Theoretical results on the global GMRES method for solving generalized Sylvester matrix‎ ‎equations

‎The global generalized minimum residual (Gl-GMRES)‎ ‎method is examined for solving the generalized Sylvester matrix equation‎ ‎\[\sum\limits_{i = 1}^q {A_i } XB_i = C.\]‎ ‎Some new theoretical results are elaborated for‎ ‎the proposed method by employing the Schur complement‎. ‎These results can be exploited to establish new convergence properties‎ ‎of the Gl-GMRES method for solving general (coupled) linear matrix‎ ‎equations‎. ‎In addition‎, ‎the Gl-GMRES method for solving the generalized‎ ‎Sylvester-transpose matrix equation is briefly studied‎. ‎Finally‎, ‎some numerical experiments are presented to illustrate‎ ‎the efficiently of the Gl-GMRES method for solving the general‎ ‎linear matrix equations‎.