Document Type
Article
Department/Program
Mathematics
Journal Title
Linear Algebra and Its Applications
Pub Date
2012
Volume
436
Issue
10
First Page
4027
Abstract
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 general results concerning rank one perturbations. (C) 2010 Elsevier Inc. All rights reserved.
Recommended Citation
Mehl, Christian; Mehrmann, Volker; Ran, Andre C. M.; and Rodman, Leiba, Perturbation theory of selfadjoint matrices and sign characteristics under generic structured rank one perturbations (2012). Linear Algebra and Its Applications, 436(10), 4027-4042.
10.1016/j.laa.2010.04.008
DOI
10.1016/j.laa.2010.04.008
