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\(\left\|\overset{\check{}}{X}\right\|=\min_{X\in C_{A}}\left\|X\right\|.\)