Positive definite solutions of the nonstrict Lyapunov inequality

Recently we have provided a full test for verifying the solvability of the nonstrict Lyapunov inequality A*X + XA + Q ≥ 0 where A and Q = Q* are arbitrary. In this paper we reveal how to test the existence of a positive definite solution of the inequality. This is achieved by determining the largest subspace on which the quadratic form defined by X remains constant if X varies in the solution set. The proof exploits duality or Farkas-type results from the theory of linear matrix inequalities.