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On the asymptotic behavior of the eigenvectors of large banded Toeplitz matrices
Author(s) -
Böttcher A.,
Grudsky S.,
Ramírez de Arellano E.
Publication year - 2006
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310349
Subject(s) - toeplitz matrix , eigenvalues and eigenvectors , mathematics , truncation (statistics) , matrix (chemical analysis) , combinatorics , pure mathematics , statistics , chemistry , physics , quantum mechanics , chromatography
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)