Perturbed spectra of defective matrices

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 $E\ne 0$ 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 $\Vert E\Vert =\rho >0$. We partially characterize the leading exponents noting that the description of the set of all leading exponents remains an open problem.