P-(skew)symmetric common solutions to a pair of quaternion matrix equations - Wang, Qing-Wen; Chang, Hai-Xia; Lin, Chun-Yan (Applied mathematics and computation, 2008)
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 A 1 X 1 = C 1 , A 2 X 2 = C 2 , A 3 X 1 B 1 + A 4 X 2 B 2 = C b , 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 A a X = C a and A b XB b = C b . Furthermore, we establish representations of P-symmetric and P-skewsymmetric quaternion matrices. A numerical example is presented to illustrate the results of this paper.