Yet another characterization of solutions of the Algebraic Riccati Equation

This paper deals with a characterization of the solution set of algebraic Riccati equation (ARE) (over reals) for both controllable and uncontrollable systems. We characterize all solutions using simple linear algebraic arguments. It turns out that solutions of ARE of maximal rank have lower rank solutions encoded within it. We demonstrate how these lower rank solutions are encoded within the full rank solution and how one can retrieve the lower rank solutions from the maximal rank solution. We characterize situations where there are no full rank solutions to the ARE. We also characterize situations when the number of solutions to the ARE is finite, when they are infinite and when they are bounded. We also explore the poset structure on the solution set of ARE, which in some specific cases turns out to be a lattice which is isomorphic to lattice of invariant subspaces of a certain matrix. We provide several examples that bring out the essence of these results.