The special least squares solutions of the complex matrix equation
\begin{equation} \left(\mathbf{AXB,CXD}\right)\mathbf{=(E,F)}\nonumber \\ \end{equation}
Yanzhen Zhang1, Shanshan Yang2, Ying Li1 *
1.School of Mathematical Science, Liaocheng University, Liaocheng Shandong 252000
2. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081
Abstract In this paper, we study the problem of least square solutions of the complex matrix equation\(\left(\mathbf{\text{AXB}},\mathbf{\text{CXD}}\right)=\left(\mathbf{E},\mathbf{F}\right)\). First, by using the real representation method of the complex matrix, we prosent the real representation form of the complex matrix equation\(\left(\mathbf{\text{AXB}},\mathbf{\text{CXD}}\right)=\left(\mathbf{E},\mathbf{F}\right)\). In combination with the special structure of the real representation matrix of the complex matrix, the vec operator of the matrix, the Kronecker product of the matrix and the MP inverse property of the matrix, we obtain the Hermitian least squares solution of the complex matrix equation\(\left(\mathbf{\text{AXB}},\mathbf{\text{CXD}}\right)=\left(\mathbf{E},\mathbf{F}\right)\),and derive the Hermitian minimum norm least square solution, the real symmetric minimum norm least square solution and the real dissymmetric minimum norm least square solution of the complex matrix equation\(\left(\mathbf{\text{AXB}},\mathbf{\text{CXD}}\right)=\left(\mathbf{E},\mathbf{F}\right)\). At last, we give the expressions of three minimum norm least squares solutions and their corresponding algorithms, respectively.
Keywords Matrix equation; Hermitian least squares solution; The real symmetric least square solution; The real opposition least square solution; Real representation matrix.