Advertisement

Advances in Applied Clifford Algebras

, Volume 9, Issue 1, pp 61–76 | Cite as

Similarity and consimilarity of elements in the real Cayley-Dickson algebras

  • Yongge Tian
Article

Abstract

The similarity and consimilarity of elements in the real quaternion, octonion and sedenion algebras, as well as in the general real Cayley-Dickson algebras are considered by solving the two fundamental equationsax=xb and\(ax = \bar xb\) in these algebras. Some consequences are also presented.

Key words and phrases

quaternions octonions sedenions equation similarity consimilarity square root 

AMS subject classifications

17A05 17A35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Althoen S. S. and J. F. Weidner, Real division algebras and Dickson construction,Amer. Math. Monthly,48, 369–371 (1978).MathSciNetGoogle Scholar
  2. [2]
    Brenner J. L., Matrices of quaternions,Pacific J. Math. 1, 329–335 (1951).zbMATHMathSciNetGoogle Scholar
  3. [3]
    Brown R. B., On generalized Cayley-Dickson algebras,Pacific J. Math.,20, 415–422 (1967).zbMATHMathSciNetGoogle Scholar
  4. [4]
    Dray T. and C. A. Manague, The octonionic eigenvalue problem, preprint.Google Scholar
  5. [5]
    Dray T. and C. A. Manague, Finding octonionic eigenvectors using mathematica, preprint.Google Scholar
  6. [6]
    Horn R. A. and C. R. Johnson, Matrix Analysis, Cambridge U. P., New York, (1985).zbMATHGoogle Scholar
  7. [7]
    Michael P. R., Quaternionic linear and quadratic equations,J. Natur. Geom.,11, 101–106 (1997).zbMATHMathSciNetGoogle Scholar
  8. [8]
    Moreno G., The zero divisors of the Cayley-Dickson algebras over the real numbers,Bol. Soc. Mat. Mex., to appear.Google Scholar
  9. [9]
    Neven I., Equations in quaternions,Amer. Math. Monthly,48, 654–661 (1941)CrossRefMathSciNetGoogle Scholar
  10. [10]
    Osborn J. M., Quadratic division algebras,Tans. Amer. Math. Soc.,115, 202–221 (1962).CrossRefMathSciNetGoogle Scholar
  11. [11]
    Porteous I. R., Clifford Algebras and the Classical Groups, Cambridge U. P., Cambridge, (1995).zbMATHGoogle Scholar
  12. [12]
    Sorgsepp L. and J. Lôhmus, About nonassociativity in physics and Cayley-Graves’ octonions.Hadronic J.,2, 1388–1459 (1979).zbMATHMathSciNetGoogle Scholar
  13. [13]
    Sorgsepp L. and J. Lôhmus, Binary and ternary sedenions,Hadronic J.,4, 327–353 (1981).zbMATHGoogle Scholar
  14. [14]
    Schafer R. D., On the algebras formed by the Cayley-Dickson process,Amer. J. Math.,76, 435–446 (1954).zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Schafer R. D., An Introduction to Nonassociative Algebras, Academic Press, New York, (1966).zbMATHGoogle Scholar
  16. [16]
    Wene G. P., Construction relating Clifford algebras and Cayley-Dickson algebras,J. Math. Phys.,25, 2351–2353 (1984).zbMATHCrossRefADSMathSciNetGoogle Scholar
  17. [17]
    Zhang F., Quaternions and matrices of quaternions,Linear Algebra Appl.,251, 21–57 (1997).zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser-Verlag AG 1999

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsConcordia UniversityMontréalCanada

Personalised recommendations