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

Abstract

Let H m × n denote the set of all m × n matrices over the quaternion algebra H and P ∈ H m × m , Q ∈ H n × n be involutions. We say that A ∈ H m × 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.