Structural Stability of Matrix Pencils and Matrix Pairs Under Contragredient Equivalence

  • M. I. García-Planas
  • T. KlymchukEmail author

A complex matrix pencil A−B is called structurally stable if there exists its neighborhood in which all pencils are strictly equivalent to this pencil. We describe all complex matrix pencils that are structurally stable. It is shown that there are no pairs (M,N) of m × n and n × m complex matrices (m, n ≥ 1) that are structurally stable under the contragredient equivalence (S1MR,R1NS) in which S and R are nondegenerate.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. A. Andronov and L. S. Pontryagin, “Structurally stable systems,” Dol. Akad. Nauk SSSR, 14, No. 5, 247–250 (1937).Google Scholar
  2. 2.
    A. Dmytryshyn and F. M. Dopico, “Generic skew-symmetric matrix polynomials with fixed rank and fixed odd grade,” Linear Algebra Appl., 536 1–18 ((2018)).MathSciNetCrossRefGoogle Scholar
  3. 3.
    N. M. Dobrovol’skaya and V. A. Ponomarev, “A pair of counter-operators,” Usp. Mat. Nauk, 20, No. 6, 81–86 (1965).MathSciNetGoogle Scholar
  4. 4.
    A. Edelman, E. Elmroth, and B. Kågström, “A geometric approach to perturbation theory of matrices and matrix pencils. Part II: A stratification-enhanced staircase algorithm,” SIAM J. Matrix Anal. Appl., 20, 667–699 (1999).MathSciNetCrossRefGoogle Scholar
  5. 5.
    P. Gabriel and A.V. Roiter, Representations of Finite-Dimensional Algebras, Encyclopedia of Mathematical Sciences, Vol. 73, Springer, Berlin (1999).Google Scholar
  6. 6.
    M. I. García-Planas and M. D. Magret, “A generalized Sylvester equation: a criterion for structural stability of triples of matrices,” Linear Multilinear Algebra, 44, No. 2, 93–109 (1998).Google Scholar
  7. 7.
    M. I. García-Planas, M. D. Magret, V. V. Sergeichuk, and N. A. Zharko, “Rigid systems of second-order linear differential equations,” Linear Algebra Appl., 414, 517–532 (2006).MathSciNetCrossRefGoogle Scholar
  8. 8.
    M. I. García-Planas and V. V. Sergeichuk, “Simplest miniversal deformations of matrices, matrix pencils, and contragredient matrix pencils,” Linear Algebra Appl., 302/303, 45–61 (1999).Google Scholar
  9. 9.
    M. I. García-Planas and V. V. Sergeichuk, “Generic families of matrix pencils and their bifurcation diagrams,” Linear Algebra Appl., 332/334, 165–179 (2001).Google Scholar
  10. 10.
    R. A. Horn and D. I. Merino, “Contragredient equivalence: A canonical form and some applications,” Linear Algebra Appl., 214, 43–92 (1995).MathSciNetCrossRefGoogle Scholar
  11. 11.
    L. Klimenko and V. V. Sergeichuk, “Block triangular miniversal deformations of matrices and matrix pencils,” in: Matrix Methods: Theory, Algorithms and Applications, World Scientific, Hackensack (2010), pp. 69–84.CrossRefGoogle Scholar
  12. 12.
    S. López de Medrano, “Topological aspects of matrix problems,” in: Representations of Algebras (Puebla, 1980) (1981), pp. 196–210.Google Scholar
  13. 13.
    A. Pokrzywa, “On perturbations and the equivalence orbit of a matrix pencil,” Linear Algebra Appl., 82, 99–121 (1986).MathSciNetCrossRefGoogle Scholar
  14. 14.
    J. W. Robbin, “Topological conjugacy and structural stability for discrete dynamical systems,” Bull. Amer. Math. Soc., 78, 923–952 (1972).MathSciNetCrossRefGoogle Scholar
  15. 15.
    J. C. Willems, “Topological classification and structural stability of linear systems,” J. Different. Equat., 35, No. 3, 306–318 (1980).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Universitat Politècnica de CatalunyaBarcelonaSpain

Personalised recommendations