Advances in Applied Clifford Algebras

, Volume 23, Issue 1, pp 253–267 | Cite as

On the Heights of Quaternionic Vectors

  • Bu-Liao Wang
  • Liang-Gui Feng


In this paper, we introduce a new numerical character, namely the height, for a quaternionic vector. According to this new concept (height), we give a complete classification of quaternionic vectors and enumerate all possible row echelon forms of the associated matrices for each type of the quaternionic vectors. By observing a link which the height has with the system of Cartesian frames, we give the character descriptions for several types of the systems of Cartesian frames.


Quaternion rank row echelon form rotation 


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  1. 1.
    Shuster M.D.: A survey of attitude representations. J. of Astr. Sci. 41, 439–517 (1993)MathSciNetGoogle Scholar
  2. 2.
    Arribas M., Elipe A., Palacios M.: Quaternions and the rotation of a rigid body. Celestial Mech. Dyn. Astr. 96, 239–251 (2006)MathSciNetADSMATHCrossRefGoogle Scholar
  3. 3.
    Miura K.T.: Unit quaternion integral curve: a new type of fair free-form curves. Computer Aided Geom. Design 17, 39–58 (1999)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Adler S.L.: Quaternionic Quantum Mechanics and Quantum Field. Oxford University Press, New York (1995)Google Scholar
  5. 5.
    Wood R.M.W.: Quaternionic eigenvalues. Bull. of London Math. Soc. 17, 137–138 (1985)MATHCrossRefGoogle Scholar
  6. 6.
    Zhang F.: Quaternions and matrices of quaternions. Linear Alg. and its Appl. 251, 21–57 (1997)ADSMATHCrossRefGoogle Scholar
  7. 7.
    Altmann S.L.: Rotations, Quaternions, and Double Groups. Clarendon Press, Oxford, England (1986)MATHGoogle Scholar
  8. 8.
    A. Gsponer and J.-P. Hurni, Quaternions in Mathematical Physics (2): Analytical Bibliography. Independent Scientific Research Institute Report ISRI-05- 05.26, 2008; also available online at
  9. 9.
    K. Hoffman and R. Kunze, Linear Algebra. 2nd edition, Prentice-Hall, Englewood Cliffs, New Jersey, 1971.Google Scholar
  10. 10.
    P.M. Cohn, Algebra, vol. 3, 2nd edition, John Wiley & Sons, Baffins Lane, Chichester, England, 1991.Google Scholar
  11. 11.
    Wang B.L., Feng L.G.: Complex representation of quaternionic differential equation and attitude computation for a spacecraft. IEEE Proceedings of 2011, The 4th International Conference on Computer Science and Information Technology 8, 692–695 (2011)Google Scholar
  12. 12.
    Wang B.L., Feng L.G.: On the rank of quaternionic matrices. IEEE Proceedings of 2011, World Congress on Engineering and Technology 2, 583–586 (2011)MathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of Mathematics and Systems ScienceNational University of Defense TechnologyChangshaP.R. China

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