A Pair of Mixed Generalized Sylvester Matrix Equations

In this paper,some necessary and sufficient solvability conditions for the system of mixed generalized Sylvester matrix equations A_1X-YB_1=C_1,A_2Y-ZB_2 = C_2 are derived,and an expression of the general solution to this system is given when it is solvable.Admissible ranks of the solution,and admissible ranks and inertias of the Hermitian part of the solution are investigated,respectively.As an application of the above system,solvability conditions and the general Hermitian solution to the generalized Sylvester matrix equation are obtained.Moreover,we provide an algorithm and an example to illustrate our results.