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Asymptotics of Eigenvectors of Large Symmetric Banded Toeplitz Matrices

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Abstract

The paper studies the asymptotic behavior of all eigenvectors of banded symmetric (in general non-Hermitian) Toeplitz matrices as the dimension of the matrices tends to infinity. The main result describes, given certain assumptions, the structure of the eigenvectors in terms of the Laurent polynomial which generates the matrices, up to an error term that decays exponentially. The effectiveness of the formulas is confirmed by some numerical examples.

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References

  1. Batalshikov A., Grudsky S., Stukopin V.: Asymptotic of eigenvalues of large symmetric banded Toeplitz matrices. Linear Algebra Appl. 469, 464–486 (2015)

    Article  MathSciNet  Google Scholar 

  2. Bogoya J.M., Bottcher A., Grudsky S.M., Maksimenko E.A.: Eigenvectors of Hessenberg Toeplitz matrices and a problem by Dai Geary and Kadanoff. Linear Algebra Appl. 436, 3480–3492 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bogoya J.M., Bottcher A., Grudsky S.M.: Asymptotics of individual eigenvalues of a class of large Hessenberg Toeplitz matrices. Oper. Theory Adv. Appl. 220, 77–95 (2012)

    MathSciNet  Google Scholar 

  4. Böttcher, A., Grudsky, S.: Spectral Properties of Banded Toeplitz Matrices, p. 422. SIAM, Philadelphia, (2005)

  5. Böttcher A., Grudsky S., Iserles A.: Spectral theory of large Wiener–Hopf operators with complex-symmetric kernels and rational symbols. Math. Proc. Camb. Philos. Soc. 151, 161–191 (2011)

    Article  MATH  Google Scholar 

  6. Böttcher A., Grudsky S.M., Maksimenko E.A.: Inside the eigenvalues of certain Hermitian Toeplitz band matrices. J. Comput. Appl. Math. 233, 2245–2264 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Böttcher A., Grudsky S.M., Maksimenko E.A.: Of the structure of the eigenvectors of large Hermitian Toeplitz band matrices. Oper. Theory Adv. Appl. 210, 15–36 (2010)

    Google Scholar 

  8. Böttcher A., Silbermann B.: Introduction to Large Truncated Toeplitz Matrices. Universitext, Springer, New York (1999)

    Book  MATH  Google Scholar 

  9. Dai, H., Geary, Z., Kadanoff, L.P.: Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices. J. Stat. Mech. Theory Exp. 5, P05012 (2009)

  10. Deift P., Its A., Krasovsky I.: Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher–Hartwig singularities. Ann. Math. 174, 1243–1299 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. Deift P., Its A., Krasovsky I.: Eigenvalues of Toeplitz matrices in the bulk of the spectrum. Bull. Inst. Math. Acad. Sin. (N.S.) 7, 437–461 (2012)

    MATH  MathSciNet  Google Scholar 

  12. Deift, P., Its, A., Krasovsky, I.: Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model. Some history and some recent results. arXiv:1207.4990v3 [math.FA] (2012)

  13. Diaconis P.: Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture. Bull. Am. Math. Soc. 40, 155–178 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Grenander U., Szegő G.: Toeplitz Forms and Their Applications. University of California Press, Berkeley (1958)

    MATH  Google Scholar 

  15. Kac M., Murdock W.L., Szegő G.: On the eigenvalues of certain Hermitian forms. J. Rat. Mech. Anal. 2, 787–800 (1953)

    Google Scholar 

  16. Kadanoff L.P.: Spin–spin correlations in the two-dimensional ising model. Nuovo Cimento 44, 276–305 (1966)

    Article  Google Scholar 

  17. McCoy B.M., Wu T.T.: The two-dimensional Ising model. Harvard University Press, Cambridge (1973)

    Book  MATH  Google Scholar 

  18. Parter S.V.: On the extreme eigenvalues of Toeplitz matrices. Trans. Am. Math. Soc. 100, 263–276 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  19. Schmidt P., Spitzer F.: The Toeplitz matrices of an arbitrary Laurent polynomial. Math. Scand. 8, 15–28 (1960)

    MATH  MathSciNet  Google Scholar 

  20. Trench W.F.: Explicit inversion formula for Toeplitz band matrices. SIAM J. Algebraic Discrete Methods 6, 546–554 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  21. Widom H.: On the eigenvalues of certain Hermitian operators. Trans. Am. Math. Soc. 88, 491–522 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  22. Widom H.: Eigenvalue distribution of nonselfadjoint Toeplitz matrices and the asymptotics of Toeplitz determinants in the case of nonvanishing index. Oper. Theory Adv. Appl. 48, 387–421 (1990)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. STEL Computer Systems Ltd., Bolshaya Pochtovaya. 55/59, 105082, Moscow, Russia

    A. Batalshchikov

  2. Departamento de Mathematicas, CINVESTAV, Av. Instituto Politécnico Nacional 2508, Col. San Pedro Zacatenco, 07360, Mexico, D.F., Mexico

    S. Grudsky & E. Ramírez de Arellano

  3. Department of Applied Mathematics, Don State Technical University, Gagarin Sq., 344010, Rostov-on-Don, Russia

    V. Stukopin

  4. South Mathematical Institute, Markusa St., 22, 362027, Vladikavkaz, Russia

    V. Stukopin

Authors
  1. A. Batalshchikov
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  2. S. Grudsky
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  3. E. Ramírez de Arellano
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  4. V. Stukopin
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Corresponding author

Correspondence to A. Batalshchikov.

Additional information

Research of the second author is partially supported by CONACYT Grant 180049 and Grant EPSRC: EP/M009475/1.

Research of the fourth author is supported by the Project 2941 of Russian Ministry of Education and Science.

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Batalshchikov, A., Grudsky, S., de Arellano, E.R. et al. Asymptotics of Eigenvectors of Large Symmetric Banded Toeplitz Matrices. Integr. Equ. Oper. Theory 83, 301–330 (2015). https://doi.org/10.1007/s00020-015-2257-y

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  • DOI: https://doi.org/10.1007/s00020-015-2257-y

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