A real representation method for special least squares solutions of the quaternion matrix equation $(AXB,DXE)=(C,F)$

In this article, our interest is the quaternion matrix equation $(AXB,DXE)=(C,F)$, and we study its minimal norm centrohermitian least squares solution and skew centrohermitian least squares solution. By applying of the real representation matrices of quaternion matrices and relative properties, we convert the quaternion least squares problems with constrained variables into the corresponding real least squares problems with free variables, and then we obtain the solutions of corresponding problems. The final results can be expressed only by real matrices and vectors, and thus the corresponding algorithms only involves real operations and avoid complex quaternion operations. Therefore, they are portable and convenient. In the end, we give two examples to verify the effectiveness of the purposed algorithms.