\(P\)-(skew)symmetric common solutions to a pair of quaternion matrix equations (Q2470181)

Necessary and sufficient conditions are given for the solvability of the system of quaternion \(n\times n\) matrix equations \(A_1X_1 = C_1\), \(A_2X_2 = C_2\), \(A_3X_1B_1 + A_4X_2B_2 = C_b\) in \(X_1\), \(X_2\) and an expression for the general solution is derived. The same is done for the \(P\)-symmetric (\(X = PXP\)) and \(P\)-skewsymmetric (\(X=-PXP\)) solutions of the system \(A_aX = C_a\), \(A_bXB_b = C_b\), where \(P^2 = I\), \(P\neq I\) and \(I\) is the \(n\times n\) identity matrix.