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

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.