Properties of Lyapunov equations with a nonnegative free term matrices

For a matrix Lyapunov equation $A'P+PA+Q=0$ where $Q=C'C\geqslant0$ and $(A, C)$ is an observable pair a theorem is proved that positive definiteness of the matrix $P$ is the necessary and sufficient condition for stability of the matrix $A$. In the case of a stablematrix $A$ the solution (matrix $P$) of the Lyapunov equation is positive definite if and only if the pair $(A, C)$ is observable.