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BIT Numerical Mathematics

, Volume 54, Issue 1, pp 219–255 | Cite as

Eigenvalue perturbation theory of symplectic, orthogonal, and unitary matrices under generic structured rank one perturbations

  • Christian MehlEmail author
  • Volker Mehrmann
  • André C. M. Ran
  • Leiba Rodman
Article

Abstract

We study the perturbation theory of structured matrices under structured rank one perturbations, with emphasis on matrices that are unitary, orthogonal, or symplectic with respect to an indefinite inner product. The rank one perturbations are not necessarily of arbitrary small size (in the sense of norm). In the case of sesquilinear forms, results on selfadjoint matrices can be applied to unitary matrices by using the Cayley transformation, but in the case of real or complex symmetric or skew-symmetric bilinear forms additional considerations are necessary. For complex symplectic matrices, it turns out that generically (with respect to the perturbations) the behavior of the Jordan form of the perturbed matrix follows the pattern established earlier for unstructured matrices and their unstructured perturbations, provided the specific properties of the Jordan form of complex symplectic matrices are accounted for. For instance, the number of Jordan blocks of fixed odd size corresponding to the eigenvalue 1 or −1 have to be even. For complex orthogonal matrices, it is shown that the behavior of the Jordan structures corresponding to the original eigenvalues that are not moved by perturbations follows again the pattern established earlier for unstructured matrices, taking into account the specifics of Jordan forms of complex orthogonal matrices. The proofs are based on general results developed in the paper concerning Jordan forms of structured matrices (which include in particular the classes of orthogonal and symplectic matrices) under structured rank one perturbations. These results are presented and proved in the framework of real as well as of complex matrices.

Keywords

Symplectic matrix Orthogonal matrix Unitary matrix Indefinite inner product Cayley transformation Perturbation analysis Generic perturbation Rank one perturbation 

Mathematics Subject Classification (2010)

15A63 15A21 15A57 47A55 93B10 93B35 93C73 

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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Christian Mehl
    • 1
    Email author
  • Volker Mehrmann
    • 1
  • André C. M. Ran
    • 2
    • 3
  • Leiba Rodman
    • 4
  1. 1.Institut für Mathematik, MA 4-5, TU BerlinBerlinGermany
  2. 2.Afdeling Wiskunde, Faculteit der Exacte WetenschappenVU AmsterdamAmsterdamThe Netherlands
  3. 3.Unit for BMINorth-West UniversityPotchefstroomSouth Africa
  4. 4.Department of MathematicsCollege of William and MaryWilliamsburgUSA

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