The Resultant Matrix and its Generalizations. I. The Resultant Operator for Matrix Polynomials

  • Israel Gohberg
  • Georg Heinig
Part of the Operator Theory: Advances and Applications book series (OT, volume 206)


Let a(λ)=a 0+a 1λ+ ... +a n λ n and b(λ)=b 0+b 1λ+ ... +b m λ m be two polynomials with coefficients in ℂ1. The determinant of the following matrix is said to be the resultant of these polynomials.


English Translation Matrix Polynomial Toeplitz Matrice Common Root Canonical System 
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  1. [1]
    A.G. Kurosh, A Course in Higher Algebra, 9th edition, Nauka, Moscow, 1968 (in Russian). English translation from the 10th Russian edition: Higher Algebra. Mir Publishers, Moscow, 1975. MR0384363 (52 #5240).Google Scholar
  2. [2]
    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. Khar’kov, 1936 (in Russian). English translation: Linear and Multilinear Algebra 10 (1981), no. 4, 265–308. MR0638124 (84i:12016), Zbl 0584.12018.Google Scholar
  3. [3]
    F.I. Lander, The Bezoutian and the inversion of Hankel and Toeplitz matrices. Matem. Issled. 9 (1974), no. 2(32), 69–87 (in Russian). MR0437559 (55 #10483), Zbl 0331.15017.MATHMathSciNetGoogle Scholar
  4. [4]
    I.C. Gohberg and A.A. Semencul, The inversion of finite Toeplitz matrices and their continual analogues. Matem. Issled. 7 (1972), no. 2(24), 201–223 (in Russian). MR0353038 (50 #5524), Zbl 0288.15004.MATHMathSciNetGoogle Scholar
  5. [5]
    I.C. Gohberg and G. Heinig, Inversion of finite Toeplitz matrices consisting of elements of a noncommutative algebra. Rev. Roumaine Math. Pures Appl. 19 (1974), 623–663 (in Russian). English translation: this volume. MR0353040 (50 #5526), Zbl 0337.15005.MATHMathSciNetGoogle Scholar
  6. [6]
    I.C. Gohberg and G. Heinig, Matrix integral operators on a finite interval with kernels depending on the difference of the arguments. Rev. Roumaine Math. Pures Appl. 20 (1975), 55–73 (in Russian). English translation: this volume. MR0380495 (52 #1395), Zbl 0327.45009.MathSciNetGoogle Scholar
  7. [7]
    M.G. Krein, Distribution of roots of polynomials orthogonal on the unit circle with respect to a sign-alternating weight. Teor. Funkts., Funkts. Anal. Prilozh. (Khar’kov) 2 (1966), 131–137 (in Russian). MR0201702 (34 #1584), Zbl 0257.30002.MATHMathSciNetGoogle Scholar
  8. [8]
    I.C. Gohberg and M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert space. Nauka, Moscow, 1965 (in Russian). MR0220070 (36 #3137), Zbl 0138.07803. English translation: Introduction to the Theory of Linear Nonselfadjoint Operators. Amer. Math. Soc., Providence, R.I. 1969. MR0246142 (39 #7447), Zbl 0181.13504. French translation: Introduction à la Théorie des Opérateurs Linéaires non Auto-Adjoints Dans un Espace Hilbertien. Dunod, Paris, 1971. MR0350445 (50 #2937).Google Scholar
  9. [9]
    I.C. Gohberg and E.I. Sigal, An operator generalization of the logarithmic residue theorem and the theorem of Rouché. Matem. Sbornik, New Ser. 84(126) (1971), 607–629 (in Russian). English translation: Math. USSR Sbornik 13 (1971), 603–625. MR0313856 (47 #2409), Zbl 0254.47046.MathSciNetGoogle Scholar

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  • Israel Gohberg
  • Georg Heinig

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