NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE OF A POSITIVE-DEFINITE SOLUTION OF THE MATRIX EQUATION X+A-ASTERISK-X-1A=Q

We consider the problem of when the matrix equation X + A*X-1A = Q has a positive definite solution. Here Q is positive definite. We study both the real and the complex case. This equation plays a crucial role in solving a special case of the discrete-time Riccati equation. We present both necessary and sufficient conditions for its solvability. This result is obtained by using an analytic factorization approach. Moreover, we present algebraic recursive algorithms to compute the largest and smallest the solution of the equation, respectively. Finally, we discuss the number of solutions.