Matrix Representation of Dual Quaternions

After a review of some properties of dual quaternions, De Moivre's and Euler's formulas for the matrices associated with these quaternions are studied. In special case, De Moivre's formula implies that there are uncountably many matrices of unit dual quaternions satisfying 4 n A I = for n≥3. Also; we give the relation between the powers of matrices of dual quaternions. 

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  • Adler, S. L.,“Quaternionic Quantum Mechanicsand Quantum Fields”, Oxford University Pressinc., New York, (1995).
  • Agrawal O. P.,“Hamilton Operatorsand Dual- number-quaternions in Spatial Kinematics”, Mech. Mach. Theory, 22 (1987) no.6, 569-575.
  • Ata, E.,Yayli, Y., “Dual Unitary Matrices and Unit Dual Dynamical System”, 10:1-12(2008). Geometry
  • Cho, E.,“De-Moivre Formula forQuaternions”, Appl. Math. Lett.,11(6): 33-35(1998). [5] Clifford,
  • W.,“Preliminary Sketch of
  • Biquaternions”, Proc. London Math. Soc.,4: 381- 395(1873).
  • Gungor, M.A., Sarduvan, M., “A Note on Dual Quaternions and Matrices of Dual Quaternions”, Scientia Magna, 7(1): 1-11(2011). [7] Gro, B.J., Trenkler, G., Troschke, S., “Quaternions: Futher Contributionsto a Matrix Oriented Approach”, Linear Algebra and its Appl., 326: 205-213(2001).
  • Jafari, M., Mortazaasl, H., Yayli, Y., “De Moivre's Formula for Matrices of Quaternions”, JP J. of Algebra, Number Theory and appl., 21(1):57-67 (2011).
  • Kabadayi, H., Yayli, Y., “De-Moivre's Formula for Dual Quaternions”, Kuwait J. Of Sci.&Tech., 38 (1):15-23(2011).
  • KotelNikov,A.P., “VintovoeSchislenie i Niko toriya Prilozheniyeevo k geometrie i mechaniki”, Kazan, (1895).
  • Ozdemir, M.,“The Roots of a Split Quaternion”, Applied Math. Lett., 22: 258-263(2009). [12] Study E.,“Von Den Bewegungenund Umlegungen”, Mathematische Annalen, 39: 441- 564(1891).
  • Ward,J.P., “Quaternions and Cayley Numbers Algebra and Applications”, Kluwer Academic Publishers, London, (1997).
  • Whittlesey, J., Whittlesey K., “Some Geometrical Generalizations of Euler's Formula”, Int. J. Of math. Edu. in Sci. &Tech., 21(3): 461-468(1990).
  • Yang, A.T., Freudensterin, F., “Application of Dual-number Quaternion Algebra to the Analysis of Spatial Mechanisms”, ASME Journal of applied Mechnics 86E (2):300-308(1964).
  • Yayli, Y.,“Homothetic Motions at E⁴”, Mech. Mach. Theory, 27( 3): 303-305(1992). [17] Zhang,
  • F.,Quaternions and Matrices of
  • Quaternions, Linea rAlgebra and its Appl., 251: 21-57(1997).