2023 Volume 13 Issue 5
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Mohammed Ahmed Adam Abdalrahman, Jiu Ding, Qianglian Huang. SOLVING THE YANG-BAXTER-LIKE MATRIX EQUATION FOR A RANK-ONE MATRIX[J]. Journal of Applied Analysis & Computation, 2023, 13(5): 2987-2994. doi: 10.11948/20230115
Citation: Mohammed Ahmed Adam Abdalrahman, Jiu Ding, Qianglian Huang. SOLVING THE YANG-BAXTER-LIKE MATRIX EQUATION FOR A RANK-ONE MATRIX[J]. Journal of Applied Analysis & Computation, 2023, 13(5): 2987-2994. doi: 10.11948/20230115

SOLVING THE YANG-BAXTER-LIKE MATRIX EQUATION FOR A RANK-ONE MATRIX

  • We reduce the problem of solving the Yang-Baxter-like matrix equation $ AXA = XAX $, where $ A $ is a rank-one matrix, to that of solving linear matrix equations, obtaining all solutions. We use a direct and unified approach for the both cases that $ A $ is diagonalizable or otherwise, instead of seeking the help of the Jordan canonical form or factorization of $ A $. Based on the characterizations for the solutions, we derive a perturbation result when $ A $ is not diagonalizable.

    MSC: 15A18
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  • [1] M. Adam, J. Ding, Q. Huang and L. Zhu, Solving a class of quadratic matrix equations, Applied Math. Lett., 2018, 82, 58–63. doi: 10.1016/j.aml.2018.02.017

    CrossRef Google Scholar

    [2] R. J. Baxter, Partition function of the eight-vertex lattice model, Ann. Phys., 1972, 70,193–228. doi: 10.1016/0003-4916(72)90335-1

    CrossRef Google Scholar

    [3] S. L. Campbell and C. D. Meyer, Generalized Inverses of Linear Transformations, Pitman, 1979.

    Google Scholar

    [4] J. Ding and N. Rhee, Spectral solutions of the Yang-Baxter matrix equation, J. Math. Anal. Appl., 2013,402,567–573. doi: 10.1016/j.jmaa.2013.01.054

    CrossRef Google Scholar

    [5] J. Ding, C. Zhang and N. Rhee, Commuting solutions of the Yang-Baxter matrix equation, Applied Math. Lett., 2015, 44, 1–4. doi: 10.1016/j.aml.2014.11.017

    CrossRef Google Scholar

    [6] Q. Dong and J. Ding, Complete commuting solutions of the Yang-Baxter-like matrix equation for diagonalizable matrices, Computers Math. Appl., 2016, 72(1), 194–201. doi: 10.1016/j.camwa.2016.04.047

    CrossRef Google Scholar

    [7] Q. Dong and J. Ding, All commuting solutions of a quadratic matrix equation for general matrices, J. Nonlinear Modeling Anal., 2020, 2(1), 111–123.

    Google Scholar

    [8] F. Felix, Nonlinear Equations, Quantum Groups and Duality Theorems: A Primer on the Yang-Baxter Equation, VDM Verlag, 2009.

    Google Scholar

    [9] Q. Huang, M. Adam, J. Ding and L. Zhu, All non-commuting solutions of the Yang-Baxter matrix equation for a class of diagonalizable matrices, Operators and Matrices, 2019, 1,187–195.

    Google Scholar

    [10] L. Lu, Manifold expressions of all solutions of the Yang-Baxter-like matrix equation for rank-one matrices, Appl. Math. Lett., 2022, 108175.

    Google Scholar

    [11] H. Mukherjee and A. M, Solutions to the matrix Yang-Baxter equation, arXiv: 2209. 04605, 2022.

    Google Scholar

    [12] D. Shen and M. Wei, All solutions of the Yang-Baxter-like matrix equation for diagonalizable coefficient matrix with two different eigenvalues, Applied Math. Lett., 2020,101, 106048. doi: 10.1016/j.aml.2019.106048

    CrossRef Google Scholar

    [13] D. Shen, M. Wei and Z. Jia, On commuting solutions of the Yang-Baxter-like matrix equation, J. Math. Anal. Appl., 2018,462(1), 665–696. doi: 10.1016/j.jmaa.2018.02.030

    CrossRef Google Scholar

    [14] H. Tian, All solutions of the Yang-Baxter-like matrix equation for rank-one matrices, Appl. Math. Lett., 2016, 51, 55–59. doi: 10.1016/j.aml.2015.07.009

    CrossRef Google Scholar

    [15] C. Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett., 1967, 19, 1312–1315. doi: 10.1103/PhysRevLett.19.1312

    CrossRef Google Scholar

    [16] C. Yang and M. Ge, Braid Group, Knot Theory, and Statistical Mechanics, World Scientific, 1989.

    Google Scholar

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