The Mathematical Institute, University of Oxford, Eprints Archive

The Spectra of Large Toeplitz Band Matrices with a Randomly Perturbed Entry

Boettcher, A. and Embree, Mark and Sokolov, V. I. (2001) The Spectra of Large Toeplitz Band Matrices with a Randomly Perturbed Entry. Technical Report. Unspecified. (Submitted)

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Abstract

This report is concerned with the union $sp_{\Omega}^{(j,k)}T_{n}(a)$ 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 $sp_{\Omega}^{(j,k)}T_{n}(a)$ 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 $sp_{\Omega}^{(j,k)}T_{n}(a)$ does not converge to something close to $sp_{\Omega}^{(j,k)}T_{n}(a)$ as $n \to \infty$, where $T(a)$ is the corresponding infinite Toeplitz matrix. Illustrations are provided for tridiagonal Toeplitz matrices, a notable special case.

The second author was supported by UK Enginering and Physical Sciences Research Council Grant GR/M12414

Item Type:Technical Report (Technical Report)
Subjects:H - N > Numerical analysis
Research Groups:Numerical Analysis Group
ID Code:1235
Deposited By:Lotti Ekert
Deposited On:20 May 2011 09:20
Last Modified:20 May 2011 09:20

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