Asymptotic distribution of the even and odd spectra of real symmetric Toeplitz matrices
type: Item
instance of: scholarly article
title: Asymptotic distribution of the even and odd spectra of real symmetric Toeplitz matrices
zbMATH Open document ID: 0947.15009
DOI: 10.1016/S0024-3795(99)00070-1
author: William F. Trench
published in: Linear Algebra and its Applications
publication date: 2000-11-07T00:00:00Z
review text: Let \(T\) and \(J\) denote a given real symmetric Toeplitz matrix and the flip matrix of the same order, respectively. The eigenvectors \(e_i\) of \(T\) are known to satisfy either \(Je_i= e_i\) or \(Je_i= -e_i\), so that the spectrum of \(T\) may be split into the even spectrum comprising the eigenvalues \(\lambda_i\) whose eigenvectors satisfy \(Je_i=e_i\) and the odd spectrum comprising the other eigenvalues. In this paper, the author presents the results that describe the asymptotic distribution of both parts of the spectrum of the matrices \(T_n\) of order \(n\) with entries \(t_{ij}= {1\over\pi} \int^\pi_0 f(x)\cos (i-j)x dx\) where \(f\in L^2 [0,\pi]\) is a given even function.
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 1419887
zbMATH Keywords: asymptotic distribution of spectra
zbMATH Keywords: real symmetric Toeplitz matrix
zbMATH Keywords: eigenvectors
zbMATH Keywords: even spectrum
zbMATH Keywords: odd spectrum
zbMATH Keywords: eigenvalues
reviewed by: Z. Dostál
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1419887