AN EQUILIBRIUM PROBLEM FOR THE LIMITING EIGENVALUE DISTRIBUTION OF RATIONAL TOEPLITZ MATRICES

We consider the asymptotic behavior of the eigenvalues of Toeplitz matrices with rational symbol as the size of the matrix goes to infinity. Our main result is that the weak limit of the normalized eigenvalue counting measure is a particular component of the unique solution to a vector equilibrium p...

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Bibliographic Details
Published in:SIAM journal on matrix analysis and applications 2010-01, Vol.31 (4), p.1894-1914
Main Authors: DELVAUX, Steven, DUITS, Maurice
Format: Article
Language:English
Subjects:
(vector) potential theory
Algebra
Asymptotic methods
Asymptotic properties
Constraining
Counting
Eigenvalues
Exact sciences and technology
generalized eigenvalues
Limit theorems
Linear and multilinear algebra, matrix theory
Mathematical analysis
Mathematics
Matrices
Matrix
Matrix methods
Nonlinear algebraic and transcendental equations
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Probability and statistics
Probability theory and stochastic processes
rational function
Sciences and techniques of general use
Studies
Toeplitz matrix
Vector space
Vectors (mathematics)
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Summary:We consider the asymptotic behavior of the eigenvalues of Toeplitz matrices with rational symbol as the size of the matrix goes to infinity. Our main result is that the weak limit of the normalized eigenvalue counting measure is a particular component of the unique solution to a vector equilibrium problem. Moreover, we show that the other components describe the limiting behavior of certain generalized eigenvalues. In this way, we generalize recent results by Kuijlaars and one of the authors [SIAM J. Matrix Anal. Appl., 30 (2008), pp. 173-196] that were concerned with banded Toeplitz matrices. [PUBLICATION ABSTRACT]
ISSN:0895-4798
1095-7162
1095-7162
DOI:10.1137/090778468