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Common Solutions to the Matrix Equations \(AX=B\) and \(XC=D\) on a Subspace

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Abstract

Let \( \mathbb{S}\mathbb{R}_{{\Omega }}^{n \times n}\) be the set of all \(n \times n\) symmetric matrices on subspace \({\Omega }\), where

$$\begin{aligned} {\Omega }=\{ z \in {\mathbb {R}}{^n}|Gz=0,\,G\in {\mathbb {R}}^{k \times n}\}. \end{aligned}$$

The necessary and sufficient conditions for the matrix equations \(AX=B\) and \(XC=D\) to have a common solution in \(\mathbb{S}\mathbb{R}_{{\Omega }}^{n \times n}\) and also an expression for the general common solution are obtained. Further, the associated optimal approximate problem to a given matrix \({\tilde{X}} \in {\mathbb {R}}^{n\times n}\) is discussed and the optimal approximate solution is elucidated. Finally, a numerical experiment is presented to validate the accuracy of our result.

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Acknowledgements

The authors would like to express their gratitude to the editor and the anonymous reviewers for their helpful comments and suggestions, which have improved the presentation of the paper.

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  1. School of Mathematics and Statistics, Hubei Normal University, Huangshi, 435002, People’s Republic of China

    Shanshan Hu & Yongxin Yuan

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  1. Shanshan Hu
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  2. Yongxin Yuan
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Correspondence to Yongxin Yuan.

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Communicated by Firdaus E. Udwadia.

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Hu, S., Yuan, Y. Common Solutions to the Matrix Equations \(AX=B\) and \(XC=D\) on a Subspace. J Optim Theory Appl 198, 372–386 (2023). https://doi.org/10.1007/s10957-023-02247-8

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