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.

In this paper, we are interested in the minimal norm of least squares Hermitian solution and the minimal norm of least squares anti-Hermitian solution for the complex generalized Sylvester matrix equation $CXD+EXF=G$. By utilizing of the real vector representations of complex matrices and the semi-tensor product of matrices, we first transform solving special least squares solutions of the above matrix equation into solving the general least squares solutions of the corresponding real matrix equations, and then obtain the expressions of the minimal norm of least squares Hermitian solution and the minimal norm of least squares anti-Hermitian solution. Further, we give two numerical algorithms and two numerical examples, and numerical examples illustrate that our proposed algorithms are more efficient and accurate.