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An Equilibrium Problem for the Limiting Eigenvalue Distribution of Rational Toeplitz Matrices

Delvaux, Steven and Duits, Maurice (2010) An Equilibrium Problem for the Limiting Eigenvalue Distribution of Rational Toeplitz Matrices. SIAM Journal on Matrix Analysis and Applications, 31 (4). pp. 1894-1914. ISSN 0895-4798. http://resolver.caltech.edu/CaltechAUTHORS:20100719-112202959

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Abstract

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.


Item Type:Article
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http://dx.doi.org/10.1137/090778468DOIUNSPECIFIED
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=SJMAEL000031000004001894000001&idtype=cvips&gifs=yesPublisherUNSPECIFIED
Additional Information:© 2010 Society for Industrial and Applied Mathematics. Received by the editors November 30, 2009; accepted for publication (in revised form) by H. J. Woerdeman February 19, 2010; published electronically April 21, 2010. The authors thank professor Arno Kuijlaars for stimulating discussions.
Subject Keywords:Toeplitz matrix; rational function; generalized eigenvalues; (vector) potential theory
Classification Code:AMS subject classifications: 15B05, 31C99
Record Number:CaltechAUTHORS:20100719-112202959
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20100719-112202959
Official Citation:An Equilibrium Problem for the Limiting Eigenvalue Distribution of Rational Toeplitz Matrices Steven Delvaux and Maurice Duits, SIAM. J. Matrix Anal. & Appl. 31, 1894 (2010), DOI:10.1137/090778468
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19107
Collection:CaltechAUTHORS
Deposited By: Jason Perez
Deposited On:19 Jul 2010 20:55
Last Modified:26 Dec 2012 12:15

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