On the perturbation of singular analytic matrix functions: A generalization of Langer and Najman's results

Given a singular n×n matrix function A(λ) , analytic in a neighborhood of an eigenvalue λ0 ∈ C , and perturbations, B(λ ,e) , such that B(λ ,0) ≡ 0 and analytic in λ and e near (λ0,0) , we provide sufficient conditions on these perturbations for the existence of eigenvalue expansions of the perturbed matrix A(λ) + B(λ ,e) near λ0 . We also describe the first order term of these expansions. This extends to the singular case some results by Langer and Najman. Mathematics subject classification (2010): 15A18, 47A56.