A Complete Algebraic Solvability Test for the Nonstrict Lyapunov Inequality

For arbitrary equally sized square complex matrices A and Q (Q Hermitian), the paper provides a complete 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. Our approach involves the validation problem for the linear matrix inequality P k j=1 (A j X j B j + B j X j A j) + Q > 0 in X j , for which we provide an algebraic solvability test and a procedure to construct solutions if the kernels of A j or, dually, those of B j form an isotonic sequence. 0 C + is the complex plane, partitioned into the open left-half plane, the imaginary axis, and the open right-half plane. For a subspace S C n , A ? S denotes the preimage fx 2 C m j A x 2 Sg of S under the-th power A of A 2 C nn .