  We give a rank characterization of the solution set of algebraic Riccati
inequality (ARI) for both controllable and uncontrollable systems. Assuming an
existence of a solution of the corresponding algebraic Riccati equation (ARE),
we characterize the boundedness/unboundedness properties of solutions of ARI
for controllable/uncontrollable systems without any assumption on sign
controllability. As a consequence of our observations, we obtain Willems'
result $K_{min}\leq K\leq K_{max}$ for an ARI in the case of controllable
systems and explore some structure on the extremal solutions. We also consider
the curious case of uncontrollable purely imaginary eigenvalues and the
behavior of the solution set of ARI. In particular, we show that a system is
controllable if and only if the set of solutions of an ARI is bounded. In
addition, we study the effect of the position of eigenvalues of the system
matrix in the complex plane on the behavior of the solution set of ARIs.
Furthermore, we obtain a rank parametrization for solutions of ARI for
controllable systems.
