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Common Solutions to the Matrix Equations $$AX=B$$ A X = B and $$XC=D$$ X C = D on a Subspace

Author

Listed:
  • Shanshan Hu

    (Hubei Normal University)

  • Yongxin Yuan

    (Hubei Normal University)

Abstract

Let $$ \mathbb{S}\mathbb{R}_{{\Omega }}^{n \times n}$$ S R Ω n × n be the set of all $$n \times n$$ n × n symmetric matrices on subspace $${\Omega }$$ Ω , where $$\begin{aligned} {\Omega }=\{ z \in {\mathbb {R}}{^n}|Gz=0,\,G\in {\mathbb {R}}^{k \times n}\}. \end{aligned}$$ Ω = { z ∈ R n | G z = 0 , G ∈ R k × n } . The necessary and sufficient conditions for the matrix equations $$AX=B$$ A X = B and $$XC=D$$ X C = D to have a common solution in $$\mathbb{S}\mathbb{R}_{{\Omega }}^{n \times n}$$ S R Ω n × n and also an expression for the general common solution are obtained. Further, the associated optimal approximate problem to a given matrix $${\tilde{X}} \in {\mathbb {R}}^{n\times n}$$ X ~ ∈ R n × n is discussed and the optimal approximate solution is elucidated. Finally, a numerical experiment is presented to validate the accuracy of our result.

Suggested Citation

  • Shanshan Hu & Yongxin Yuan, 2023. "Common Solutions to the Matrix Equations $$AX=B$$ A X = B and $$XC=D$$ X C = D on a Subspace," Journal of Optimization Theory and Applications, Springer, vol. 198(1), pages 372-386, July.
  • Handle: RePEc:spr:joptap:v:198:y:2023:i:1:d:10.1007_s10957-023-02247-8
    DOI: 10.1007/s10957-023-02247-8
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    References listed on IDEAS

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    1. Magnus, J.R. & Neudecker, H., 1980. "The elimination matrix : Some lemmas and applications," Other publications TiSEM 0e3315d3-846c-4bc5-928e-f, Tilburg University, School of Economics and Management.
    2. Yonghui Liu & Yongge Tian, 2011. "Max-Min Problems on the Ranks and Inertias of the Matrix Expressions A−BXC±(BXC)∗ with Applications," Journal of Optimization Theory and Applications, Springer, vol. 148(3), pages 593-622, March.
    3. Kumar, Ashim & Cardoso, João R., 2018. "Iterative methods for finding commuting solutions of the Yang–Baxter-like matrix equation," Applied Mathematics and Computation, Elsevier, vol. 333(C), pages 246-253.
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