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Year 2018, Volume: 1 Issue: 1, 20 - 27, 14.12.2018

Abstract

References

  • [1] B. L. van der Waerden, Hamilton’s discovery of quaternions, Math. Mag., 49(5) (1976), 227-234.
  • [2] W. R. Hamilton, Lectures on quaternions, Hodges and Smith, Dublin, 1853.
  • [3] T. Y. Lam, The algebraic theory of quadratic forms, Benjamin, Newyork, 1973.
  • [4] M. L. Mehta, Matrix theory, selected topics and useful results, Hindustan P. Co., India, 1989.
  • [5] R. S. Pierce, Associative algebras, Springer-Verlag, Newyork, 1982.
  • [6] B. L. van der Waerden, A history of algebra from al-Khwarizmi to Emmy Noether, Springer-Verlag, Newyork, 1985.
  • [7] M. Bekar, Y. Yaylı, Involutions of complexified quaternions and split quaternions, Adv. Appl. Clifford Algebr., 23(2) (2013), 283-299.
  • [8] M. A. Güngör, M. Sarduvan, A note on dual quaternions and matrices of dual quaternions, Sci. Magna, 7(1) (2011), 1-11.
  • [9] I. A. Kösal, A note on hyperbolic quaternions, Univ. J. Math. Appl., 1(3) (2018), 155-159.
  • [10] G. Cerda-Morales, On fourth-order jacobsthal quaternions, J. Math. Sci. Model., 1(2) (2018), 73-79.
  • [11] Y. Tian, Universal similarity factorization equalities over real clifford algebras, Adv. Appl. Clifford Algebr., 8(2) (1998), 365-402.
  • [12] Y. Tian, Universal factorization equalities for quaternion matrices and their applications, Math. J. Okoyama Univ., 41(1) (1999), 45-62.
  • [13] Y. Tian, Universal similarity factorization equalities over generalized clifford algebras, Acta Math. Sin., 22(1) (2006), 289-300.
  • [14] Y. Tian, Biquaternions and their complex matrix representations, Beitrage Algebra Geom., 54(2) (2013), 575-592.
  • [15] H. H. Kösal, M. Tosun, Universal similarity factorization equalities for commutative quaternions and their matrices, Linear Multilinear Algebra, (2018), doi:10.1080/03081087.2018.1439878.
  • [16] K. E. Özen, M. Tosun, Elliptic biquaternion algebra, AIP Conf. Proc., 1926 (2018), 020032-1–020032-6.
  • [17] K. E. Özen, M. Tosun, A note on elliptic biquaternions, AIP Conf. Proc., 1926 (2018), 020033-1–020033-6.
  • [18] K. E. Özen, M. Tosun, p-Trigonometric approach to elliptic biquaternions, Adv. Appl. Clifford Algebr., (2018), doi:10.1007/s00006-018-0878-3.
  • [19] K. E. Özen, M. Tosun, Elliptic matrix representations of elliptic biquaternions and their applications, Int. Electron. J. Geom., 11(2) (2018), 96-103.
  • [20] K. E. Özen, M. Tosun, On the matrix algebra of elliptic biquaternions, (Submitted Article).
  • [21] H. H. Kösal, On Commutative quaternion matrices, Ph.D. Thesis, Sakarya University, 2016.
  • [22] A. Ben-Israel, T. N. E. Greville, Generalized inverses: theory and applications, Springer-Verlag, Newyork, 2003.

Further Results for Elliptic Biquaternions

Year 2018, Volume: 1 Issue: 1, 20 - 27, 14.12.2018

Abstract

In this study, we show that the elliptic biquaternion algebra is algebraically isomorphic to the $2\times 2$ total elliptic matrix algebra and so, we get a faithful $2\times 2$ elliptic matrix representation of an elliptic biquaternion. Also, we investigate the similarity and the Moore-Penrose inverses of elliptic biquaternions by means of these matrix representations. Moreover, we establish universal similarity factorization equality (USFE) over the elliptic biquaternion algebra which reveals a deeper relationship between an elliptic biquaternion and its elliptic matrix representation. This equality and these representations can serve as useful tools for discussing many problems concerned with the elliptic biquaternions, especially for solving various elliptic biquaternion equations.

References

  • [1] B. L. van der Waerden, Hamilton’s discovery of quaternions, Math. Mag., 49(5) (1976), 227-234.
  • [2] W. R. Hamilton, Lectures on quaternions, Hodges and Smith, Dublin, 1853.
  • [3] T. Y. Lam, The algebraic theory of quadratic forms, Benjamin, Newyork, 1973.
  • [4] M. L. Mehta, Matrix theory, selected topics and useful results, Hindustan P. Co., India, 1989.
  • [5] R. S. Pierce, Associative algebras, Springer-Verlag, Newyork, 1982.
  • [6] B. L. van der Waerden, A history of algebra from al-Khwarizmi to Emmy Noether, Springer-Verlag, Newyork, 1985.
  • [7] M. Bekar, Y. Yaylı, Involutions of complexified quaternions and split quaternions, Adv. Appl. Clifford Algebr., 23(2) (2013), 283-299.
  • [8] M. A. Güngör, M. Sarduvan, A note on dual quaternions and matrices of dual quaternions, Sci. Magna, 7(1) (2011), 1-11.
  • [9] I. A. Kösal, A note on hyperbolic quaternions, Univ. J. Math. Appl., 1(3) (2018), 155-159.
  • [10] G. Cerda-Morales, On fourth-order jacobsthal quaternions, J. Math. Sci. Model., 1(2) (2018), 73-79.
  • [11] Y. Tian, Universal similarity factorization equalities over real clifford algebras, Adv. Appl. Clifford Algebr., 8(2) (1998), 365-402.
  • [12] Y. Tian, Universal factorization equalities for quaternion matrices and their applications, Math. J. Okoyama Univ., 41(1) (1999), 45-62.
  • [13] Y. Tian, Universal similarity factorization equalities over generalized clifford algebras, Acta Math. Sin., 22(1) (2006), 289-300.
  • [14] Y. Tian, Biquaternions and their complex matrix representations, Beitrage Algebra Geom., 54(2) (2013), 575-592.
  • [15] H. H. Kösal, M. Tosun, Universal similarity factorization equalities for commutative quaternions and their matrices, Linear Multilinear Algebra, (2018), doi:10.1080/03081087.2018.1439878.
  • [16] K. E. Özen, M. Tosun, Elliptic biquaternion algebra, AIP Conf. Proc., 1926 (2018), 020032-1–020032-6.
  • [17] K. E. Özen, M. Tosun, A note on elliptic biquaternions, AIP Conf. Proc., 1926 (2018), 020033-1–020033-6.
  • [18] K. E. Özen, M. Tosun, p-Trigonometric approach to elliptic biquaternions, Adv. Appl. Clifford Algebr., (2018), doi:10.1007/s00006-018-0878-3.
  • [19] K. E. Özen, M. Tosun, Elliptic matrix representations of elliptic biquaternions and their applications, Int. Electron. J. Geom., 11(2) (2018), 96-103.
  • [20] K. E. Özen, M. Tosun, On the matrix algebra of elliptic biquaternions, (Submitted Article).
  • [21] H. H. Kösal, On Commutative quaternion matrices, Ph.D. Thesis, Sakarya University, 2016.
  • [22] A. Ben-Israel, T. N. E. Greville, Generalized inverses: theory and applications, Springer-Verlag, Newyork, 2003.
There are 22 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Kahraman Esen Özen 0000-0002-3299-6709

Murat Tosun 0000-0002-4888-1412

Publication Date December 14, 2018
Acceptance Date November 19, 2018
Published in Issue Year 2018 Volume: 1 Issue: 1

Cite

APA Özen, K. E., & Tosun, M. (2018). Further Results for Elliptic Biquaternions. Conference Proceedings of Science and Technology, 1(1), 20-27.
AMA Özen KE, Tosun M. Further Results for Elliptic Biquaternions. Conference Proceedings of Science and Technology. December 2018;1(1):20-27.
Chicago Özen, Kahraman Esen, and Murat Tosun. “Further Results for Elliptic Biquaternions”. Conference Proceedings of Science and Technology 1, no. 1 (December 2018): 20-27.
EndNote Özen KE, Tosun M (December 1, 2018) Further Results for Elliptic Biquaternions. Conference Proceedings of Science and Technology 1 1 20–27.
IEEE K. E. Özen and M. Tosun, “Further Results for Elliptic Biquaternions”, Conference Proceedings of Science and Technology, vol. 1, no. 1, pp. 20–27, 2018.
ISNAD Özen, Kahraman Esen - Tosun, Murat. “Further Results for Elliptic Biquaternions”. Conference Proceedings of Science and Technology 1/1 (December 2018), 20-27.
JAMA Özen KE, Tosun M. Further Results for Elliptic Biquaternions. Conference Proceedings of Science and Technology. 2018;1:20–27.
MLA Özen, Kahraman Esen and Murat Tosun. “Further Results for Elliptic Biquaternions”. Conference Proceedings of Science and Technology, vol. 1, no. 1, 2018, pp. 20-27.
Vancouver Özen KE, Tosun M. Further Results for Elliptic Biquaternions. Conference Proceedings of Science and Technology. 2018;1(1):20-7.