P-(skew)symmetric common solutions to a pair of quaternion matrix equations

An n × n quaternion matrix A is termed P-symmetric (or P-skewsymmetric) if A = PAP (or A = −PAP), where P is an n × n   nontrivial quaternion involution. In this paper, we first establish necessary and sufficient conditions for the existence and the expression of the general solution to the system of quaternion matrix equations A1X1=C1,A2X2=C2,A3X1B1+A4X2B2=CbA1X1=C1,A2X2=C2,A3X1B1+A4X2B2=Cb, then use the results on the system mentioned above to give necessary and sufficient conditions for the existence and the representations of P-symmetric and P-skewsymmetric solutions to the system of quaternion matrix equations AaX = Ca and AbXBb = Cb. Furthermore, we establish representations of P-symmetric and P-skewsymmetric quaternion matrices. A numerical example is presented to illustrate the results of this paper.