On the Solutions of Linear Elliptic Biquaternion Equations

Year 2021, Volume: 3 Issue: 1, 9 - 15, 30.08.2021

Kahraman Esen Özen

Abstract

The real and complex quaternion algebras are isomorphic to real matrix algebras including the special types 4x4 and 8x8 real matrices, respectively. These situations are based on the fact that a finite dimensional associative algebra L over any field K is isomorphic to a subalgebra of M_n(K) where dimension of L equals n over the field K. Considering this fact and using the left Hamilton operator, we get 8x8 real matrix representations of elliptic biquaternions in this study. Then a numerical method is developed to solve the linear elliptic biquaternion equations with the aid of the aforesaid representations. Also, an illustrative example and an algorithm are provided to show how this method works.

Keywords

Field of elliptic numbers, Quaternion equation, Real representation, Solution

References

• Van der Waerden, B. L. (1976). Hamilton’s discovery of quaternions. Math. Mag., 49(5), 227-234.
• Hamilton, W. R. (1853). Lectures on Quaternions. Hodges and Smith, Dublin.
• Flaut, C., & Shpakivskyi, V. (2013). Real matrix representations for the complex quaternions. Adv. Appl. Clifford Algebras, 23(3), 657-671.
• Tian, Y. (2013). Biquaternions and their complex matrix representations. Beitr. Algebra Geom., 54(2), 575-592.
• Johnson, R. E. (1944). On the equation ca = gc +b over algebraic division ring. Bull. Amer. Math. Soc., 50(4), 202-207.
• Longxuan, C. (1991). Definition of determinant and Cramer solutions over the quaternion field. Acta Mathematica Sinica, 7(2), 171-180.
• Shpakivskyi, V. S. (2011). Linear quaternionic equations and their systems. Adv. Appl. Clifford Algebras, 21(3), 637-645.
• Özen, K. E., & Tosun, M. (2018). Elliptic biquaternion algebra. AIP Conf. Proc. 1926, 020032.
• Yu, C. E., Liu, X., & Zhang, Y. (2021). On Elliptic Biquaternion Matrices. Adv. Appl. Clifford Algebras,31(1), 1-14.
• Derin, Z. & Güngör, M. A. (2020). On Lorentz transformations with elliptic biquaternions. In: Hvedri, I. (ed.) TBILISI-MATHEMATICS, pp 121-140. Sciendo, Berlin.
• Özen, K. E. (2020). A general method for solving linear matrix equations of elliptic biquaternions with applications. AIMS Mathematics, 5(3), 2211-2225.
• Özen, K. E., & Tosun, M. (2021). A general method for solving linear elliptic biquaternion equations. Complex Variables and Elliptic Equations, 66(4), 708-719.
• Özen, K. E., & Tosun, M. (2018). Elliptic matrix representations of elliptic biquaternions and their applications, International Electronic Journal of Geometry, 11(2), 96-103.
• Kösal, H. H. (2016). On commutative quaternion matrices, Sakarya University, Graduate School of Natural and Applied Sciences, Sakarya, Ph.D. Thesis.