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

Let H-mxn denote the set of all m x n matrices over the quaternion algebra H and P is an element of H-mxm; Q is an element of H-nxn be involutions. We say that A is an element of H-mxn 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. (C) 2011 Elsevier Inc. All rights reserved.