The symmetric solution of the matrix equation $AXB=D$ on subspace

Abstract

Let $\Omega =\{ z \in {\mathbb {R}}{^n}|Gz=0,G\in {\mathbb {R}}^{k \times n}\}$ and $\mathbb{S}\mathbb{R}_\Omega ^{n \times n}=\{A\in {\mathbb {R}}^{n\times n}|(Ax,y)=(x,Ay),\forall x,y \in \Omega \}$. In this paper, we first consider the following problem (Problem 1): Given $A\in {\mathbb {R}}^{m\times n}$, $B\in \mathbb {R}^{n \times q }$ and $D\in \mathbb {R}^{m \times q }$, find $X\in \mathbb{S}\mathbb{R}_\Omega ^{n \times n}$ such that $AXB=D$. Further, we consider an associated optimal approximation problem: Given ${\tilde{X}} \in {\mathbb {R}}^{n\times n}$, find ${\hat{X}}\in S_E$ such that $\Vert \hat{X} - \tilde{X}\Vert ={\min \limits _{X\in S_E}}\Vert X - {\tilde{X}}\Vert$, where $S_E$ is the solution set of Problem 1. The solvability conditions and the representation of the general solution of Problem 1 are derived by using the generalized inverses, and then, the unique approximation solution ${\hat{X}}$ of the optimal approximation problem are deduced by applying the Kronecker product of matrices. Finally, two numerical examples are presented to show the correctness of our results.