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

Let ${‐{H}}^{m\times n}$ denote the set of all m × n matrices over the quaternion algebra $‐{H}$ and $P\in {‐{H}}^{m\times m}\text{,}Q\in {‐{H}}^{n\times n}$ be involutions. We say that $A\in {‐{H}}^{m\times n}$ 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.