Eigenvalue perturbation theory of symplectic, orthogonal, and unitary matrices under generic structured rank one perturbations
Ran, André C.M.
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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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Mehl, Christian; Ran, André C. M.; Mehrmann, Volker; Rodman, Leiba (Element, Zagreb, 2013)New perturbation results for the behavior of eigenvalues and Jordan forms of real and complex matrices under generic rank one perturbations are discussed. Several results that are available in the complex case are proved ...
A singular M-matrix perturbed by a nonnegative rank one matrix has positive principal minors; is it D-stable? Bierkens, Joris; Ran, André (Elsevier, 2014)The positive stability and D-stability of singular M-matrices, perturbed by (non-trivial) nonnegative rank one perturbations, is investigated. In special cases positive stability or D-stability can be established. In full ...
Batzke, Leonhard; Ran, André C.M.; Rodman, Leiba; Mehl, Christian (Springer, 2016)This paper deals with the effect of generic but structured low rank perturbations on the Jordan structure and sign characteristic of matrices that have structure in an indefinite inner product space. The paper is a follow-up ...