This paper aims to present, in a unified manner, algebraic techniques for linear equations which are valid on both the algebras of quaternions and split quaternions. This paper, introduces a concept of v-quaternion, studies the problem of v-quaternionic linear equations by means of a complex representation and a real representation of v-quaternion matrices, and gives two algebraic methods for solving v-quaternionic linear equations. This paper also gives a unification of algebraic techniques for quaternionic and split quaternionic linear equations in quaternionic and split quaternionic mechanics.
References
[1]
Hamilton, W.R. (1843) On a New Species of Imaginary Quantities Connected with a Theory of Quaternions. Proceedings of the Royal Irish Academy, 2, 424-434.
[2]
Özdemir, M. and Ergin, A.A. (2006) Rotations with Unit Timelike Quaternions in Minkowski 3-Space. Journal of Geometry and Physics, 56, 322-336.
https://doi.org/10.1016/j.geomphys.2005.02.004
[3]
Schmidt, J. and Nieman, H. (2001) Using Quaternions for Parametrizing 3-D Rotations in Unconstrained Nonlinear Optimization. Vision Modeling, and Visualization, Stuttgart, 21-23 November 2001, 399-406.
[4]
Cockle, J. (1849) On Systems of Algebra Involving More than One Imaginary and on Equations of the Fifth Degree. Philosophical Magazine, 35, 434-437.
https://doi.org/10.1080/14786444908646384
[5]
Kula, L. and Yayli, Y. (2007) Split Quaternions and Rotations in Semi Euclidean Space . Journal of the Korean Mathematical Society, 44, 1313-1327.
https://doi.org/10.4134/JKMS.2007.44.6.1313
[6]
Özkaldi, S. and Gündogan, H. (2011) Dual Split Quaternions and Screw Motion in 3-Dimensional Lorentzian Space. Advances in Applied Clifford Algebras, 21, 193-202. https://doi.org/10.1007/s00006-010-0236-6
[7]
Özdemir, M., Erdogdu, M. and Simsek, H. (2014) On the Eigenvalues and Eigenvectors of a Lorentzian Rotation Matrix by Using Split Quaternions. Advances in Applied Clifford Algebras, 24, 179-192. https://doi.org/10.1007/s00006-013-0424-2
[8]
Adler, S.L. (1995) Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press, New York.
[9]
Özdemir, M. and Ergin, A.A. (2005) Some Geometric Applications of Timelike Quaternions. International Conference of the Jangjeon Mathematical Society, Vol. 16, 108-115.
[10]
Ramis, C. and Yayli, Y. (2013) Dual Split Quaternions and Chasles’ Theorem in 3-Dimensional Minkowski Space E13. Advances in Applied Clifford Algebras, 23, 951-964. https://doi.org/10.1007/s00006-013-0405-5
[11]
Aslan, S. and Yayli, Y. (2016) Split Quaternions and Canal Surfaces in Minkowski 3-Space. International Journal of Geometry, 5, 51-61.
[12]
Brody, D.C. and Graefe, E.M. (2011) On Complexified Mechanics and Coquaternions. Journal of Physics A: Mathematical and Theoretical, 44, Article ID: 072001.
https://doi.org/10.1088/1751-8113/44/7/072001
[13]
Janovská, D. and Opfer, G. (2013) Linear Equations and the Kronecker Product in Coquaternions. Mitteilungen der Mathematische Gesellschaft in Hamburg, 33, 181-196.
[14]
Jiang, T. (2004) An Algorithm for Quaternionic Linear Equations in Quaternionic Quantum Theory. Journal of Mathematical Physics, 45, 4218-4222.
https://doi.org/10.1063/1.1794368
[15]
Zhang, Z., Jiang, Z. and Jiang, T. (2015) Algebraic Methods for Least Squares Problem in Split Quaternionic Mechanics. Applied Mathematics and Computation, 269, 618-625. https://doi.org/10.1016/j.amc.2015.07.072
