This paper is devoted to the perturbation theory for defective matrices. We consider the asymptotic expansions of the perturbed spectrum when a matrix is changed to , where and is a small parameter. In particular, we analyse the rational exponents that may occur when the matrix varies over the sphere . We partially characterize the leading exponents noting that the description of the set of all leading exponents remains an open problem.
Mihail Konstantinov. Volker Mehrmann. Petko Petkov. "Perturbed spectra of defective matrices." J. Appl. Math. 2003 (3) 115 - 140, 5 March 2003. https://doi.org/10.1155/S1110757X0320512X