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Commuting Outer Inverse-Based Solutions to the Yang–Baxter-like Matrix Equation

Author

Listed:
  • Ashim Kumar

    (Department of Mathematical Sciences, I.K. Gujral Punjab Technical University Jalandhar, Kapurthala 144603, India)

  • Dijana Mosić

    (Faculty of Sciences and Mathematics, University of Niš, 18000 Niš, Serbia)

  • Predrag S. Stanimirović

    (Faculty of Sciences and Mathematics, University of Niš, 18000 Niš, Serbia
    Laboratory “Hybrid Methods of Modelling and Optimization in Complex Systems”, Siberian Federal University, Prosp. Svobodny 79, 660041 Krasnoyarsk, Russia)

  • Gurjinder Singh

    (Department of Mathematical Sciences, I.K. Gujral Punjab Technical University Jalandhar, Kapurthala 144603, India)

  • Lev A. Kazakovtsev

    (Laboratory “Hybrid Methods of Modelling and Optimization in Complex Systems”, Siberian Federal University, Prosp. Svobodny 79, 660041 Krasnoyarsk, Russia)

Abstract

This paper investigates new solution sets for the Yang–Baxter-like (YB-like) matrix equation involving constant entries or rational functional entries over complex numbers. Towards this aim, first, we introduce and characterize an essential class of generalized outer inverses (termed as { 2 , 5 } -inverses) of a matrix, which commute with it. This class of { 2 , 5 } -inverses is defined based on resolving appropriate matrix equations and inner inverses. In general, solutions to such matrix equations represent optimization problems and require the minimization of corresponding matrix norms. We decided to analytically extend the obtained results to the derivation of explicit formulae for solving the YB-like matrix equation. Furthermore, algorithms for computing the solutions are developed corresponding to the suggested methods in some computer algebra systems. The main features of the proposed approach are highlighted and illustrated by numerical experiments.

Suggested Citation

  • Ashim Kumar & Dijana Mosić & Predrag S. Stanimirović & Gurjinder Singh & Lev A. Kazakovtsev, 2022. "Commuting Outer Inverse-Based Solutions to the Yang–Baxter-like Matrix Equation," Mathematics, MDPI, vol. 10(15), pages 1-16, August.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:15:p:2738-:d:878671
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    References listed on IDEAS

    as
    1. Duanmei Zhou & Jiawen Ding, 2020. "All Solutions of the Yang–Baxter-Like Matrix Equation for Nilpotent Matrices of Index Two," Complexity, Hindawi, vol. 2020, pages 1-7, June.
    2. Stanimirović, Predrag S. & Ćirić, Miroslav & Lastra, Alberto & Sendra, Juan Rafael & Sendra, Juana, 2021. "Representations and symbolic computation of generalized inverses over fields," Applied Mathematics and Computation, Elsevier, vol. 406(C).
    3. Predrag S. Stanimirović & Miroslav Ćirić & Igor Stojanović & Dimitrios Gerontitis, 2017. "Conditions for Existence, Representations, and Computation of Matrix Generalized Inverses," Complexity, Hindawi, vol. 2017, pages 1-27, June.
    4. 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.
    Full references (including those not matched with items on IDEAS)

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