The exact solution of a system of quaternion matrix equations involving η-Hermicity

A quaternion matrix A is called η-Hermitian if A=Aη∗=-ηA∗η,η∈{i,j,k}, where A∗ is the conjugate transpose of A. In this paper, we consider the following system of linear real quaternion matrix equations involving η-Hermicity:A1X=C1,A2Y=C2,ZB2=D2,A3X+(A3X)η∗+B3YB3η∗+C3ZC3η∗=D3where Y and Z are required to be η-Hermitian. We obtain some necessary and sufficient conditions for the existence of a solution (X,Y,Z) to above system, and give an expression of the general solution when the system is solvable. Our results include the main result of He and Wang (2013) [2] and other well-known results as special cases.