The main objective of this paper is to solve the following stabilizing output feedback control problem : given matrices (ii, B-2, C-2) with appropriate dimensions, find (if one exists) a static output feedback gain L such that the closed-loop matrix A-B2LC2 is asymptotically stable, It is known that the existence of L is equivalent to the existence of a positive definite matrix belonging to a convex set such that its inverse belongs to another convex set, Conditions are provided for the convergence of an algorithm which decomposes the determination of the aforementioned matrix in a sequence of convex programs, Hence, this paper provides a new sufficient (but not necessary) condition for the solvability of the above stabilizing output feedback control problem, As a natural extension, we also discuss a simple procedure for the determination of a stabilizing output feedback gain assuring good suboptimal performance with respect to a given quadratic index. Some examples borrowed from the literature are solved to illustrate the theoretical results.