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

Please always quote using this URN:urn:nbn:de:0296-matheon-12013
  • 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.

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Metadaten
Author:Christian Mehl, Volker Mehrmann, Andre C.M. Ran, Leiba Rodman
URN:urn:nbn:de:0296-matheon-12013
Referee:Fredi Tröltzsch
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2013/02/27
Release Date:2013/02/27
Tag:indefinite inner product; orthogonal matrix; perturbation analysis; rank one perturbation; symplectic matrix
Institute:Technische Universität Berlin
MSC-Classification:15-XX LINEAR AND MULTILINEAR ALGEBRA; MATRIX THEORY / 15Axx Basic linear algebra / 15A21 Canonical forms, reductions, classification
15-XX LINEAR AND MULTILINEAR ALGEBRA; MATRIX THEORY / 15Axx Basic linear algebra / 15A63 Quadratic and bilinear forms, inner products [See mainly 11Exx]
Preprint Number:1001
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