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Abstract

In the present paper we consider classes of matrices the entries of which are in a given field F. These matrices have a special structure, they are Bezoutians. Historically, Bezoutians were at first introduced in connection with the elimination for the solution of systems of nonlinear algebraic equations and in connection with root localization problems. Only much later their importance for Hankel and Toeplitz matrix inversion became clear.

Mathematics Subject Classification (2000)

Primary 15–01 Secondary 15A09 15A23 65F05 

Keywords

Bezoutian Toeplitz matrix inverses Hankel matrix inverses resultant matrix Toeplitz-plus-Hankel matrix inverses 

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References

  1. [1]
    E. Bezout. Recherches sur le degré des équations résultants de l’évanouissement des inconnues, et sur les moyens qu’il convenient d’employer pour trouver ces equations. Mem. Acad. Roy. Sci. Paris, pages 288–338, 1764.Google Scholar
  2. [2]
    D. Bini and V.Y. Pan. Polynomial and matrix computations. Vol. 1. Progress in Theoretical Computer Science. Birkhäuser Boston Inc., Boston, MA, 1994. Fundamental algorithms.zbMATHGoogle Scholar
  3. [3]
    A. Cayley. Note sur la méthode d’élimination de Bezout. J. Reine Angew. Math., 53:366–367, 1857.zbMATHGoogle Scholar
  4. [4]
    I.Z. Emiris and B. Mourrain. Matrices in elimination theory. J. Symbolic Comput., 28(1–2):3–44, 1999. Polynomial elimination — algorithms and applications.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    L. Euler. Introductio in analysin infinitorum. Tomus primus. Sociedad Andaluza de Educación Matemática “Thales”, Seville, 2000. Reprint of the 1748 original.Google Scholar
  6. [6]
    P.A. Fuhrmann. A polynomial approach to linear algebra. Universitext. Springer-Verlag, New York, 1996.zbMATHGoogle Scholar
  7. [7]
    P.A. Fuhrmann and B.N. Datta. On Bezoutians, Vandermonde matrices, and the Liénard-Chipart stability criterion. In Proceedings of the Fourth Haifa Matrix Theory Conference (Haifa, 1988), volume 120, pages 23–37, 1989.zbMATHMathSciNetGoogle Scholar
  8. [8]
    F.R. Gantmacher. Matrizentheorie. Hochschulbücher für Mathematik [University Books for Mathematics], 86. VEB Deutscher Verlag derWissenschaften, Berlin, 1986. With a foreword by D.P. ≤Zelobenko, Translated from the Russian by Helmut Boseck, Dietmar Soyka and Klaus Stengert.Google Scholar
  9. [9]
    I. Gohberg, M.A. Kaashoek, L. Lerer, and L. Rodman. Common multiples and common divisors of matrix polynomials. II. Vandermonde and resultant matrices. Linear and Multilinear Algebra, 12(3):159–203, 1982/83.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    I. Gohberg, P. Lancaster, and L. Rodman. Matrix polynomials. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982. Computer Science and Applied Mathematics.Google Scholar
  11. [11]
    I. Gohberg and T. Shalom. On Bezoutians of nonsquare matrix polynomials and inversion of matrices with nonsquare blocks. Linear Algebra Appl., 137/138:249–323, 1990.CrossRefMathSciNetGoogle Scholar
  12. [12]
    I.C. Gohberg and G. Heinig. The inversion of finite Toeplitz matrices. Mat. Issled., 8(3(29)):151–156, 183, 1973.zbMATHMathSciNetGoogle Scholar
  13. [13]
    I.C. Gohberg and G. Heinig. Inversion of finite Toeplitz matrices consisting of elements of a noncommutative algebra. Rev. Roumaine Math. Pures Appl., 19:623–663, 1974.zbMATHMathSciNetGoogle Scholar
  14. [14]
    I.C. Gohberg and G. Heinig. The resultant matrix and its generalizations. I. The resultant operator for matrix polynomials. Acta Sci. Math. (Szeged), 37:41–61, 1975.zbMATHMathSciNetGoogle Scholar
  15. [15]
    I.C. Gohberg and G. Heinig. The resultant matrix and its generalizations. II. The continual analogue of the resultant operator. Acta Math. Acad. Sci. Hungar., 28(3-4):189–209, 1976.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    G. Heinig. The notion of Bezoutian and of resultant for operator pencils. Funkcional. Anal. i Prilo≤zen., 11(3):94–95, 1977.zbMATHMathSciNetGoogle Scholar
  17. [17]
    G. Heinig. Über Block-Hankelmatrizen und den Begriff der Resultante für Matrixpolynome. Wiss. Z. Techn. Hochsch. Karl-Marx-Stadt, 19(4):513–519, 1977.zbMATHMathSciNetGoogle Scholar
  18. [18]
    G. Heinig. Verallgemeinerte Resultantenbegriffe bei beliebigen Matrixbüscheln. I. Einseitiger Resultantenoperator. Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt, 20(6): 693–700, 1978.zbMATHMathSciNetGoogle Scholar
  19. [19]
    G. Heinig. Verallgemeinerte Resultantenbegriffe bei beliebigen Matrixbüscheln. II. Gemischter Resultantenoperator. Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt, 20(6):701–703, 1978.zbMATHMathSciNetGoogle Scholar
  20. [20]
    G. Heinig. Bezoutiante, Resultante und Spektralverteilungsprobleme für Operatorpolynome. Math. Nachr., 91:23–43, 1979.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    G. Heinig. Generalized resultant operators and classification of linear operator pencils up to strong equivalence. In Functions, series, operators, Vol. I, II (Budapest, 1980), volume 35 of Colloq. Math. Soc. János Bolyai, pages 611–620. North-Holland, Amsterdam, 1983.Google Scholar
  22. [22]
    G. Heinig. Chebyshev-Hankel matrices and the splitting approach for centrosymmetric Toeplitz-plus-Hankel matrices. Linear Algebra Appl., 327(1-3):181–196, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    G. Heinig and F. Hellinger. On the Bezoutian structure of the Moore-Penrose inverses of Hankel matrices. SIAM J. Matrix Anal. Appl., 14(3):629–645, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    G. Heinig and F. Hellinger. Displacement structure of generalized inverse matrices. Linear Algebra Appl., 211:67–83, 1994. Generalized inverses (1993).zbMATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    G. Heinig and F. Hellinger. Moore-Penrose inversion of square Toeplitz matrices. SIAM J. Matrix Anal. Appl., 15(2):418–450, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    G. Heinig and U. Jungnickel. Zur Lösung von Matrixgleichungen der Form AX — XB = C. Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt, 23(4):387–393, 1981.zbMATHMathSciNetGoogle Scholar
  27. [27]
    G. Heinig and U. Jungnickel. On the Routh-Hurwitz and Schur-Cohn problems for matrix polynomials and generalized Bezoutians. Math. Nachr., 116:185–196, 1984.zbMATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    G. Heinig and U. Jungnickel. On the Bezoutian and root localization for polynomials. Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt, 27(1):62–65, 1985.zbMATHMathSciNetGoogle Scholar
  29. [29]
    G. Heinig and U. Jungnickel. Hankel matrices generated by Markov parameters, Hankel matrix extension, partial realization, and Padé-approximation. In Operator theory and systems (Amsterdam, 1985), volume 19 of Oper. Theory Adv. Appl., pages 231–253. Birkhäuser, Basel, 1986.Google Scholar
  30. [30]
    G. Heinig and U. Jungnickel. Hankel matrices generated by the Markov parameters of rational functions. Linear Algebra Appl., 76:121–135, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    G. Heinig and U. Jungnickel. Lyapunov equations for companion matrices. Linear Algebra Appl., 76:137–147, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  32. [32]
    G. Heinig and K. Rost. Algebraic methods for Toeplitz-like matrices and operators, volume 19 of Mathematical Research. Akademie-Verlag, Berlin, 1984.zbMATHGoogle Scholar
  33. [33]
    G. Heinig and K. Rost. Algebraic methods for Toeplitz-like matrices and operators, volume 13 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 1984.zbMATHGoogle Scholar
  34. [34]
    G. Heinig and K. Rost. On the inverses of Toeplitz-plus-Hankel matrices. Linear Algebra Appl., 106:39–52, 1988.zbMATHCrossRefMathSciNetGoogle Scholar
  35. [35]
    G. Heinig and K. Rost. Inversion of matrices with displacement structure. Integral Equations Operator Theory, 12(6):813–834, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  36. [36]
    G. Heinig and K. Rost. Matrices with displacement structure, generalized Bezoutians, and Moebius transformations. In The Gohberg anniversary collection, Vol. I (Calgary, AB, 1988), volume 40 of Oper. Theory Adv. Appl., pages 203–230. Birkhäuser, Basel, 1989.Google Scholar
  37. [37]
    G. Heinig and K. Rost. Matrix representations of Toeplitz-plus-Hankel matrix inverses. Linear Algebra Appl., 113:65–78, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  38. [38]
    G. Heinig and K. Rost. DFT representations of Toeplitz-plus-Hankel Bezoutians with application to fast matrix-vector multiplication. Linear Algebra Appl., 284(1-3):157–175, 1998. ILAS Symposium on Fast Algorithms for Control, Signals and Image Processing (Winnipeg, MB, 1997).zbMATHCrossRefMathSciNetGoogle Scholar
  39. [39]
    G. Heinig and K. Rost. Representations of Toeplitz-plus-Hankel matrices using trigonometric transformations with application to fast matrix-vector multiplication. In Proceedings of the Sixth Conference of the International Linear Algebra Society (Chemnitz, 1996), volume 275/276, pages 225–248, 1998.MathSciNetGoogle Scholar
  40. [40]
    G. Heinig and K. Rost. Hartley transform representations of inverses of real Toeplitzplus-Hankel matrices. In Proceedings of the International Conference on Fourier Analysis and Applications (Kuwait, 1998), volume 21, pages 175–189, 2000.zbMATHMathSciNetGoogle Scholar
  41. [41]
    G. Heinig and K. Rost. Hartley transform representations of symmetric Toeplitz matrix inverses with application to fast matrix-vector multiplication. SIAM J. Matrix Anal. Appl., 22(1):86–105 (electronic), 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  42. [42]
    G. Heinig and K. Rost. Representations of inverses of real Toeplitz-plus-Hankel matrices using trigonometric transformations. In Large-scale scientific computations of engineering and environmental problems, II (Sozopol, 1999), volume 73 of Notes Numer. Fluid Mech., pages 80–86. Vieweg, Braunschweig, 2000.Google Scholar
  43. [43]
    G. Heinig and K. Rost. Efficient inversion formulas for Toeplitz-plus-Hankel matrices using trigonometric transformations. In Structured matrices in mathematics, computer science, and engineering, II (Boulder, CO, 1999), volume 281 of Contemp. Math., pages 247–264. Amer. Math. Soc., Providence, RI, 2001.Google Scholar
  44. [44]
    G. Heinig and K. Rost. Centro-symmetric and centro-skewsymmetric Toeplitz matrices and Bezoutians. Linear Algebra Appl., 343/344:195–209, 2002. Special issue on structured and infinite systems of linear equations.CrossRefMathSciNetGoogle Scholar
  45. [45]
    G. Heinig and K. Rost. Centrosymmetric and centro-skewsymmetric Toeplitz-plus-Hankel matrices and Bezoutians. Linear Algebra Appl., 366:257–281, 2003. Special issue on structured matrices: analysis, algorithms and applications (Cortona, 2000).zbMATHCrossRefMathSciNetGoogle Scholar
  46. [46]
    G. Heinig and K. Rost. Fast algorithms for centro-symmetric and centro-skewsymmetric Toeplitz-plus-Hankel matrices. Numer. Algorithms, 33(1-4):305–317, 2003. International Conference on Numerical Algorithms, Vol. I (Marrakesh, 2001).zbMATHCrossRefMathSciNetGoogle Scholar
  47. [47]
    G. Heinig and K. Rost. Split algorithms for Hermitian Toeplitz matrices with arbitrary rank profile. Linear Algebra Appl., 392:235–253, 2004.zbMATHCrossRefMathSciNetGoogle Scholar
  48. [48]
    G. Heinig and K. Rost. Split algorithms for centrosymmetric Toeplitz-plus-Hankel matrices with arbitrary rank profile. In The extended field of operator theory, volume 171 of Oper. Theory Adv. Appl., pages 129–146. Birkhäuser, Basel, 2007.CrossRefGoogle Scholar
  49. [49]
    C. Hermite. Extrait d’une lettre de Mr. Ch. Hermite de Paris `a Mr. Borchard de Berlin, sur le nombre des racines d’une équation algébrique comprises entre des limites données. J. Reine Angew. Math., 52:39–51, 1856.zbMATHGoogle Scholar
  50. [50]
    A.S. Householder. Bezoutiants, elimination and localization. SIAM Rev., 12:73–78, 1970.zbMATHCrossRefMathSciNetGoogle Scholar
  51. [51]
    C. Jacobi. eliminatione variablis e duabus aequatione algebraicis. J. Reine Angew. Math., 15:101–124, 1836.zbMATHGoogle Scholar
  52. [52]
    M.G. Krein and M.A. Naimark. The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations. Linear and Multilinear Algebra, 10(4):265–308, 1981. Translated from the Russian by O. Boshko and J.L. Howland.zbMATHCrossRefMathSciNetGoogle Scholar
  53. [53]
    B. Krishna and H. Krishna. Computationally efficient reduced polynomial based algorithms for Hermitian Toeplitz matrices. SIAM J. Appl. Math., 49(4):1275–1282, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  54. [54]
    P. Lancaster and M. Tismenetsky. The theory of matrices. Computer Science and Applied Mathematics. Academic Press Inc., Orlando, FL, second edition, 1985.Google Scholar
  55. [55]
    F.I. Lander. The Bezoutian and the inversion of Hankel and Toeplitz matrices (in Russian). Mat. Issled., 9(2 (32)):69–87, 249–250, 1974.zbMATHMathSciNetGoogle Scholar
  56. [56]
    A. Lascoux. Symmetric functions and combinatorial operators on polynomials, volume 99 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2003.Google Scholar
  57. [57]
    L. Lerer and L. Rodman. Bezoutians of rational matrix functions. J. Funct. Anal., 141(1):1–36, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  58. [58]
    L. Lerer and M. Tismenetsky. The Bezoutian and the eigenvalue-separation problem for matrix polynomials. Integral Equations Operator Theory, 5(3):386–445, 1982.zbMATHCrossRefMathSciNetGoogle Scholar
  59. [59]
    B. Mourrain and V.Y. Pan. Multivariate polynomials, duality, and structured matrices. J. Complexity, 16(1):110–180, 2000. Real computation and complexity (Schloss Dagstuhl, 1998).zbMATHCrossRefMathSciNetGoogle Scholar
  60. [60]
    A. Olshevsky and V. Olshevsky. Kharitonov’s theorem and Bezoutians. Linear Algebra Appl., 399:285–297, 2005.zbMATHCrossRefMathSciNetGoogle Scholar
  61. [61]
    V. Olshevsky and L. Sakhnovich. An operator identities approach to bezoutians. A general scheme and examples. In Proc.of the MTNS’ 04 Conference. 2004.Google Scholar
  62. [62]
    V.Y. Pan. Structured matrices and polynomials. Birkhäuser Boston Inc., Boston, MA, 2001. Unified superfast algorithms.zbMATHGoogle Scholar
  63. [63]
    M.M. Postnikov. Ustoichivye mnogochleny (Stable polynomials). “Nauka”, Moscow, 1981.Google Scholar
  64. [64]
    K. Rost. Toeplitz-plus-Hankel Bezoutians and inverses of Toeplitz and Toeplitz-plus-Hankel matrices. Oper. Matrices, 2(3):385–406, 2008.zbMATHMathSciNetGoogle Scholar
  65. [65]
    I. Sylvester. On a theory of syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm’s functions, and that of the greatest algebraical common measure. Phylos. Trans. Roy. Soc. London, 143:407–548, 1853.CrossRefGoogle Scholar
  66. [66]
    H.K. Wimmer. On the history of the Bezoutian and the resultant matrix. Linear Algebra Appl., 128:27–34, 1990.zbMATHCrossRefMathSciNetGoogle Scholar

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© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  • Karla Rost
    • 1
  1. 1.Dept. of MathematicsChemnitz University of TechnologyChemnitzGermany

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