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2005 | 15 | 3 | 339-349
Tytuł artykułu

On the computation of the minimal polynomial of a polynomial matrix

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The main contribution of this work is to provide two algorithms for the computation of the minimal polynomial of univariate polynomial matrices. The first algorithm is based on the solution of linear matrix equations while the second one employs DFT techniques. The whole theory is illustrated with examples.
Rocznik
Tom
15
Numer
3
Strony
339-349
Opis fizyczny
Daty
wydano
2005
otrzymano
2005-03-18
poprawiono
2005-05-19
Twórcy
  • Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki 54006, Greece
  • Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki 54006, Greece
Bibliografia
  • Atiyah M.F. and McDonald I.G. (1964): Introduction to Commutative Algebra. - Reading, MA: Addison-Wesley.
  • Augot D. and Camion P. (1997): On the computation of minimal polynomials, cyclic vectors, and Frobenius forms. - Lin. Alg. App., Vol. 260, pp. 61-94.
  • Ciftcibaci T. and Yuksel O. (1982): On the Cayley-Hamllton theorem for two-dimensional systems. - IEEE Trans. Automat.Contr., Vol. 27, No. 1, pp. 193-194.
  • Coppersmith D. and Winograd S. (1990): Matrix multiplication via arithmetic progressions. - J. Symb. Comput., Vol. 9, No.3, pp. 251-280.
  • Faddeev D.K. and Faddeeva V.N. (1963): Computational Methods of Linear Algebra. - San Francisco: Freeman.
  • Fragulis G.F., Mertzios B. and Vardulakis A.I.G. (1991): Computation of the inverse of a polynomial matrix and evaluation of its Laurent expansion. - Int. J. Contr., Vol. 53, No. 2, pp. 431-443.
  • Fragulis G.F. (1995): Generalized Cayley-Hamilton theoremfor polynomial matrices with arbitrary degree. - Int. J. Contr., Vol. 62,No. 6, pp. 1341-1349.
  • Gałkowski K. (1996): Matrix description ofpolynomials multivariable. - Lin. Alg. Applic., Vol. 234,pp. 209-226.
  • Gantmacher F.R. (1959): The Theory of Matrices. - New York: Chelsea.
  • Givone D.D. and Roesser R.P. (1973): Minimization of multidimensional linear iterative circuits. - IEEE Trans. Comput., Vol. C-22.pp. 673-678.
  • Helmberg G., Wagner P. and Veltkamp G. (1993): On Faddeev-Leverrier's method for the computation of the characteristic polynomial of a matrix and of eigenvectors. - Lin. Alg. Applic., Vol. 185, pp. 219-223.
  • Kaczorek T. (1989): Existence and uniqueness of solutions and Cayley-Hamilton theorem for a general singular model of 2-D systems. - Bull. Polish Acad. Sci. Techn. Sci., Vol. 37, No. 5-6, pp. 275-284.
  • Kaczorek T. (1995a): An extension of the Cayley-Hamilton theorem for 2-D continuous-discrete linear systems. - Appl. Math. Comput. Sci., Vol. 4, No. 4, pp. 507-515.
  • Kaczorek T. (1995b): Generalization of the Cayley-Hamilton theorem for nonsquare matrices. - Proc. Int. Conf. Fundamentals of Electrotechnics and Circuit Theory XVIII-SPETO, Gliwice-Ustron, Poland, pp. 77-83.
  • Kaczorek T. (1995c): An extension of the Cayley-Hamilton theorem for nonsquare block matrices and computation of the left and right inverses of matrices. - Bull. Polish Acad. Sci. Techn. Sci., Vol. 43, No. 1,pp. 49-56.
  • Kaczorek T. (1995d): An extension of the Cayley-Hamilton theorem for singular 2D linear systems with non-square matrices. - Bull. Polish Acad. Sci., Techn. Sci., Vol. 43, No. 1, pp. 39-48.
  • Kaczorek T. (1998): An extension of the Cayley-Hamiltontheorem for a standard pair of block matrices. - Appl. Math. Comput. Sci.,Vol. 8, No. 3, pp. 511-516.
  • Kaczorek T. (2005): Generalization of Cayley-Hamilton theorem for n-D polynomial matrices. - IEEE Trans. Automat. Contr., Vol. 50,No. 5, pp.671-674.
  • Kitamoto T. (1999): Efficient computation of the characteristic polynomial of a polynomial matrix. - IEICE Trans. Fundament., Vol. E83-A, No.5, pp. 842-848.
  • Lewis F.L. (1986): Further remarks on the Cayley-Hamilton theorem and Leverrier's method for the matrix pencil (sE-A). - IEEE Trans. Automat. Contr., Vol. 31, No. 9, pp. 869-870.
  • Mertzios B. and Christodoulou M. (1986): On the generalized Cayley-Hamilton theorem. - IEEE Trans. Automat. Contr.,Vol. 31, No. 2, pp. 156-157.
  • Paccagnella L. E. and Pierobon G. L. (1976): FFT calculation of a determinantal polynomial. - IEEE Trans. Automat. Contr.,Vol. 21, No. 3, pp. 401-402.
  • Schuster A. and Hippe P. (1992): Inversion of polynomial matrices by interpolation. - IEEE Trans. Automat. Contr., Vol. 37, No. 3, pp. 363-365.
  • Theodorou N.J. (1989): M-dimensional Cayley-Hamilton theorem. - IEEE Trans. Automat. Contr., Vol. 34, No. 5, pp. 563-565.
  • Tzekis P. and Karampetakis N. (2005): On the computation of the minimal polynomial of a two-variable polynomial matrix. - Proc. 4th Int. Workshop Multidimensional (nD) Systems, Wuppertal, Germany.
  • Victoria J. (1982): A block Cayley-Hamilton theorem. - Bull. Math. Soc. Sci. Math. Roum., Vol. 26, No. 1, pp. 93-97.
  • Vilfan B. (1973): Another proof of the two-dimensional Cayley-Hamilton theorem. - IEEE Trans. Comput., Vol. C-22, pp. 1140.
  • Yu B. and Kitamoto T. (2000): The CHACM method for computing the characteristic polynomial of a polynomial matrix. - IEICE Trans. Fundament., E83-A, No.7, pp. 1405-1410.
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Bibliografia
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