Journal of Applied Mathematics
Volume 2003 (2003), Issue 3, Pages 115-140
doi:10.1155/S1110757X0320512X

Perturbed spectra of defective matrices

Mihail Konstantinov,1 Volker Mehrmann,2 and Petko Petkov3

1Department of Automatics, University of Architecture and Civil Engineering, 1 Christo Smirnenski Blvd., Sofia 1046, Bulgaria
2Institut für Mathematik, MA 4–5, Technische Universitfät Berlin, Strasse des 17.Juni 136, Berlin D-10623, Germany
3Department of Systems and Control, Faculty of Automatics, Technical University of Sofia, Sofia 1756, Bulgaria

Received 22 May 2002; Revised 7 October 2002

Copyright © 2003 Mihail Konstantinov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is devoted to the perturbation theory for defective matrices. We consider the asymptotic expansions of the perturbed spectrum when a matrix A is changed to A+tE, where E≠0 and t>0 is a small parameter. In particular, we analyse the rational exponents that may occur when the matrix E varies over the sphere ‖E‖ =ρ>0. We partially characterize the leading exponents noting that the description of the set of all leading exponents remains an open problem.