Prediction and the Inverse of Toeplitz Matrices

  • I. Gohberg
  • H. J. Landau
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 119)


Toeplitz matrices represent the discrete analogue of convolutions, and the problem of inverting them is often encountered. The inverse of a Toeplitz matrix is no longer Toeplitz. However, thanks to a formula of Gohberg and Semençul, it can be expressed in terms of two closely related triangular Toeplitz matrices. Here we use an analogy with predicting stationary stochastic processes to motivate a simple proof of this formula, as well as of the main facts in the classical trigonometric moment problem.


Toeplitz Operator Toeplitz Matrix Toeplitz Matrice Stationary Stochastic Process Displacement Rank 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Akhiezer N. I. The Classical Moment Problem. Hafner, New York, 1965.MATHGoogle Scholar
  2. [2]
    Baxter G., Hirschman I. I. Jr. An explicit inversion formula for finite-section Wiener-Hopf operators. Bull. Amer. Math. Soc., 70: 820–823, 1964.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Gohberg I. C., Feldman I. A. Convolution Equations and Projection Methods for their Solution. Transi. Math. Monographs 41, Amer. Math. Soc., Providence, R.I., 1974.Google Scholar
  4. [4]
    Gohberg I. C., Feldman I. A. Projection methods for solving Wiener-Hopf equations. Acad. Sci. Moldovian SSR. Inst. Math, with Computing Center, Akad. Nauk Moldov. SSR, Kishiniev, 1967. ( Russian. )Google Scholar
  5. [5]
    Gohberg I., Heinig G. Inversion of finite Toeplitz matrices composed of elements of a noncommutative algebra. Revue Roumaine de Mathématiques Pures et Appliquées, 19: 623–663, 1974.MathSciNetGoogle Scholar
  6. [6]
    Gohberg I., Semençul A. A. On the inverse of finite Toeplitz matrices and their continuous analogs. Math. Issled., 7:238–253, 1972. (Russian.)MATHGoogle Scholar
  7. [7]
    Heinig G., Rost K. Algebraic methods for Toeplitz-like matrices and operators. In Operator Theory: Advances and Applications, 13, Birkhäuser, Basel, Boston, Berlin, 1984.Google Scholar
  8. [8]
    Kailath T., Kung S. Y., Morf M. Displacement ranks of matrices and linear equations. J. Math. Anal. Appl., 68:395–407, 1979; Bull. Amer. Math. Soc., 1: 769–773, 1979.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Kailath T., Viera A., Morf M. Inverses of Toeplitz operators, innovations, and orthogonal polynomials. SIAM Rev., 20: 106–119, 1978.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Kutikov L. M. Inversion of correlation matrices and some problems of selftuning. Izv. Akad. Nauk SSSR, Ser. Tech.-Cybernetics, 5, 1965. (Russian.)Google Scholar
  11. [11]
    Kutikov L. M. On the structure of matrices, inverse to correlation matrices for vector random processes. J. of Computational Math, and Math. Physics, 7(4), 1967. (Russian.)MathSciNetGoogle Scholar
  12. [12]
    Landau H. J. Maximum entropy and the moment problem. Bull. Amer. Math. Soc., 16: 47–77, 1987.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Moments in Mathematics. Proc. Symp. Appl. Math., 37. Amer. Math. Soc., Providence, R.I., 1987. H.J. Landau, ed.Google Scholar
  14. [14]
    Levinson N. The Wiener rms error criterion in filter design and prediction. J. Math. Phys., 25: 261–278, 1947.MathSciNetGoogle Scholar
  15. [15]
    Trench W. F. An algorithm for the inversion of finite Toeplitz matrices. J. SIAM, 12: 515–522, 1964.MathSciNetMATHGoogle Scholar
  16. [16]
    Trench W. F. A note on a Toeplitz inversion formula. Lin. Alg. Appl., 129: 55–61, 1990.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Yaglom A. M. A Introduction to the Theory of Stationary Random Functions. Prentice-Hall, Englewood Cliffs, NJ, 1962.MATHGoogle Scholar

Copyright information

© Birkhäuser 1994

Authors and Affiliations

  • I. Gohberg
    • 1
  • H. J. Landau
    • 2
  1. 1.Tel-Aviv UniversityTel-AvivIsrael
  2. 2.AT&T Bell LaboratoriesMurray HillUSA

Personalised recommendations