Perturbation Theory for Analytic Matrix Functions: The Semisimple Case

The eigenvalue problem for non-self-adjoint, analytic matrix functions of two variables, $L(\lambda,\alpha)$, is examined with emphasis on the case when, at a fixed $\al_0$, $L(\lambda,\alpha_0)$ has a multiple, semisimple eigenvalue $\lambda_0$. New sufficient conditions for analytic dependence of eigenvalue functions, $\lambda(\alpha)$, on $\alpha$ in a neighborhood of $\alpha_0$ are obtained. An algorithm for generating Taylor coefficients of perturbed eigenvalues and eigenvectors is studied and the existence of positive radii of convergence is established. Connections with known results on self-adjoint problems are made.