Periodic Schur forms and some matrix equations

We propose an elegant and conceptually simple method for computing the periodic solution of three classes of periodic matrix equations --- Riccati, Lyapunov and Sylvester. Such equations arise naturally in several problems of linear system theory. Our approach is very attractive from a numerical point of view, since it is based on the periodic Schur form of matrix sequences G i ; H i ; i = 0; : : : ; K Gamma 1, which utilizes stable numerical techniques involving unitary (orthogonal in the real case) transformations only. Our approach readily extends to more general situations, such as when the equations are given in implicit or descriptor form.