The spectra of large Toeplitz band matrices with a randomly perturbed entry

Böttcher, A.; Embree, M.; Sokolov, V.

doi:10.1090/S0025-5718-03-01505-9

The spectra of large Toeplitz band matrices with a randomly perturbed entry
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by A. Böttcher, M. Embree and V. I. Sokolov PDF

Math. Comp. 72 (2003), 1329-1348 Request permission

Abstract:

This paper is concerned with the union $\operatorname {sp}_\Omega ^{(j,k)} T_n(a)$ of all possible spectra that may emerge when perturbing a large $n \times n$ Toeplitz band matrix $T_n(a)$ in the $(j,k)$ site by a number randomly chosen from some set $\Omega$. The main results give descriptive bounds and, in several interesting situations, even provide complete identifications of the limit of $\operatorname {sp}_\Omega ^{(j,k)} T_n(a)$ as $n \to \infty$. Also discussed are the cases of small and large sets $\Omega$ as well as the “discontinuity of the infinite volume case”, which means that in general $\operatorname {sp}_\Omega ^{(j,k)} T_n(a)$ does not converge to something close to $\operatorname {sp}_\Omega ^{(j,k)} T(a)$ as $n \to \infty$, where $T(a)$ is the corresponding infinite Toeplitz matrix. Illustrations are provided for tridiagonal Toeplitz matrices, a notable special case.

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