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.