Generic low rank perturbations of structured regular matrix pencils and structured matrices

Abstract

In this thesis, the spectral behavior of several classes of structured regular matrix pencils is studied under structure-preserving low-rank perturbations. Some of the principles that are proved are the following. For certain classes of structured matrix pencils, that have a symmetry in their spectrum, not all canonical forms are allowed when the matrix pencil is subjected to a structure-preserving low-rank perturbation. Therefore, for T -alternating, palindromic, Hermitian, symmetric, and skew-symmetric matrix pencils one observes the following effects at each eigenvalue λ under a generic, structure-preserving rank-1 or rank-2 perturbation: (a) The largest one or two, respectively, Jordan blocks at λ are destroyed. (b) If hereby the eigenvalue pairing imposed by the structure is violated, then also the largest remaining Jordan block at λ will grow in size by one. (c) If λ is a single (double) eigenvalue of the perturbing pencil, then one (two) new Jordan blocks of size one will be created at λ. Hermitian and real structured matrix pencils have signs attached to certain (real or purely imaginary) Jordan blocks in the canonical form, so it is not only the Jordan structure but also this so-called sign characteristic that needs to be examined under perturbation. If λ has a sign characteristic (otherwise the canonical form at λ is completely described by the previous paragraph), then the observed effects under a generic structure-preserving rank-1 or rank-2 perturbation are as follows: (a) All signs from the sign characteristic at λ but one or two, respectively, that correspond to the destroyed blocks, are preserved under perturbation. (b) If λ is not a semisimple eigenvalue of the original matrix pencil, then the sign of the potential new block at λ is prescribed to be the sign that is attached to the eigenvalue λ in the perturbation. Matrices that are structured with respect to an indefinite inner product will be studied under structure-preserving perturbations of arbitrary rank. ForH-selfadjoint, H-symmetric, and J-Hamiltonian matrices, the following results are proved: Let A be a matrix from one of these structure classes and let B be a matrix of rank k such that A+B is from the same structure class as A. Then, generically the Jordan structure and sign characteristic of A+ B is the same that one would obtain by performing a sequence of k generic structured rank-1 perturbations on A.