An Iterative Algorithm for the Generalized Reflexive Solutions of the Generalized Coupled Sylvester Matrix Equations
Feng Yin, Guang-Xin Huang · Journal of Applied Mathematics · 2012
An iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations (AXB − CYD, EXF − GYH) = (M, N), which includes Sylvester and Lyapunov matrix equations as special cases, over generalized reflexive matrices X and Y. When the matrix equations are consistent, for any initial generalized reflexive matrix pair [X1, Y1], the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round‐off errors, and the least Frobenius norm generalized reflexive solutions can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation generalized reflexive solution pair to a given matrix pair [X0, Y0] in Frobenius norm can be derived by finding the least‐norm generalized reflexive solution pair of a new corresponding generalized coupled Sylvester matrix equation pair , where . Several numerical examples are given to show the effectiveness of the presented iterative algorithm.