dc.contributor.author Mehl, Christian dc.contributor.author Ran, André C.M. dc.contributor.author Mehrmann, Volker dc.contributor.author Rodman, Leiba dc.date.accessioned 2016-01-15T06:33:26Z dc.date.available 2016-01-15T06:33:26Z dc.date.issued 2014 dc.identifier.citation Mehl, C. et al. 2014. Eigenvalue perturbation theory of symplectic, orthogonal, and unitary matrices under generic structured rank one perturbations. BIT numerical mathematics, 54(1):219-255. [http://link.springer.com/journal/10543] en_US dc.identifier.issn 0006-3835 dc.identifier.issn 1572-9125 (Online) dc.identifier.uri http://hdl.handle.net/10394/15867 dc.description.abstract We study the perturbation theory of structured matrices under structured en_US 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 skewsymmetric 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 dc.description.sponsorship Supported by Deutsche Forschungsgemeinschaft, through the DFG Research en_US Center MATHEON Mathematics for key technologies in Berlin dc.description.uri http://dx.doi.org/10.1007/s10543-013-0451-3 dc.description.uri http://link.springer.com/article/10.1007/s10543-013-0451-3 dc.description.uri http://link.springer.com/journal/10543 dc.language.iso en en_US dc.publisher Springer Verlag en_US dc.subject Symplectic matrix en_US dc.subject orthogonal matrix en_US dc.subject unitary matrix en_US dc.subject indefinite inner product en_US dc.subject Cayley transformation en_US dc.subject perturbation analysis en_US dc.subject generic perturbation en_US dc.subject rank one perturbation en_US dc.title Eigenvalue perturbation theory of symplectic, orthogonal, and unitary matrices under generic structured rank one perturbations en_US dc.type Article en_US
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