Expansions for eigenfunction and eigenvalues of large-$n$ Toeplitz matrices

Leo P Kadanoff

doi:10.4279/pip.020003

Abstract

This paper constructs methods for finding convergent expansions for eigenvectors and eigenvalues of large-$n$ Toeplitz matrices based on a situation in which the analogous infinite-$n$ matrix would be singular. It builds upon work done by Dai, Geary, and Kadanoff [H Dai et al., J. Stat. Mech. P05012 (2009)] on exact eigenfunctions for Toeplitz operators which are infinite-dimension Toeplitz matrices. One expansion for the finite-$n$ case is derived from the operator eigenvalue equations obtained by continuing the finite-$n$ Toeplitz matrix to plus infinity. A second expansion is obtained by continuing the finite-$n$ matrix to minus infinity. The two expansions work together to give an apparently convergent expansion for the finite-$n$ eigenvalues and eigenvectors, based upon a solvability condition for determining eigenvalues. The expansions involve an expansion parameter expressed as an inverse power of $n$. A variational principle is developed, which gives an approximate expression for determining eigenvalues. The lowest order asymptotics for eigenvalues and eigenvectors agree with the earlier work [H Dai et al., J. Stat. Mech. P05012 (2009)]. The eigenvalues have a $(\ln n)/n$ term as their leading finite-$n$ correction in the central region of the spectrum. The $1/n$ correction in this region is obtained here for the first time.Received: 19 October 2009; Accepted: 29 September 2010; Edited by: A. G. Green; Reviewed by: T. Ehrhardt, Math. Dept., Univ. California, Santa Cruz, USA; DOI: 10.4279/PIP.020003

Highlights

Introduction iHistory write dz a(z)Tj,k = Tj−k = 2πiz zj−k (1)This paper is a continuation of recent work by Dai, Geary, and Kadano [1] and Lee, Dai and Bettleheim [2] on the spectrum of eigenvalues and eigenfunctions for singular Toeplitz matrices
The Toeplitz matrix is dened by having the indices j and k live in the interval [0, n − 1]. (Note that I use the subscript notation to describe behavior in coordinate space, and argument notation to describe behavior in Fourier space.)
The basic problem under consideration here is the denition of a good method for calculating the eigenvalues and eigenfunctions of Toeplitz matrices for large values of n

Summary

Introduction

Situations and for large n, the solution of the second equation provided an excellent approximation for the eigenfunction of the rst one, at least in the situation in which one is given the correct eigenvalue. In the same case I, situation in which Eq (5b) has eigenvalue solutions, Eq (5c) has none so that the forcing term produces a unique, nite result. The equation for the Toeplitz matrix's eigenvector can be cast in terms of three dierent kinds of functions which are respectively indicated by superscripts −, 0, and +.

Results

Conclusion