Matrices with Displacement Structure, Generalized Bezoutians, and Moebius Transformations

  • Georg Heinig
  • Karla Rost
Part of the Operator Theory: Advances and Applications book series (OT, volume 40)


Matrices are considered the entries of which fulfil a difference equation (called ω-structured matrices) and a class of generalized Bezoutians is introduced. It is shown that A is a generalized Bezoutian iff its inverse is ω-structured. This result generalizes the Gohberg/Semencul theorem and other facts concerning Toeplitz matrices.


Full Rank Inversion Formula Scalar Case Hermitian Matrice Toeplitz Matrice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ab]
    Abukov, V.M., Kernel structure and the inversion of Toeplitz and Hankel matrices. (in Russian) Izvestija vuzov (Mat.) 7 (290), 1986, 3–8.Google Scholar
  2. [AJ]
    Anderson, B. and Jury, E., Generalized Bezoutian and Sylvester matrices in multivariable linear control. IEEE Trans. on a.c., AC-21, 4 (1976), 551–556.CrossRefGoogle Scholar
  3. [BH]
    Baxter, G. and Hirschman, I.I., An explicit inversion formula for finite-section Wiener-Hopf operators. Bull. AMS 70 (1964), 820–823.CrossRefGoogle Scholar
  4. [BAS1]
    Ben-Artzi, A. and Shalom, T., On inversion of Toeplitz and close to Toeplitz matrices. Linear Algebra and Appl. 75 (1986), 173–192.CrossRefGoogle Scholar
  5. [BAS2]
    Ben-Artzi, A. and Shalom, T., On inversion of block Toeplitz matrices. Int. Equ. and Op. Theory 8,6 (1985), 751–779.CrossRefGoogle Scholar
  6. [GH]
    Gohberg, I. and Heinig, G., Inversion of finite Toeplitz matrices composed of elements of a noncommutative algebra. (in Russian) Rev. Roumaine Math. Pures and Appl. 19,5 (1974), 623–663.Google Scholar
  7. [GK]
    Gohberg, I. and Krupnik, N.Ja., A formula for the inversion of finite-section Toeplitz matrices. (in Russian) Mat.Issled. 7,2 (1972), 272–284.Google Scholar
  8. [GS]
    Gohberg, I. and Semencul, A.A., On inversion of finite-section Toeplitz matrices and their continuous analogues. (in Russian) Mat.Issled. 7,2 (1972), 201–224.Google Scholar
  9. [HR1]
    Heinig, G. and Rost, K., Algebraic methods for Toeplitz-like matrices and operators. Akademie-Verlag, Berlin 1984 and Operator theory, Vol.13, Birkhäuser Basel 1984.Google Scholar
  10. [HR2]
    Heinig, G. and Rost, K., On the inverses of Toeplitz-plus-Hankel matrices. Lin.Alg.and Appl.(to appear 1988)Google Scholar
  11. [HR3]
    Heinig, G. and Rost, K., Inversion of matrices with displacement structure (to appear)Google Scholar
  12. [HT]
    Heinig, G. and Tewodros, A., On the inverse of Hankel and Toeplitz mosaic matrices. Seminar Analysis, Teubner-Verlag Leipzig (to appear 1989).Google Scholar
  13. [Ioh]
    Iohvidov, I.S., Hankel and Toeplitz matrices and forms. Birkhäuser Basel 1982.Google Scholar
  14. [KK]
    Kailath, T. and Koltracht, I., Matrices with block Toeplitz inverses. Lin.Alg.and Appl. 75 (1986), 145–153.CrossRefGoogle Scholar
  15. [L]
    Lander, F.I., The Bezoutian and the inversion of Hankel and Toeplitz matrices. Mat.Issled. 9,2 (1974), 69–87.Google Scholar
  16. [LT]
    Lerer, L. and Tismenetsky, M., Generalized Bezoutian and the inversion problem for block Toeplitz matrices. Int.Equ. and Op.Theory 9,6 (1986), 790–819.CrossRefGoogle Scholar
  17. [LABK]
    Lev-Ari, H., Bistritz, Y. and Kailath, T., Generalized Bezoutians and families of efficient root-location procedures. IEEE Trans, on Circ. and Syst.(to appear)Google Scholar
  18. [LAK]
    Lev-Ari, H. and Kailath T., Triangular factorization of structured Hermitian matrices. Operator Theory, Vol.18, Birkhäuser Basel 1986.Google Scholar
  19. [T1]
    Trench, W.F., An algorithm for the inversion of finite Toeplitz matrices. SIAM J.Appl.Math. 12 (1964), 515–522.CrossRefGoogle Scholar
  20. [T2]
    Trench, W.F., A note on Toeplitz inversion formulas, (to appear)Google Scholar

Copyright information

© Birkhäuser Verlag Basel 1989

Authors and Affiliations

  • Georg Heinig
    • 1
  • Karla Rost
    • 1
  1. 1.Sektion MathematikTechnische Universität Karl-Marx-StadtKarl-Marx-StadtGerman Democratic Republic

Personalised recommendations