The semi-tensor product method for special least squares solutions of the complex generalized Sylvester matrix equation

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.