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Jordan Canonical Form with Parameters from Frobenius Form with Parameters

  • Robert M. Corless
  • Marc Moreno Maza
  • Steven E. Thornton
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10693)

Abstract

The Jordan canonical form (JCF) of a square matrix is a foundational tool in matrix analysis. If the matrix A is known exactly, symbolic computation of the JCF is possible though expensive. When the matrix contains parameters, exact computation requires either a potentially very expensive case discussion, significant expression swell or both. For this reason, no current computer algebra system (CAS) of which we are aware will compute a case discussion for the JCF of a matrix \(A(\alpha )\) where \(\alpha \) is a (vector of) parameter(s). This problem is extremely difficult in general, even though the JCF is encountered early in most curricula.

In this paper we make some progress towards a practical solution. We base our computation of the JCF of \(A(\alpha )\) on the theory of regular chains and present an implementation built on the RegularChains library of the Maple CAS. Our algorithm takes as input a matrix in Frobenius (rational) canonical form where the entries are (multivariate) polynomials in the parameter(s). We do not solve the problem in full generality, but our approach is useful for solving some examples of interest.

Keywords

Jordan form Rational canonical form Parametric linear algebra Regular chains Triangular decomposition 

References

  1. 1.
    Hogben, L.: Handbook of Linear Algebra. CRC Press, Boca Raton (2016)zbMATHGoogle Scholar
  2. 2.
    O’Meara, K., Clark, J., Vinsonhaler, C.I.: Advanced Topics in Linear Algebra: Weaving Matrix Problems Through the Weyr Form. Oxford University Press, New York (2011)zbMATHGoogle Scholar
  3. 3.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. CUP (2012)Google Scholar
  4. 4.
    Giesbrecht, M.: Nearly optimal algorithms for canonical matrix forms. SIAM J. Comput. 24(5), 948–969 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kaltofen, E., Krishnamoorthy, M., Saunders, B.D.: Fast parallel algorithms for similarity of matrices. In: Proceedings of the Fifth ACM Symposium on Symbolic and Algebraic Computation, pp. 65–70. ACM (1986)Google Scholar
  6. 6.
    Higham, N.J.: Functions of Matrices: Theory and Computation. SIAM, Philadelphia (2008)CrossRefzbMATHGoogle Scholar
  7. 7.
    Moler, C., Van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45(1), 3–49 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Frank, J., Huang, W., Leimkuhler, B.: Geometric integrators for classical spin systems. J. Comput. Phys. 133(1), 160–172 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, vol. 31. Springer, Heidelberg (2006).  https://doi.org/10.1007/3-540-30666-8 zbMATHGoogle Scholar
  10. 10.
    Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations Analysis and Numerical Solution. EMS Publishing House, Zürich (2006)CrossRefzbMATHGoogle Scholar
  11. 11.
    Corless, R.M., Jeffrey, D.J.: Well...it isn’t quite that simple. ACM SIGSAM Bull. 26(3), 2–6 (1992)CrossRefGoogle Scholar
  12. 12.
    Van Gils, S., Krupa, M., Langford, W.F.: Hopf bifurcation with non-semisimple 1:1 resonance. Nonlinearity 3(3), 825 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Beelen, T., Van Dooren, P.: Computational Aspects of the Jordan Canonical Form, pp. 57–72. Oxford Science Publication, Oxford University Press, New York (1990)zbMATHGoogle Scholar
  14. 14.
    Gil, I.: Computation of the Jordan canonical form of a square matrix (using the Axiom programming language), pp. 138–145. ACM (1992)Google Scholar
  15. 15.
    Ozello, P.: Calcul exact des formes de Jordan et de Frobenius d’une matrice. Ph.D. thesis, Université Joseph-Fourier-Grenoble I (1987)Google Scholar
  16. 16.
    Arnol’d, V.I.: On matrices depending on parameters. Russian Math. Surv. 26(2), 29–43 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Chen, G., Della Dora, J.: Rational normal form for dynamical systems by Carleman linearization. In: Proceedings of ISSAC 1999, pp. 165–172. ACM (1999)Google Scholar
  18. 18.
    Storjohann, A.: An \(\cal{O}(n^3)\) algorithm for the Frobenius normal form. In: Proceedings of ISSAC 1998, pp. 101–105. ACM (1998)Google Scholar
  19. 19.
    Chen, G.: Computing the normal forms of matrices depending on parameters. In: Proceedings of ISSAC 1989, pp. 242–249. ACM (1989)Google Scholar
  20. 20.
    Storjohann, A.: Algorithms for matrix canonical forms. Ph.D. thesis, Swiss Federal Institute of Technology Zurich (2013)Google Scholar
  21. 21.
    Ballarin, C., Kauers, M.: Solving parametric linear systems. ACM SIGSAM Bull. 38(2), 33–46 (2004)CrossRefzbMATHGoogle Scholar
  22. 22.
    Broadbery, P.A., Gómez-Díaz, T., Watt, S.M.: On the implementation of dynamic evaluation. In: Proceedings of ISSAC 1995, pp. 77–84. ACM (1995)Google Scholar
  23. 23.
    Diaz-Toca, G.M., Gonzalez-Vega, L., Lombardi, H.: Generalizing Cramer’s rule. SIAM J. Matrix Anal. Appl. 27(3), 621–637 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kapur, D.: An approach for solving systems of parametric polynomial equations. In: Principles and Practices of Constraint Programming, pp. 217–244 (1995)Google Scholar
  25. 25.
    Sit, W.Y.: An algorithm for solving parametric linear systems. J. Symb. Comput. 13(4), 353–394 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Corless, R.M., Thornton, S.E.: A package for parametric matrix computations. In: Mathematical Software - ICMS 2014–4th International Congress, Seoul, South Korea, 5–9 August 2014, Proceedings, pp. 442–449 (2014)Google Scholar
  27. 27.
    Yang, L., Hou, X., Xia, B.: A complete algorithm for automated discovering of a class of inequality-type theorems. Sci. China Ser. F Inf. Sci. 44(1), 33–49 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lazard, D., Rouillier, F.: Solving parametric polynomial systems. J. Symb. Comput. 42(6), 636–667 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Moreno Maza, M., Xia, B., Xiao, R.: On solving parametric polynomial systems. Math. Comput. Sci. 6(4), 457–473 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Chen, C., Moreno Maza, M.: Algorithms for computing triangular decomposition of polynomial systems. J. Symb. Comput. 47(6), 610–642 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Landau, S.: Factoring polynomials over algebraic number fields. SIAM J. Comput. 14(1), 184–195 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Trager, B.M.: Algebraic factoring and rational function integration. In: Jenks, R.D. (ed.) SYMSAC 1976, Proceedings of the Third ACM Symposium on Symbolic and Algebraic Manipulation, Yorktown Heights, New York, USA, 10–12 August 1976. ACM, pp. 219–226 (1976)Google Scholar
  33. 33.
    Gantmacher, F.R.: The Theory of Matrices, vol. 1. Chelsea Publishing Company, New York (1960)zbMATHGoogle Scholar
  34. 34.
    Gohberg, I., Lancaster, P., Rodman, L.: Matrix Polynomials. SIAM, Philadelphia (2009)CrossRefzbMATHGoogle Scholar
  35. 35.
    Trench, W.F.: Properties of some generalizations of KAC-Murdock-Szegö matrices. In: Structured Matrices in Mathematics, Computer Science II Control, Signal and Image Processing (AMS Contemporary Mathematics Series), vol. 281 (2001)Google Scholar
  36. 36.
    Gray, C.R.: An analysis of the Belousov-Zhabotinskii reaction. Rose-Hulman Undergraduate Math. J. 3(1) (2002)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Robert M. Corless
    • 1
  • Marc Moreno Maza
    • 1
  • Steven E. Thornton
    • 1
  1. 1.ORCCAUniversity of Western OntarioLondonCanada

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