A PARAMETRIZATION OF SOLUTIONS OF THE DISCRETE-TIME ALGEBRAIC RICCATI EQUATION BASED ON PAIRS OF OPPOSITE UNMIXED SOLUTIONS∗

Abstract. The paper describes the set of solutions of the discrete-time algebraic Riccati equa-tion. It is shown that each solution is a combination of a pair of opposite unmixed solutions. There is a one-to-one correspondence between solutions and invariant subspaces of the closed loop matrix of an unmixed solution. The results of the paper provide an extended counterpart of the parametrization theory of continuous-time algebraic Riccati equations by Willems, Coppel, and Shayman. Key words. discrete-time algebraic Riccati equation, unmixed solutions, opposite solutions, parametrization by invariant subspaces, shorted operators, Schur complement