Expansions for eigenfunction and eigenvalues of large-n toeplitz matrices

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 [11 Dai et ai, J. Stat. Mech. P05012 (2009)] on exact eigenfuiictions 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 ai. J. Stat. Mech. P05012 (2009)]. The eigenvalues have a (Inn)/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.