A system of generalized Sylvester quaternion matrix equations and its applications

Let Hm×nHm×n be the set of all m × n   matrices over the real quaternion algebra. We call that A∈Hn×nA∈Hn×n is η  -Hermitian if A=−ηA*η,A=−ηA*η,η ∈ {i, j, k}, where i, j, k   are the quaternion units. Denote Aη*=−ηA*ηAη*=−ηA*η. In this paper, we derive some necessary and sufficient conditions for the solvability to the system of generalized Sylvester real quaternion matrix equations AiXi+YiBi+CiZDi=Ei,(i=1,2),AiXi+YiBi+CiZDi=Ei,(i=1,2), and give an expression of the general solution to the above mentioned system. As applications, we give some solvability conditions and general solution for the generalized Sylvester real quaternion matrix equation A1X+YB1+C1ZD1=E1,A1X+YB1+C1ZD1=E1, where Z is required to be η-Hermitian. We also present some solvability conditions and general solution for the system of real quaternion matrix equations involving η  -Hermicity AiXi+(AiXi)η*+BiYBiη*=Ci,(i=1,2), where Y is required to be η-Hermitian. Our results include some well-known results as special cases.