Elliptic Matrix Representations of Elliptic Biquaternions and Their Applications

Kahraman Esen Özen; Murat Tosun

doi:10.36890/iejg.545136

Research Article

Elliptic Matrix Representations of Elliptic Biquaternions and Their Applications

Year 2018, Volume: 11 Issue: 2, 96 - 103, 30.11.2018

Kahraman Esen Özen Murat Tosun

https://doi.org/10.36890/iejg.545136

Cited By: 5

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.

Keywords

Complex quaternions, Hamilton matrices, Matrix representations, Elliptic biquaternions

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, Volume: 11 Issue: 2, 96 - 103, 30.11.2018

Kahraman Esen Özen Murat Tosun

https://doi.org/10.36890/iejg.545136

Cited By: 5

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 Volume: 11 Issue: 2

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.