Research Article
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Elliptic Matrix Representations of Elliptic Biquaternions and Their Applications

Year 2018, , 96 - 103, 30.11.2018
https://doi.org/10.36890/iejg.545136

Abstract

In this study, we obtain the 4 x 4 elliptic matrix representations of elliptic biquaternions with the
aid of the left and right Hamilton operators. Afterwards, we show that the space of 4 x 4 matrices
generated by left Hamilton operator is isomorphic to the space of elliptic biquaternions. Then, we
study the De-Moivre’s and Euler formulas for the matrices of this matrix space. Additionally, the
powers of these matrices are obtained with the aid of the De-Moivre’s formula.

References

  • [1] van der Waerden, B. L., Hamilton’s discovery of quaternions. Math. Mag. 49 (1976), no. 5, 227-234.
  • [2] Zhang, F., Quaternions and matrices of quaternions. Linear Algebra and its Applications 251 (1997), 21-57.
  • [3] Grob, J., Trenkler, G. and Troschke, S.-O., Quaternions: further contributions to a matrix oriented approach. Linear Algebra and its Applications 326 (2001), 205-213.
  • [4] Farebrother, R.W., Grob, J. and Troschke, S.-O., Matrix representation of quaternions. Linear Algebra and its Applications 362 (2003), 251-255. [5] Cho, E., De-Moivre’s formula for Quaternions. Appl. Math. Lett. 11 (1998), no. 6, 33-35.
  • [6] Jafari, M., Mortazaasl, H. and Yaylı, Y., De Moivre’s formula for matrices of quaternions. JP Journal of Algebra, Number Theory and Applications 21 (2011), no. 1, 57-67.
  • [7] Hamilton, W. R., Lectures on quaternions. Hodges and Smith, Dublin, 1853.
  • [8] Jafari, M., On the matrix algebra of complex quaternions. Accepted for publication in TWMS Journal of Pure and Applied Mathematics (2016), DOI: 10.13140/RG.2.1.3565.2321.
  • [9] Agrawal, O. P., Hamilton operators and dual-number-quaternions in spatial kinematics. Mech. Mach. Theory 22 (1987), no. 6, 569-575.
  • [10] Yaylı, Y., Homothetic motions at E4. Mech. Mach. Theory 27 (1992), no. 3, 303-305.
  • [11] Güngör, M. A. and Sarduvan, M., A note on dual quaternions and matrices of dual quaternions. Scientia Magna 7 (2011), no. 1, 1-11.
  • [12] Kösal, H. H. and Tosun, M., Commutative quaternion matrices. Adv. Appl. Clifford Alg. 24 (2014), no. 3, 769-779.
  • [13] Akyi˜ git, M., Kösal, H. H. and Tosun, M., A Note on matrix representations of split quaternions. Journal of Advanced Research in Applied Mathematics 7 (2015), no. 2, 26-39.
  • [14] Özen, K. E. and Tosun, M., Elliptic biquaternion algebra. AIP Conf. Proc. 1926 (2018), 020032-1–020032-6, https://doi.org/10.1063/1.5020481.
  • [15] Özen, K. E. and Tosun, M., A note on elliptic biquaternions. AIP Conf. Proc. 1926 (2018), 020033-1–020033-6, https://doi.org/10.1063/1.5020482.
  • [16] Özen, K. E. and Tosun, M., p-Trigonometric approach to elliptic biquaternions. Adv. Appl. Clifford Alg. 28:62 (2018), https://doi.org/10.1007/s00006-018-0878-3.
  • [17] Harkin, A. A. and Harkin, J. B., Geometry of generalized complex numbers. Math. Mag. 77 (2004), no. 2, 118-129.
  • [18] Kösal, H. H., On commutative quaternion matrices. Sakarya University, Graduate School of Natural and Applied Sciences, Sakarya, Ph.D.
Year 2018, , 96 - 103, 30.11.2018
https://doi.org/10.36890/iejg.545136

Abstract

References

  • [1] van der Waerden, B. L., Hamilton’s discovery of quaternions. Math. Mag. 49 (1976), no. 5, 227-234.
  • [2] Zhang, F., Quaternions and matrices of quaternions. Linear Algebra and its Applications 251 (1997), 21-57.
  • [3] Grob, J., Trenkler, G. and Troschke, S.-O., Quaternions: further contributions to a matrix oriented approach. Linear Algebra and its Applications 326 (2001), 205-213.
  • [4] Farebrother, R.W., Grob, J. and Troschke, S.-O., Matrix representation of quaternions. Linear Algebra and its Applications 362 (2003), 251-255. [5] Cho, E., De-Moivre’s formula for Quaternions. Appl. Math. Lett. 11 (1998), no. 6, 33-35.
  • [6] Jafari, M., Mortazaasl, H. and Yaylı, Y., De Moivre’s formula for matrices of quaternions. JP Journal of Algebra, Number Theory and Applications 21 (2011), no. 1, 57-67.
  • [7] Hamilton, W. R., Lectures on quaternions. Hodges and Smith, Dublin, 1853.
  • [8] Jafari, M., On the matrix algebra of complex quaternions. Accepted for publication in TWMS Journal of Pure and Applied Mathematics (2016), DOI: 10.13140/RG.2.1.3565.2321.
  • [9] Agrawal, O. P., Hamilton operators and dual-number-quaternions in spatial kinematics. Mech. Mach. Theory 22 (1987), no. 6, 569-575.
  • [10] Yaylı, Y., Homothetic motions at E4. Mech. Mach. Theory 27 (1992), no. 3, 303-305.
  • [11] Güngör, M. A. and Sarduvan, M., A note on dual quaternions and matrices of dual quaternions. Scientia Magna 7 (2011), no. 1, 1-11.
  • [12] Kösal, H. H. and Tosun, M., Commutative quaternion matrices. Adv. Appl. Clifford Alg. 24 (2014), no. 3, 769-779.
  • [13] Akyi˜ git, M., Kösal, H. H. and Tosun, M., A Note on matrix representations of split quaternions. Journal of Advanced Research in Applied Mathematics 7 (2015), no. 2, 26-39.
  • [14] Özen, K. E. and Tosun, M., Elliptic biquaternion algebra. AIP Conf. Proc. 1926 (2018), 020032-1–020032-6, https://doi.org/10.1063/1.5020481.
  • [15] Özen, K. E. and Tosun, M., A note on elliptic biquaternions. AIP Conf. Proc. 1926 (2018), 020033-1–020033-6, https://doi.org/10.1063/1.5020482.
  • [16] Özen, K. E. and Tosun, M., p-Trigonometric approach to elliptic biquaternions. Adv. Appl. Clifford Alg. 28:62 (2018), https://doi.org/10.1007/s00006-018-0878-3.
  • [17] Harkin, A. A. and Harkin, J. B., Geometry of generalized complex numbers. Math. Mag. 77 (2004), no. 2, 118-129.
  • [18] Kösal, H. H., On commutative quaternion matrices. Sakarya University, Graduate School of Natural and Applied Sciences, Sakarya, Ph.D.
There are 17 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

Kahraman Esen Özen This is me

Murat Tosun

Publication Date November 30, 2018
Published in Issue Year 2018

Cite

APA Özen, K. E., & Tosun, M. (2018). Elliptic Matrix Representations of Elliptic Biquaternions and Their Applications. International Electronic Journal of Geometry, 11(2), 96-103. https://doi.org/10.36890/iejg.545136
AMA Özen KE, Tosun M. Elliptic Matrix Representations of Elliptic Biquaternions and Their Applications. Int. Electron. J. Geom. November 2018;11(2):96-103. doi:10.36890/iejg.545136
Chicago Özen, Kahraman Esen, and Murat Tosun. “Elliptic Matrix Representations of Elliptic Biquaternions and Their Applications”. International Electronic Journal of Geometry 11, no. 2 (November 2018): 96-103. https://doi.org/10.36890/iejg.545136.
EndNote Özen KE, Tosun M (November 1, 2018) Elliptic Matrix Representations of Elliptic Biquaternions and Their Applications. International Electronic Journal of Geometry 11 2 96–103.
IEEE K. E. Özen and M. Tosun, “Elliptic Matrix Representations of Elliptic Biquaternions and Their Applications”, Int. Electron. J. Geom., vol. 11, no. 2, pp. 96–103, 2018, doi: 10.36890/iejg.545136.
ISNAD Özen, Kahraman Esen - Tosun, Murat. “Elliptic Matrix Representations of Elliptic Biquaternions and Their Applications”. International Electronic Journal of Geometry 11/2 (November 2018), 96-103. https://doi.org/10.36890/iejg.545136.
JAMA Özen KE, Tosun M. Elliptic Matrix Representations of Elliptic Biquaternions and Their Applications. Int. Electron. J. Geom. 2018;11:96–103.
MLA Özen, Kahraman Esen and Murat Tosun. “Elliptic Matrix Representations of Elliptic Biquaternions and Their Applications”. International Electronic Journal of Geometry, vol. 11, no. 2, 2018, pp. 96-103, doi:10.36890/iejg.545136.
Vancouver Özen KE, Tosun M. Elliptic Matrix Representations of Elliptic Biquaternions and Their Applications. Int. Electron. J. Geom. 2018;11(2):96-103.