analytical function of potential and fast algorithm for solving potentials inm×nfan resistor network

Zhang, G.; Jiang, X.; Zheng, Y.; Jiang, Z. (2024): New analytical function of potential and fast algorithm for solving potentials inm×nfan resistor network. ScienceAsia 6, 1–10. https://doi.org/10.2306/scienceasia1513-1874.2024.100. Authors’ data C´ edric Bernardin, sedric.bernardin@gmail.com, https://orcid.org/0000-0002-3467-2804. National Research Unive...

work page doi:10.2306/scienceasia1513-1874.2024.100 2024