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Eigenvalues of tridiagonal Hermitian Toeplitz matrices with perturbations in the off-diagonal corners

Sergei M. Grudsky, Egor A. Maximenko, Alejandro Soto-González

Abstract

In this paper we study the eigenvalues of Hermitian Toeplitz matrices with the entries $2,-1,0,\ldots,0,-α$ in the first column. Notice that the generating symbol depends on the order $n$ of the matrix. If $|α|\le 1$, then the eigenvalues belong to $[0,4]$ and are asymptotically distributed as the function $g(x)=4\sin^2(x/2)$ on $[0,π]$. The situation changes drastically when $|α|>1$ and $n$ tends to infinity. Then the two extreme eigenvalues (the minimal and the maximal one) lay out of $[0,4]$ and converge rapidly to certain limits determined by the value of $α$, whilst all others belong to $[0,4]$ and are asymptotically distributed as $g$. In all cases, we transform the characteristic equation to a form convenient to solve by numerical methods, and derive asymptotic formulas for the eigenvalues.

Eigenvalues of tridiagonal Hermitian Toeplitz matrices with perturbations in the off-diagonal corners

Abstract

In this paper we study the eigenvalues of Hermitian Toeplitz matrices with the entries in the first column. Notice that the generating symbol depends on the order of the matrix. If , then the eigenvalues belong to and are asymptotically distributed as the function on . The situation changes drastically when and tends to infinity. Then the two extreme eigenvalues (the minimal and the maximal one) lay out of and converge rapidly to certain limits determined by the value of , whilst all others belong to and are asymptotically distributed as . In all cases, we transform the characteristic equation to a form convenient to solve by numerical methods, and derive asymptotic formulas for the eigenvalues.

Paper Structure

This paper contains 9 sections, 25 theorems, 88 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Main results for weak perturbations
  4. Main results for strong perturbations
  5. Proofs for the case of weak perturbations
  6. Case |alpha|=1
  7. Proofs for the case of strong perturbations
  8. Numerical experiments
  9. Acknowledgements.

Key Result

Theorem 1

Let $\alpha\in\mathbb{C}$, $|\alpha|\le 1$, $\alpha\notin\{-1,1\}$, and $n\ge 3$. Then the matrix $A_{\alpha,n}$ has $n$ different eigenvalues belonging to $(0,4)$. More precisely, for every $j$ in $\{1,\ldots,n\}$,

Figures (4)

  • Figure 1: Functions \ref{['eq:eta']} for $\alpha=0.7+0.6i$ (left) and $\alpha=2+i$ (right).
  • Figure 2: The points $\vartheta_{\alpha,n,j}$ and the corresponding values of $\lambda_{\alpha,n,j}$ for $\alpha=2+i$, $n=6$. Different scales are used for the axis.
  • Figure 3: Points on the complex plane that yield the eigenvalues $\lambda_{\alpha,n,j}$ after applying the function $g$. In this example, $\alpha=2+i$ and $n=6$.
  • Figure 4: Left-hand sides (black) and right-hand sides (blue) of equations \ref{['eq:weak_equation_tan']} for $\alpha=0.7+0.6i$, $n=8$, $1\le j\le n$. Notice that $[0,\pi]$ is divided into $8$ equal subintervals, and each subinterval corresponds to its proper equation.

Theorems & Definitions (44)

  • Theorem 1: localization of the eigenvalues for weak perturbations
  • Theorem 2: characteristic equation for weak perturbations
  • Theorem 3: asymptotic expansion of the eigenvalues for weak perturbations
  • Theorem 4: localization of the eigenvalues for strong perturbations
  • Theorem 5: characteristic equations for strong perturbations
  • Theorem 6: asymptotic expansion of the eigenvalues for strong perturbations
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • ...and 34 more