Inside the eigenvalues of certain Hermitian Toeplitz band matrices

Authors:
A. Böttcher;S. M. Grudsky;E. A. Maksimenko
Affiliations:
Fakultät für Mathematik, TU Chemnitz, 09107 Chemnitz, Germany;Departamento de Matemáticas, CINVESTAV del I.P.N., Apartado Postal 14-740, 07000 México, D.F., Mexico;Departamento de Matemáticas, CINVESTAV del I.P.N., Apartado Postal 14-740, 07000 México, D.F., Mexico
Venue:
Journal of Computational and Applied Mathematics
Year:
2010

Citing 3
Cited 1

A Finer Aspect of Eigenvalue Distribution of Selfadjoint Band Toeplitz Matrices

SIAM Journal on Matrix Analysis and Applications
On the Complexity of the Preconditioned Conjugate Gradient Algorithm for Solving Toeplitz Systems with a Fisher-Hartwig Singularity

SIAM Journal on Matrix Analysis and Applications
Spectral Properties of Banded Toeplitz Matrices

Spectral Properties of Banded Toeplitz Matrices

The inverse of banded matrices

Journal of Computational and Applied Mathematics

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Abstract

While extreme eigenvalues of large Hermitian Toeplitz matrices have been studied in detail for a long time, much less is known about individual inner eigenvalues. This paper explores the behavior of the jth eigenvalue of an n-by-n banded Hermitian Toeplitz matrix as n tends to infinity and provides asymptotic formulas that are uniform in j for 1@?j@?n. The real-valued generating function of the matrices is assumed to increase strictly from its minimum to its maximum, and then to decrease strictly back from the maximum to the minimum, having nonzero second derivatives at the minimum and the maximum. The results, which are of interest in numerical analysis, probability theory, or statistical physics, for example, are illustrated and underpinned by numerical examples.