An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation

The generalized coupled Sylvester matrix equations (AY-ZB,CY-ZD)=(E,F)(AY-ZB,CY-ZD)=(E,F) with unknown matrices Y,ZY,Z are encountered in many systems and control applications. Also these matrix equations have several applications relating to the problem of computing stable eigendecompositions of matrix pencils. In this work, we construct an iterative algorithm to solve the generalized coupled Sylvester matrix equations over reflexive matrices Y,ZY,Z. And when the matrix equations are consistent, for any initial matrix pair [Y0,Z0][Y0,Z0], a reflexive solution pair can be obtained within finite iteration steps in the absence of roundoff errors, and the least Frobenius norm reflexive solution pair can be obtained by choosing a special kind of initial matrix pair. Also we obtain the optimal approximation reflexive solution pair to a given matrix pair [Y¯,Z¯] in the reflexive solution pair set of the generalized coupled Sylvester matrix equations (AY-ZB,CY-ZD)=(E,F)(AY-ZB,CY-ZD)=(E,F). Moreover, several numerical examples are given to show the efficiency of the presented iterative algorithm.