The (P, Q)-(skew)symmetric extremal rank solutions to a system of quaternion matrix equations

Let Hm×n denote the set of all m × n matrices over the quaternion algebra H and P∈Hm×m,Q∈Hn×n be involutions. We say that A∈Hm×n is (P, Q)-symmetric (or (P, Q)-skewsymmetric) if A = PAQ (or A = − PAQ). We in this paper mainly investigate the (P, Q)-(skew)symmetric maximal and minimal rank solutions to the system of quaternion matrix equations AX = B, XC = D. We present necessary and sufficient conditions for the existence of the maximal and minimal rank solutions with (P, Q)-symmetry and (P, Q)-skewsymmetry to the system. The expressions of such solutions to this system are also given when the solvability conditions are satisfied. A numerical example is presented to illustrate our results. The findings of this paper extend some known results in this literature.