On the asymptotic behavior of the eigenvectors of large banded Toeplitz matrices

Let λ be an eigenvalue of an infinite Toeplitz band matrix A and let λ n be an eigenvalue of the n × n truncation A n of A . Suppose λ n converges to λ as n → ∞. We show that generically the eigenspaces for λ n are onedimensional and contain a vector x n whose first component is 1 if only n is large enough, and we prove that x n converges to an eigenvector x 0 of A that is independent of the particular choice of the λ n . The eigenspace of A corresponding to λ is spanned by x 0 and a finite number of shifts of x 0 . (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)