On the solutions of the dual matrix equation $ A^\top XA = B $

Abstract

Let $ \mathbb{D}^{m \times n} = \{A = A_{1}+\varepsilon A_{2}|A_{1}, A_{2}\in \mathbb{R}^{m \times n}\} $ be the set of all $ m\times n $ real dual matrices. In this paper, the following problems are considered. Problem I: Given dual matrices $ A = A_{1}+\varepsilon A_{2}\in \mathbb{D}^{m\times n} $ and $ B = B_{1}+\varepsilon B_{2}\in \mathbb{D}^{n\times n} $, find $ X\in S $ such that the dual matrix equation $ A^\top XA = B $ is satisfied, where $ S = \{X\in \mathbb{D}^{m \times m}|CX = D, C, D\in \mathbb{D}^{p \times m}\} $. Problem II: Given dual matrices $ A = A_{1}+\varepsilon A_{2}\in \mathbb{D}^{m\times n}, B = B_{1}+\varepsilon B_{2}\in \mathbb{D}^{n\times n} $ and $ \tilde{X} = \tilde{X}_{1}+\varepsilon \tilde{X}_{2}\in \mathbb{D}^{m\times m} $, with $ B_{i} = B^\top_{i}, i = 1, 2 $, find $ \hat{X}\in T $ such that $ \|\hat{X}-\tilde{X}\|_{{\rm D}} = \mathop{\min}\limits_{X\in T} \|X-\tilde{X}\|_{{\rm D}} = \mathop{\min}\limits_{X\in T}\sqrt{\Vert X_{1}-\tilde{X}_{1} \Vert^{2}+\Vert X_{2}-\tilde{X}_{2}\Vert^{2}} $, where $ T = \{X = X_{1}+\varepsilon X_{2}\in \mathbb{D}^{m \times m}|A^\top XA = B \ \ \mbox{s. t.} \ X_{i} = X^\top_{i}, i = 1, 2\} $. We derive the solvability conditions and the representation of the general solution of Problem I using the Moore-Penrose inverse. Also, we deduce the solvability conditions and the explicit formula of $ T $ and the unique approximation solution $ \hat{X} $ of Problem II by applying the Moore-Penrose inverse and Kronecker product of matrices. Finally, we give a numerical example to show the correctness of our result.

Keywords

• dual matrix equation
• optimal approximation
• linear manifold
• kronecker product