On discrete algebraic Riccati equations: A rank characterization of solutions

We give a rank characterization of the solution set of discrete algebraic Riccati equations (DAREs) involving real matrices. Our characterization involves the use of invariant subspaces of the system matrix. We observe that a DARE can be considered as a specific type of a non-symmetric continuous-time algebraic Riccati equation. We show how to extract the low rank solutions from the full rank solution of a homogeneous DARE. We also study the eigen-structure of various feedback/closed loop matrices associated with solutions of a DARE. We show as a consequence of our observations that a solution X of a DARE for controllable systems and certain specific types of uncontrollable systems satisfies the inequality Xmin≤X≤Xmax. We characterize conditions under which the solution set of DARE is bounded/unbounded. We use the method of invariant subspaces to identify some invariant sets of a difference Riccati equation.