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A new approach is presented for obtaining the solutions to Yakubovich-j-conjugate quaternionmatrix equation X−AX^B=CY based on the real representation of a quaternion matrix. Comparedto the existing results, there are no requirements on the coefficient matrix A. The closedform solution is established and the equivalent form of solution is given for this Yakubovich-j-conjugatequaternion matrix equation. Moreover, the existence of solution to complex conjugatematrix equation X−AX¯B=CY is also characterized and the solution is derived in an explicitform by means of real representation of a complex matrix. Actually, Yakubovich-conjugate matrixequation over complex field is a special case of Yakubovich-j-conjugate quaternion matrix equationX−AX^B=CY. Numerical example shows the effectiveness of the proposed results.

A Real Representation Method for Solving Yakubovich-j-Conjugate Quaternion Matrix Equation