 
Summary: Analytic perturbation of Sylvester and Lyapunov matrix
equations
Konstantin E. Avrachenkov 1 Jean B. Lasserre 2
Abstract
We consider an analytic perturbation of the Sylvester
matrix equation. Mainly we are interested in the sin
gular case, that is, when the null space of the unper
turbed Sylvester operator is not trivial, but the per
turbed equation has a unique solution. In this case, the
solution of the perturbed equation can be given in terms
of a Laurent series. Here we provide a necessary and
suOEcient condition for the existence of a Laurent series
with a ørst order pole. An eOEcient recursive procedure
for the calculation of the Laurent series' coeOEcients is
given. Finally, we show that in the particular, but prac
tically important case of semisimple eigenvalues, the re
cursive procedure can be written in a compact matrix
form.
1 Introduction
The Sylvester matrix equation AX+XB = C (or Lya
