Advertisement

Integral Equations and Operator Theory

, Volume 9, Issue 6, pp 790–819 | Cite as

Generalized Bezoutian and the inversion problem for block matrices, I. General scheme

  • L. Lerer
  • M. Tismenetsky
Article

Abstract

The unified approach to the matrix inversion problem initiated in this work is based on the concept of the generalized Bezoutian for several matrix polynomials introduced earlier by the authors. The inverse X−1 of a given block matrix X is shown to generate a set of matrix polynomials satisfying certain conditions and such that X−1 coincides with the Bezoutian associated with that set. Thus the inversion of X is reduced to determining the underlying set of polynomials. This approach provides a fruitful tool for obtaining new results as well as an adequate interpretation of the known ones.

Keywords

General Scheme Unify Approach Block Matrix Matrix Inversion Matrix Polynomial 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    B.D.O. Anderson and E.I. Jury: Generalized Bezoutian and Sylvester matrices in multivariable linear control. IEEE Trans. Autom. Control, AC-21 (1976), 551–556.Google Scholar
  2. [2]
    S. Barnett and M.J.C. Gover: Some extensions of Hankel and Toeplitz matrices. Lin. and Multilin. Alg., 14 (1983), 45–65.Google Scholar
  3. [3]
    A. Ben-Artzi and T. Shalom: On inversion of Toeplitz and close to Toeplitz matrices. Lin. Alg. and Appl. 75 (1986), 173–192.Google Scholar
  4. [4]
    A. Ben-Artzi and T. Shalom: On inversion of block-Toeplitz matrices. Integral Eq. and Oper. Th., 8 (1985), 751–779.Google Scholar
  5. [5]
    R.R. Bitmead, S.Y. Kung, B.D.O. Anderson, T. Kailath: Greatest common divisors via generalized Sylvester and Bezout matrices. IEEE Trans. Autom. Control, AC-23 (1978), 1043–1047.Google Scholar
  6. [6]
    K.F. Clancey and B.A. Kon: The Bezoutian and the algebraic Riccati equation. Lin. and Multilin. Alg., 15 (1984), 265–278.Google Scholar
  7. [7]
    B. Friedlander, M. Morf, T. Kailath, and L. Ljung: New inversion formulas for matrices classified in terms of their distance from Toeplitz matrices. Lin. Alg. and Appl., 27 (1979), 31–60.Google Scholar
  8. [8]
    P.A. Fuhrmann: On symmetric rational transfer functions. Lin. Alg. and Appl., 50 (1983), 167–250.Google Scholar
  9. [9]
    P.A. Fuhrmann: Polynomial models and algebraic stability criteria. Proc. of the Joint Workshop on Feedback and Synthesis of Linear and Nonlinear Systems, ZIF Bielefeld, June 1981.Google Scholar
  10. [10]
    P.A. Fuhrmann: Block Hankel matrix inversion-the polynomial approach. Integral Eq. and Oper. Th., to appear.Google Scholar
  11. [11]
    I.C. Gohberg and I.A. Feldman: Convolution equations and projections methods for their solutions. Translation of Math. Monographs, Vol. 41, American Mathematical Society, Providence, RI, 1974.Google Scholar
  12. [12]
    I.C. Gohberg and G. Henig: Inversion of finite Toeplitz matrices consisting of elements of a non-commutative algebra. Rev. Roum. Math. Pures et Appl., 19(5) (1974), 623–663 (Russian).Google Scholar
  13. [13]
    I.C. Gohberg and N.Ya. Krupnik: A formula for the inversion of finite Toeplitz matrices. Mat. Issled 7(2) (1972), 272–283 (Russian).Google Scholar
  14. [14]
    I. C. Gohberg and A. A. Semencul: On the inversion of finite Toeplitz matrices and their continuous analogues. Mat. Issled 7(2) (1972), 201–223 (Russian).Google Scholar
  15. [15]
    G. Heinig and K. Rost: Algebraic methods for Toeplitzlike matrices and operators. Operator Theory: Advances and Applications, Vol. 13, Birkhauser, 1984.Google Scholar
  16. [16]
    T. Kailath, S. Kung and M. Morf: Displacement rank of matrices and linear equations. J. of Math. Anal. and Appl., 68 (1979), 395–407.Google Scholar
  17. [17]
    F. I. Lander: The Bezoutian and the inversion of Hankel and Toeplitz matrices. Math. Issled., 9(2) (1974), 69–87 (Russian).Google Scholar
  18. [18]
    L. Lerer, L. Rodman, M. Tismenetsky: Bezoutian and the Schur-Cohn problem for operator polynomials. J. of Math. Anal. and Appl., 103 (1984), 83–102.Google Scholar
  19. [19]
    L. Lerer and M. Tismenetsky: The eigenvalue-separation problem for matrix polynomials. Integral Eq. and Oper. Theory, 5 (1982), 386–445.Google Scholar
  20. [20]
    L. Lerer and M. Tismenetsky: Bezoutian for several matrix polynomials and matrix equations. Technical Report 88. 145, IBM-Israel Scientific Center, Haifa, November, 1984.Google Scholar
  21. [21]
    L. Lerer and M. Tismenetsky: On the location of spectrum of matrix polynomials. Contemporary Mathematics, 47 (1985), 287–297.Google Scholar
  22. [22]
    L. Lerer and M. Tismenetsky: Bezoutian for several matrix polynomials and polynomial Lyapunov-type equation. Lin. Alg. and Appl., (1986) in press.Google Scholar
  23. [23]
    M. Tismenetsky: Bezoutians, Toeplitz and Hankel matrices in the spectral theory of matrix polynomials. Ph. D. thesis, Technion Haifa (1981).Google Scholar

Copyright information

© Birkhäuser Verlag 1986

Authors and Affiliations

  • L. Lerer
    • 1
  • M. Tismenetsky
    • 2
  1. 1.Department of MathematicsTechnion-Israel Institute of TechnologyHaifaIsrael
  2. 2.IBM-Israel Scientific Center Technion CityHaifaIsrael

Personalised recommendations