Research article

A general method for solving linear matrix equations of elliptic biquaternions with applications

  • Received: 05 November 2019 Accepted: 21 February 2020 Published: 28 February 2020
  • MSC : 11R52, 15B33, 15A24

  • In this study, we obtain the real representations of elliptic biquaternion matrices. Afterwards, with the aid of these representations, we develop a general method to solve the linear matrix equations over the elliptic biquaternion algebra. Also we apply this method to the well known matrix equations X - AXB = C and AX - XB = C over the elliptic biquaternion algebra. Then, we give some illustrative numerical examples to show how the aforementioned method and its results work. Furthermore, we provide numerical algorithms for all the problems considered in this paper. Elliptic biquaternions are generalized form of complex quaternions and so real quaternions. This relation is valid for their matrices, as well. Thus, the obtained results extend, generalize and complement some known results from the literature.

    Citation: Kahraman Esen Özen. A general method for solving linear matrix equations of elliptic biquaternions with applications[J]. AIMS Mathematics, 2020, 5(3): 2211-2225. doi: 10.3934/math.2020146

    Related Papers:

  • In this study, we obtain the real representations of elliptic biquaternion matrices. Afterwards, with the aid of these representations, we develop a general method to solve the linear matrix equations over the elliptic biquaternion algebra. Also we apply this method to the well known matrix equations X - AXB = C and AX - XB = C over the elliptic biquaternion algebra. Then, we give some illustrative numerical examples to show how the aforementioned method and its results work. Furthermore, we provide numerical algorithms for all the problems considered in this paper. Elliptic biquaternions are generalized form of complex quaternions and so real quaternions. This relation is valid for their matrices, as well. Thus, the obtained results extend, generalize and complement some known results from the literature.


    加载中


    [1] B. L. van der Waerden, Hamilton's discovery of quaternions, Math. Mag., 49 (1976), 227–234.
    [2] L. A. Wolf, Similarity of matrices in which the elements are real quaternions, Bull. Amer. Math. Soc., 42 (1936), 737–743.
    [3] Y. Tian, Universal factorization equalities for quaternion matrices and their applications, Mathematical Journal of Okayama University, 41 (1999), 45–62.
    [4] C. Song, G. Chen, Q. Liu, Explicit solutions to the quaternion matrix equations X-AXF=C and X-A$\tilde{X}$F=C, Int. J. Comput. Math., 89 (2012), 890–900.
    [5] C. Song, G. Chen, On solutions of matrix equation XF-AX=C and XF-A$\tilde{X}$=C over quaternion field, J. Appl. Math. Comput., 37 (2011), 57–68.
    [6] Q. W. Wang, J. W. Van der Woude, H. X. Chang, A system of real quaternion matrix equations with applications, Linear Algebra Appl., 431 (2009), 2291–2303.
    [7] Z. H. He, Q. W. Wang, A real quaternion matrix equation with applications, Linear Multilinear A., 61 (2013), 725–740.
    [8] F. Zhang, M. Wei, Y. Li, et al. Special least squares solutions of the quaternion matrix equation AX=B with applications, Appl. Math. Comput., 270 (2015), 425–433.
    [9] F. Zhang, W. Mu, Y. Li, et al. Special least squares solutions of the quaternion matrix equation AXB+CXD=E with applications, Comput. Math. Appl., 72 (2016), 1426–1435.
    [10] W. R. Hamilton, Lectures on quaternions, Dublin: Hodges and Smith, 1853.
    [11] Y. Tian, Biquaternions and their complex matrix representations, Beitr Algebra Geom., 54 (2013), 575–592.
    [12] Y. Huang, S. Zhang, Complex matrix decomposition and quadratic programming, Math. Oper. Res., 32 (2007), 758–768.
    [13]

    F. Zhang, M. Wei, Y. Li, et al. The minimal norm least squares Hermitian solution of the complex matrix equation AXB+CXD=E, J. Franklin I., 355 (2018), 1296–1310.

    [14] F. Zhang, M. Wei, Y. Li, et al. An efficient method for special least squares solution of the complex matrix equation (AXB, CXD)=(E, F), Comput. Math. Appl., 76 (2018), 2001–2010.
    [15] K. E. Özen, M. Tosun, Elliptic biquaternion algebra, AIP Conf. Proc., 1926 (2018), 020032.
    [16] K. E. Özen, M. Tosun, A note on elliptic biquaternions, AIP Conf. Proc., 1926 (2018), 020033.
    [17] K. E. Özen, M. Tosun, p-Trigonometric approach to elliptic biquaternions, Adv. Appl. Clifford Alg., 28 (2018), 62.
    [18] K. E. Özen, M. Tosun, Elliptic matrix representations of elliptic biquaternions and their applications, Int. Electron. J. Geom., 11 (2018), 96–103.
    [19] K. E. Özen, M. Tosun, Further results for elliptic biquaternions, Conference Proceedings of Science and Technology, 1 (2018), 20–27.
    [20] A. A. Harkin, J. B. Harkin, Geometry of generalized complex numbers, Math. Mag., 77 (2004), 118–129.
    [21] H. H. Kösal, On commutative quaternion matrices, Ph.D. Thesis, Sakarya: Sakarya University, 2016.
    [22] K. E. Özen, M. Tosun, On the matrix algebra of elliptic biquaternions, Math. Method. Appl. Sci., 2019.
    [23] I. M. Yaglom, Complex numbers in geometry, Newyork: Academic Press, 1968.
    [24] Y. Tian, Universal similarity factorication equalities over real Clifford algebras, Adv. Appl. Clifford Al., 8 (1998), 365–402.

  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2731) PDF downloads(316) Cited by(3)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog