Analytic perturbation of Sylvester and Lyapunov matrix equations

We consider an analytic perturbation of the Sylvester matrix equation. Mainly we are interested in the singular case, that is, when the null space of the unperturbed Sylvester operator is not trivial, but the perturbed equation has a unique solution. In this case, the solution of the perturbed equation can be given in terms of a Laurent series. We provide a necessary and sufficient condition for the existence of a Laurent series with a first order pole. An efficient recursive procedure for the calculation of the Laurent series´ coefficients is given. Finally, we show that in the particular, but practically important case of semisimple eigenvalues, the recursive procedure can be written in a compact matrix form