Eigenvalue perturbation theory of symplectic, orthogonal, and unitary matrices under generic structured rank one perturbations
Date
2014Author
Mehl, Christian
Ran, André C.M.
Mehrmann, Volker
Rodman, Leiba
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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 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
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