On the kernel structure of generalized resultant matrices. (English) Zbl 1259.15005
Resultant matrices are defined as square matrices associated with the coefficients of two polynomials \(\mathbf{u}(t)\), \(\mathbf{v}(t)\). Generalized resultant matrices, which are considered in this paper, have entries from a given field \(\mathbb F\). Resultant matrices are used for instance for the study of common divisors and common multiples of polynomials over \(\mathbb C\). In this paper the structure of the kernel of the generalized resultant matrices of two polynomials is studied. A procedure is proposed how to get a basis in this kernel using just the greatest common divisor of \(\mathbf{u}(t)\) and \(\mathbf{v}(t)\) and solutions of corresponding Bézout equations. The results obtained here will be useful for the computation of inverses of structured matrices and will be discussed in a forthcoming paper of authors.
Reviewer: Václav Burjan (Praha)
MSC:
| 15A09 | Theory of matrix inversion and generalized inverses |
| 12E05 | Polynomials in general fields (irreducibility, etc.) |
| 11A05 | Multiplicative structure; Euclidean algorithm; greatest common divisors |
Keywords:
generalized resultant matrix; Sylvester matrix; Bezout equations; inverse matrix; kernel; polynomials; greatest common divisorReferences:
| [1] | Anderson, B. D.O.; Jury, E. I., Generalized Bezoutian and Sylvester matrices in multivariable linear control, IEEE Trans. Automat. Control, AC-21, 4, 551-556 (1976) · Zbl 0332.93032 |
| [2] | Bini, D.; Pan, V. Y., (Polynomial and Matrix Computations, Vol. 1. Polynomial and Matrix Computations, Vol. 1, Progress in Theoretical Computer Science (1994), Birkhäuser Boston Inc.: Birkhäuser Boston Inc. Boston, MA), Fundamental algorithms · Zbl 0809.65012 |
| [3] | T. Ehrhardt, K. Rost, Resultant matrices and inversion of Bezoutians, Linear Algebra Appl. (submitted for publication).; T. Ehrhardt, K. Rost, Resultant matrices and inversion of Bezoutians, Linear Algebra Appl. (submitted for publication). · Zbl 1283.15015 |
| [4] | L. Euler, Introductio in analysin infinitorum, Tomus primus, Sociedad Andaluza de Educación Matemática Thales, Seville, 2000, Reprint of the 1748 original.; L. Euler, Introductio in analysin infinitorum, Tomus primus, Sociedad Andaluza de Educación Matemática Thales, Seville, 2000, Reprint of the 1748 original. · JFM 48.0007.02 |
| [5] | Finck, T.; Heinig, G.; Rost, K., An inversion formula and fast algorithms for Cauchy-Vandermonde matrices, Linear Algebra Appl., 183, 179-191 (1993) · Zbl 0776.65019 |
| [6] | Gohberg, I.; Heinig, G., The resultant matrix and its generalizations, I, The resultant operator for matrix polynomials [mr0380471], (Convolution Equations and Singular Integral Operators. Convolution Equations and Singular Integral Operators, Oper. Theory Adv. Appl., vol. 206 (2010), Birkhäuser Verlag: Birkhäuser Verlag Basel), 65-88 · Zbl 1207.15021 |
| [7] | Gohberg, I.; Heinig, G., The resultant matrix and its generalizations. II, the continual analogue of the resultant operator [mr0425652], (Convolution Equations and Singular Integral Operators. Convolution Equations and Singular Integral Operators, Oper. Theory Adv. Appl., vol. 206 (2010), Birkhäuser Verlag: Birkhäuser Verlag Basel), 89-109 · Zbl 1207.15022 |
| [8] | Gohberg, I.; Kaashoek, M. A.; Lerer, L., The resultant for regular matrix polynomials and quasi commutativity, Indiana Univ. Math. J., 57, 6, 2793-2813 (2008) · Zbl 1178.15014 |
| [9] | Gohberg, I.; Kaashoek, M. A.; Lerer, L.; Rodman, L., Common multiples and common divisors of matrix polynomials, II, vandermonde and resultant matrices, Linear Multilinear Algebra, 12, 3, 159-203 (1982/83) · Zbl 0496.15014 |
| [10] | Gohberg, I. C.; Lerer, L. E., Resultants of matrix polynomials, Bull. Amer. Math. Soc., 82, 4, 565-567 (1976) · Zbl 0343.15011 |
| [11] | M. Halwass, A. Böttcher, Wiener-Hopf and spectral factorization of real polynomials by Newton’s method (in preparation).; M. Halwass, A. Böttcher, Wiener-Hopf and spectral factorization of real polynomials by Newton’s method (in preparation). · Zbl 1281.65075 |
| [12] | Heinig, G., The notion of Bezoutian and of resultant for operator pencils, Funkcional. Anal. i Priložen., 11, 3, 94-95 (1977) · Zbl 0383.47016 |
| [13] | Heinig, G., Verallgemeinerte Resultantenbegriffe bei beliebigen Matrixbüscheln, II, Gemischter Resultantenoperator, Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt, 20, 6, 701-703 (1978) · Zbl 0453.15007 |
| [14] | Heinig, G., Verallgemeinerte Resultantenbegriffe bei beliebigen Matrixbüscheln, I, Einseitiger Resultantenoperator, Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt, 20, 6, 693-700 (1978) · Zbl 0453.15006 |
| [15] | Heinig, G.; Rost, K., Algebraic Methods for Toeplitz-Like Matrices and Operators, Operator Theory: Advances and Applications, vol. 13 (1984), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 0549.15013 |
| [16] | Kaashoek, M. A.; Lerer, L., Quasi commutativity of regular matrix polynomials: Resultant and Bezoutian, (Topics in Operator Theory, Volume 1, Operators, Matrices and Analytic Functions. Topics in Operator Theory, Volume 1, Operators, Matrices and Analytic Functions, Oper. Theory Adv. Appl., vol. 202 (2010), Birkhäuser Verlag: Birkhäuser Verlag Basel), 297-314 · Zbl 1207.47011 |
| [17] | Krein, M. G.; Naimark, M. A., The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations, Linear Multilinear Algebra, 10, 4, 265-308 (1981), Translated from the Russian by O. Boshko and J. L. Howland · Zbl 0584.12018 |
| [18] | P. Lancaster, Common eigenvalues, divisors, and multiples of matrix polynomials: a review, in: Proceedings of the symposium on operator theory (Athens, 1985), vol. 84, pp. 139-160, 1986.; P. Lancaster, Common eigenvalues, divisors, and multiples of matrix polynomials: a review, in: Proceedings of the symposium on operator theory (Athens, 1985), vol. 84, pp. 139-160, 1986. · Zbl 0627.15004 |
| [19] | Pan, V. Y., Structured Matrices and Polynomials (2001), Birkhäuser Boston Inc.: Birkhäuser Boston Inc. Boston, MA, Unified superfast algorithms · Zbl 0996.65028 |
| [20] | Wimmer, H. K., On the history of the Bezoutian and the resultant matrix, Linear Algebra Appl., 128, 27-34 (1990) · Zbl 0711.15028 |
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