A note on the discrete Lyapunov and Riccati matrix equations

Inequalities which are satisfied by the determinants of the positive definite solutions of the discrete algebraic Riccati and Lyapunov matrix equations are presented. The results give lower bounds for the product of the eigenvalues of the matrix solutions. Also for a discrete Lyapunov equation, an algorithm is presented to determine under what conditions a positive diagonal solution will exist. If all the conditions are satisfied, the algorithm also provides such a diagonal solution.