A real representation method for solving Yakubovich-\(j\)-conjugate quaternion matrix equation (Q1723767)

Summary: A new approach is presented for obtaining the solutions to Yakubovich-\(j\)-conjugate quaternion matrix equation \(X - A \hat{X} B = C Y\) based on the real representation of a quaternion matrix. Compared to the existing results, there are no requirements on the coefficient matrix \(A\). The closed form solution is established and the equivalent form of solution is given for this Yakubovich-\(j\)-conjugate quaternion matrix equation. Moreover, the existence of solution to complex conjugate matrix equation \(X - A \overline{X} B = C Y\) is also characterized and the solution is derived in an explicit form by means of real representation of a complex matrix. Actually, Yakubovich-conjugate matrix equation over complex field is a special case of Yakubovich-\(j\)-conjugate quaternion matrix equation \(X - A \hat{X} B = C Y\). Numerical example shows the effectiveness of the proposed results.