On a solution of the quaternion matrix equation \({A \widetilde{X} -X B = C}\) and its applications (Q2443927)

Using the real representation of a quaternion matrix, this paper investigates the solution to the quaternion matrix equation \(A\tilde{X}-XB = C\). A similarity version of Roth's theorem for the consistence of the quaternion matrix equation is derived, and an explicit expression of the closed-form solution to the quaternion matrix equation is also given. As a special case, this paper also gives some applications of the solution of the complex matrix equation \(A\bar{X}-XB = C\) and of the consimilarity of complex matrices.