IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v198y2023i1d10.1007_s10957-023-02247-8.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-023-02247-8
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s10957-023-02247-8?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    4. Magnus, J.R. & Neudecker, H., 1979. "The commutation matrix : Some properties and applications," Other publications TiSEM d0b1e779-7795-4676-ac98-1, Tilburg University, School of Economics and Management.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Liu, Shuangzhe & Leiva, Víctor & Zhuang, Dan & Ma, Tiefeng & Figueroa-Zúñiga, Jorge I., 2022. "Matrix differential calculus with applications in the multivariate linear model and its diagnostics," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    2. Christian Gische & Manuel C. Voelkle, 2022. "Beyond the Mean: A Flexible Framework for Studying Causal Effects Using Linear Models," Psychometrika, Springer;The Psychometric Society, vol. 87(3), pages 868-901, September.
    3. Turkington, Darrell A., 1998. "Efficient estimation in the linear simultaneous equations model with vector autoregressive disturbances," Journal of Econometrics, Elsevier, vol. 85(1), pages 51-74, July.
    4. D.A. Turkington, 1997. "Some results in matrix calculus and an example of their application to econometrics," Economics Discussion / Working Papers 97-07, The University of Western Australia, Department of Economics.
    5. Paulo M. D. C. Parente & Richard J. Smith, 2021. "Quasi‐maximum likelihood and the kernel block bootstrap for nonlinear dynamic models," Journal of Time Series Analysis, Wiley Blackwell, vol. 42(4), pages 377-405, July.
    6. repec:hal:journl:hal-04672521 is not listed on IDEAS
    7. Joshua C. C. Chan & Liana Jacobi & Dan Zhu, 2019. "How Sensitive Are VAR Forecasts to Prior Hyperparameters? An Automated Sensitivity Analysis," Advances in Econometrics, in: Topics in Identification, Limited Dependent Variables, Partial Observability, Experimentation, and Flexible Modeling: Part A, volume 40, pages 229-248, Emerald Group Publishing Limited.
    8. Seok Young Hong & Oliver Linton & Hui Jun Zhang, 2014. "Multivariate variance ratio statistics," CeMMAP working papers 29/14, Institute for Fiscal Studies.
    9. O. J. Boxma & E. J. Cahen & D. Koops & M. Mandjes, 2019. "Linear Stochastic Fluid Networks: Rare-Event Simulation and Markov Modulation," Methodology and Computing in Applied Probability, Springer, vol. 21(1), pages 125-153, March.
    10. Loperfido, Nicola, 2014. "Linear transformations to symmetry," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 186-192.
    11. Arismendi, J.C., 2013. "Multivariate truncated moments," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 41-75.
    12. Shriram Srinivasan & Nishant Panda, 2023. "What is the gradient of a scalar function of a symmetric matrix?," Indian Journal of Pure and Applied Mathematics, Springer, vol. 54(3), pages 907-919, September.
    13. Reinaldo B. Arellano-Valle & Adelchi Azzalini, 2022. "Some properties of the unified skew-normal distribution," Statistical Papers, Springer, vol. 63(2), pages 461-487, April.
    14. Bollerslev, Tim & Patton, Andrew J. & Quaedvlieg, Rogier, 2018. "Modeling and forecasting (un)reliable realized covariances for more reliable financial decisions," Journal of Econometrics, Elsevier, vol. 207(1), pages 71-91.
    15. Dante Amengual & Gabriele Fiorentini & Enrique Sentana, 2022. "Tests for Random Coefficient Variation in Vector Autoregressive Models," Advances in Econometrics, in: Essays in Honour of Fabio Canova, volume 44, pages 1-35, Emerald Group Publishing Limited.
    16. Dlugoszek, Grzegorz R., 2016. "Solving DSGE portfolio choice models with asymmetric countries," SFB 649 Discussion Papers 2016-009, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    17. Savas Papadopoulos, 2010. "Theory and methodology for dynamic panel data: tested by simulations based on financial data," International Journal of Computational Economics and Econometrics, Inderscience Enterprises Ltd, vol. 1(3/4), pages 239-253.
    18. Symeonides Spyridon D. & Karavias Yiannis & Tzavalis Elias, 2017. "Size corrected Significance Tests in Seemingly Unrelated Regressions with Autocorrelated Errors," Journal of Time Series Econometrics, De Gruyter, vol. 9(1), pages 1-41, January.
    19. Seok Young Hong & Oliver Linton & Hui Jun Zhang, 2014. "Multivariate Variance Ratio Statistics," Cambridge Working Papers in Economics 1459, Faculty of Economics, University of Cambridge.
    20. P. C. B. Phillips & S. N. Durlauf, 1986. "Multiple Time Series Regression with Integrated Processes," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 53(4), pages 473-495.
    21. Jiang, Bo & Tian, Yongge, 2017. "Rank/inertia approaches to weighted least-squares solutions of linear matrix equations," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 400-413.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:198:y:2023:i:1:d:10.1007_s10957-023-02247-8. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.