In this talk, I consider the problem of finding explicit recursive for- mulas to compute the perturbed eigenvalues and eigenvectors of non- selfadjoint analytic perturbations of matrices with degenerate eigenvalues. Based on some math-physics problems arising from the study of slow light in photonic crystals, we single out a class of perturbations that satisfy what I call the generic condition. It will be shown that for this class of perturbations, the problem mentioned above of finding explicit recursive formulas can be solved. Using these recursive formulas, I will list the first and second order terms for the perturbed eigenvalues and eigenvectors of perturbations belonging to this class.