The Anti-symmetric Solution of the Matrix Equation $AXA^\top =B$ on a Null Subspace

Abstract

For a given matrix $G \in {\mathbb {R}}^{k\times n}$, let the set $\Omega _G =\hbox {Null}(G),$ and let $\mathbb{A}\mathbb{S}_{{\Omega _G}}({\mathbb {R}}^{n \times n})=\{ A \in {\mathbb {R}}^{n\times n}\vert (Ax,y)=(x,-Ay),\forall x,y \in {\Omega _G}\}$, that is, $\mathbb {{AS}}_{{\Omega _G}}({\mathbb {R}}^{n \times n})$ is the set of all anti-symmetric matrices on $\Omega _G$. In this paper, we first consider the following problem (Problem 1): Given $A\in {\mathbb {R}}^{m\times n},B\in {\mathbb {R}}^{m\times m}$, find $X\in \mathbb {{AS}}_{{\Omega _G}}({\mathbb {R}}^{n \times n})$ such that $AXA^\top =B$. Then, we consider the associated optimal approximation problem: Given ${\tilde{X}} \in {\mathbb {R}}^{n\times n}$, find ${\hat{X}}\in {\mathcal {S}}_{{\Omega _G}}(A,B)$ such that $\hat{X}={\arg \min \limits _{X\in {\mathcal {S}}_{{\Omega _G}}(A,B)}}\Vert X - {\tilde{X}}\Vert$, where $\Vert \cdot \Vert$ is the Frobenius norm and ${\mathcal {S}}_{{\Omega _G}}(A,B)$ is the solution set of Problem 1. By using the generalized singular value decomposition of a matrix pair, we deduce the solvability condition and the representation of the general solution of Problem 1. Moreover, we obtain the unique approximation solution ${\hat{X}}$ of the optimal approximation problem. Finally, a numerical example is presented to show the correctness of our result.