E. A. Maximenko | Searcharxiv

arXiv2026

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

In this paper, we study the eigenvalues of the matrices $T_n(a)+γE_{n,1,1}$ 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 $γ$ is a fixed complex number such that $0<|γ|<1$. As $n\to\infty$, the eigenvalues of these matrices are asymptotically distributed as the function $2 i \sin(x)$, $x\in[0,2π]$. Our main result is an asymptotic formula for every eigenvalue with a residue of the order $O(1/n^3)$.

math.SP↗

arXiv2023

Eigenvalues of laplacian matrices of the cycles with one negative-weighted edge

We study the individual behavior of the eigenvalues of the laplacian matrices of the cyclic graph of order $n$, where one edge has weight $α\in\mathbb{C}$, with $\operatorname{Re}(α)<0$, and all the others have weights $1$. This paper is a sequel of a previous one where we considered $\operatorname{Re}(α) \in[0,1]$ (Eigenvalues of laplacian matrices of the cycles with one weighted edge, Linear Algebra Appl. 653, 2022, 86--115). We prove that for $\operatorname{Re}(α)<0$ and $n>\operatorname{Re}(α-1)/\operatorname{Re}(α)$, one eigenvalue is negative while the others belong to $[0,4]$ and are distributed as the function $x\mapsto 4\sin^2(x/2)$. Additionally, we prove that as $n$ tends to $\infty$, the outlier eigenvalue converges exponentially to $4\operatorname{Re}(α)^2/(2\operatorname{Re}(α)-1)$. We give exact formulas for the half of the inner eigenvalues, while for the others we justify the convergence of Newton's method and fixed-point iteration method. We find asymptotic expansions, as $n$ tends to $\infty$, both for the eigenvalues belonging to $[0,4]$ and the outlier. We also compute the eigenvectors and their norms.

math.SP↗