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

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. © 2013 Springer Science+Business Media Dordrecht

Perturbation theory of selfadjoint matrices and sign characteristics under generic structured rank one perturbations.

For selfadjoint matrices in an indefinite inner product, possible canonical forms are identified that arise when the matrix is subjected to a selfadjoint generic rank one perturbation. Genericity is understood in the sense of algebraic geometry. Special attention is paid to the perturbation behavior of the sign characteristic. Typically, under such a perturbation, for every given eigenvalue, the largest Jordan block of the eigenvalue is destroyed and (in case the eigenvalue is real) all other Jordan blocks keep their sign characteristic. The new eigenvalues, i.e. those eigenvalues of the perturbed matrix that are not eigenvalues of the original matrix, are typically simple, and in some cases information is provided about their sign characteristic (if the new eigenvalue is real). The main results are proved by using the well known canonical forms of selfadjoint matrices in an indefinite inner product, a version of the Brunovsky canonical form and on general results concerning rank one perturbations obtained. © 2009 Elsevier Inc. All rights reserved

Perturbation analysis of Lagrangian invariant subspaces of symplectic matrices

Lagrangian invariant subspaces for symplectic matrices play an important role in the numerical solution of discrete time, robust and optimal control problems. The sensitivity (perturbation) analysis of these subspaces, however, is a difficult problem, in particular, when the eigenvalues are on or close to some critical regions in the complex plane, such as the unit circle. We present a detailed perturbation analysis for several different cases of real and complex symplectic matrices. We analyze stability and conditional stability as well as the index of stability for these subspaces