Advances in Applied Clifford Algebras

, Volume 24, Issue 1, pp 179–192 | Cite as

On the Eigenvalues and Eigenvectors of a Lorentzian Rotation Matrix by Using Split Quaternions

  • Mustafa Özdemir
  • Melek Erdoğdu
  • Hakan Şimşek


In this paper, we examine eigenvalue problem of a rotation matrix in Minkowski 3 space by using split quaternions. We express the eigenvalues and the eigenvectors of a rotation matrix in term of the coefficients of the corresponding unit timelike split quaternion. We give the characterizations of eigenvalues (complex or real) of a rotation matrix in Minkowski 3 space according to only first component of the corresponding quaternion. Moreover, we find that the casual characters of rotation axis depend only on first component of the corresponding quaternion. Finally, we give the way to generate an orthogonal basis for \({\mathbb{E}^{3}_{1}}\) by using eigenvectors of a rotation matrix.


Quaternions Split Quaternions Rotation Matrix 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cockle J.: On Systems of Algebra Involving More than One imaginary. Philosophical Magazine 35, 434–435 (1849)Google Scholar
  2. 2.
    I. L. Kantor, A. S. Solodovnikov, Hypercomplex Numbers, An Elementary Introduction to Algebras. Springer-Verlag, 1989.Google Scholar
  3. 3.
    L. Kula, Y. Yaylı, Split Quaternions and Rotations in Semi Euclidean Space \({\mathbb{E}^{4}_{2}}\) . Journal of Korean Mathematical Society 44 (2007), 1313–1327.Google Scholar
  4. 4.
    M. Özdemir, A.A. Ergin, Rotations with unit timelike quaternions in Minkowski 3-space. Journal of Geometry and Physics 56 (2006), 322–336.ADSCrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Özdemir M.: The Roots of a Split Quaternion. Applied Mathematics Letters 22, 258–263 (2009)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Mustafa Özdemir
    • 1
  • Melek Erdoğdu
    • 2
  • Hakan Şimşek
    • 1
  1. 1.Department of MathematicsAkdeniz UniversityAntalyaTurkey
  2. 2.Department of Mathematics and Computer SciencesNecmettin Erbakan UniversityKonyaTurkey

Personalised recommendations