Algebraic solvability tests for the nonstrict Lyapunov inequality

For arbitrary complex A and Q (Q Hermitian), this paper provides an algebraic test for verifying the existence of a Hermitian solution X of the nonstrict Lyapunov inequality A^*X+XA+Q⩾0. If existing we exhibit how to construct a solution. Moreover, a necessary condition for the existence of a positive definite solution is presented which is most likely to be sufficient as well. Our approach involves the validation problem for the linear matrix inequality Σ_j=1^k(A_j*X_jB_j +B_j*X_j*A_j)+Q>0 in X_j for which we provide a (constructive) algebraic solvability test if the kernels of A_j or, dually, those of B _j form an isotonic sequence