1.Introduction
Matrix equations have a wide range of applications, which the calculation of special least squares solution is always a research hotspot in the field of numerical algebra. The least squares solutions of complex matrix equations have been studied extensively[1]-[11]. For example, [1,2] by using the real representation method of the complex matrix, Zhang et al. study the Hermitian minimum norm least square solution of\(AXB+CXD=E\ \)and\(\ (AXB,\ CXD)=(E,\ F)\), respectively; [3] by the Kronecker product of the Matrix-Vector and the MP inverse property of the matrix, Wang et al. proposed a direct method to solve the least squares solution of the complex matrix equation\((AXB,\ CXD)=(E,\ F)\); [4] Yuan et al. proposed a new method to solve the Hermitian least square solution of the complex matrix equation, and gave the expression and algorithm of the least square solution of the Hermitian least square solution and the Hermitian minimum norm least square solution, et al.
This paper study the problem of least square solutions of complex matrix equations
\(\left(AXB,CXD\right)=\left(E,F\right).\) (1)
In this paper, we use the following notations.
Let\(R^{m\times n}C^{m\times n}{HC}^{n\times n}\text{SR}^{n\times n}\text{ASR}^{n\times n}\)be the sets of \(m\times n\) real matrices, be the sets of\(m\times n\ \)real matrices, be the sets of \(m\times n\ \)Hermitian complex matrices, be the sets of \(m\times n\ \)real symmetric matrices, be the sets of \(m\times n\ \)real dissymmetric matrices.\(R^{n}\) be the sets of \(n\) column vector,
\(A^{T}{A}^{H\ }\)denote the transpose and the conjugate transpose of matrix \(A\), respectively. \(A^{+}\ \)denote the MP inverse of matrix\(A\), ,\(I_{n}=\left(e_{1},e_{2},\ldots,e_{n}\right)\) denote the identity matrix of order \(n\), \(\left\|A\right\|\) denote Frobenius norm of complex matrices
For (1), we mainly discuss the following three issues.
Problem 1 Let\(A,C\in C^{m\times n},\ B,D\in C^{n\times s}\ ,E,F\in C^{m\times s},\)the Hermitian minimum norm least squares solution of (1)\(\ \)denote by\(C_{H}\), that is
\begin{equation} \text{\ \ \ C}_{H}=\left\{X\middle|X\in\text{HC}^{n\times n},\left\|AXB-E\right\|+\left\|CXD-F\right\|=min\right\},\nonumber \\ \end{equation}
make sure \(C_{H}\), and solve the Hermitian minimum norm least square solution \(\hat{X}\in C_{S}\) satisfying\(\left\|\tilde{X}\right\|=\min_{X\in C_{H}}\left\|X\right\|.\)
Problem 2 Let\(A,C\in C^{m\times n},\ B,D\in C^{n\times s}\ ,E,F\in C^{m\times s},\)the real symmetric minimum norm least squares solution of (1)\(\ \)denote by\(\text{\ \ }C_{S}\), that is
\begin{equation} \text{\ \ \ C}_{S}=\left\{X\middle|X\in\text{SR}^{n\times n},\left\|AXB-E\right\|+\left\|CXD-F\right\|=min\right\},\nonumber \\ \end{equation}
make sure\(\ C_{S}\), and solve the symmetric minimum norm least square solution\(\ \hat{X}\in C_{S}\)satisfying\(\ \left\|\hat{X}\right\|=\min_{X\in C_{s}}\left\|X\right\|\).
Problem 3Let\(\ A,C\in C^{m\times n},\ B,D\in C^{n\times s}\ ,E,F\in C^{m\times s},\ \)the real dissymmetric minimum norm least squares solution of (1)\(\ \)denote by\(\text{\ \ }C_{A}\), that is
\({\text{\ \ \ }C}_{A}=\{X|X\in\text{ASR}^{n\times n},\left\|\text{AXB}-E\right\|+\left\|\text{CXD}-F\right\|=\min\}\),
make sure\(\ C_{A}\), and solve the symmetric minimum norm least square solution\(\ \overset{\check{}}{X}\in C_{A}\)satisfying足\(\left\|\overset{\check{}}{X}\right\|=\min_{X\in C_{A}}\left\|X\right\|.\)