Skip to main content
Log in

Quaternionic determinants

  • Article
  • Published:
The Mathematical Intelligencer Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. E. Artin,Geometric Algebra, New York: Interscience, 1957; reprinted by Wiley, New York, 1988.

    MATH  Google Scholar 

  2. H. Aslaksen,SO(2) invariants of a set of 2 x 2 matrices.Math. Scand. 65 (1989), 59–66.

    MATH  MathSciNet  Google Scholar 

  3. H. Aslaksen, E.-C. Tan, and C. Zhu, Invariant theory of special orthogonal groups,Pacific ]. Math. (in press).

  4. A. Bagazgoitia, A determinantal identity for quaternions, inProceedings of 1983 Conference on Algebra Lineal y Aplicaciones, Vitoria-Gasteiz, Spain, 1984, pp. 127–132.

    Google Scholar 

  5. R. W. Barnard and E. Hastings Moore,General analysis. Part 1, Memoirs of the American Philosophical Society, 1935.

  6. J. Brenner, Expanded matrices from matrices with complex elements,SIAM Rev. 3 (1961), 165–166.

    Article  MATH  MathSciNet  Google Scholar 

  7. J. Brenner, Applications of the Dieudonné determinant,Linear Algebra Appl. 1 (1968), 511–536.

    Article  MATH  Google Scholar 

  8. J. Brenner, Corrections to “Applications of the Dieudonné determinant,”Linear Algebra Appl. 13 (1976), 289.

    Article  MATH  Google Scholar 

  9. J. Brenner and J. De Pillis, Generalized elementary symmetric functions and quaternion matrices,Linear Algebra Appl. 4 (1971), 55–69.

    Article  MATH  Google Scholar 

  10. A. Cayley, On certain results relating to quaternions,Philos. Mag. 26 (1845), 141–145; reprinted inThe Collected Mathematical Papers Vol. 1, Cambridge: Cambridge University Press, 1989, pp. 123–126.

    Google Scholar 

  11. L. Chen, Definition of determinant and Cramer solution over the quaternion field,Acta Math. Sinica (N.S.) 7 (1991), 171–180.

    Article  MATH  MathSciNet  Google Scholar 

  12. L. Chen, Inverse matrix and properties of double determinant over quaternion field,Sci. China Ser. A 34 (1991), 528–540.

    MATH  MathSciNet  Google Scholar 

  13. P. M. Cohn, The similarity reduction of matrices over a skew field,Math. Z. 132 (1973), 151–163.

    Article  MATH  MathSciNet  Google Scholar 

  14. P. M. Cohn,Algebra, vol. 1, 2nd ed., New York: Wiley, 1991.

    Google Scholar 

  15. P. M. Cohn,Algebra, vol. 3, 2nd ed. New York: Wiley, 1991.

    Google Scholar 

  16. M. L. Curtis,Matrix Groups, New York: Springer-Verlag, 1979; 1984.

    Book  Google Scholar 

  17. J. Dieudonné, Les déterminants sur un corps non-commutatif,Bull. Soc. Math. France 71 (1943), 27–45.

    MATH  MathSciNet  Google Scholar 

  18. J. Dieudonné.Special Functions and Linear Representations of Lie Groups, CBMS 42, Providence, RI, American Mathematical Society, 1980.

    MATH  Google Scholar 

  19. R. Dimitrić and B. Goldsmith, Sir William Rowan Hamilton,Math. Intelligencer 11 (1989), no. 2, 29–30.

    Article  MathSciNet  Google Scholar 

  20. F. J. Dyson, Correlations between eigenvalues of a random matrix,Commun. Math. Phys. 19 (1970), 235–250.

    Article  MATH  MathSciNet  Google Scholar 

  21. F. J. Dyson, Quaternion determinants,Helv. Phys. Acta 45 (1972), 289–302.

    Google Scholar 

  22. I. M. Gelfand and V. S. Retakh, Determinants of matrices over noncommutative rings,Functional Anal. Appl. 25 (1991), 91–102.

    Article  MathSciNet  Google Scholar 

  23. F. Reese Harvey,Spinors and Calibrations, New York, Academic Press, 1990.

    MATH  Google Scholar 

  24. W. R. Hamilton,Elements of Quaternions, 2nd ed., London: Longman, 1889.

    Google Scholar 

  25. A. Heyting, Die Théorie der linearen Gleichungen in einer Zahlenspezies mit nichtkommutativer Multiplikation,Math. Ann. 98 (1927), 465–490.

    Article  MATH  MathSciNet  Google Scholar 

  26. M. H. Ingraham, A note on determinants,Bull. Amer. Math. Soc. 43 (1937), 579–580.

    Article  MathSciNet  Google Scholar 

  27. N. Jacobson, Normal semi-linear transformations,Amer. J. Math. 61 (1939), 45–58.

    Article  MathSciNet  Google Scholar 

  28. N. Jacobson, An application of E. H. Moore’s determinant of a Hermitian matrix,Bull. Amer. Math. Soc. 45 (1939), 745–748.

    Article  MathSciNet  Google Scholar 

  29. H. C. Lee, Eigenvalues and canonical forms of matrices with quaternionic entries,Proc. Roy. Irish Acad. Sect. A, 52 (1949), 253–260.

    MathSciNet  Google Scholar 

  30. D. W. Lewis, A determinantal identity for skewfields,Linear Algebra Appl. 7 (1985), 213–217.

    Article  Google Scholar 

  31. K. O. May, The impossibility of a division algebra of vectors in three dimensional space,Amer. Math. Monthly 73 (1966), 289–291.

    Article  MathSciNet  Google Scholar 

  32. Madan Lal Mehta, Determinants of quaternion matrices,J. Math. Phys. Sci. 8 (1974), 559–570.

    MATH  Google Scholar 

  33. Madan Lal Mehta,Elements of Matrix Theory, Dehli Hindustan Pub. Corp., 1977.

    MATH  Google Scholar 

  34. E. H. Moore, On the determinant of an hermitian matrix of quaternionic elements,Bull. Amer. Math. Soc. 28 (1922), 161–162.

    MATH  Google Scholar 

  35. Thomas Muir,The Theory of Determinants, Vol. 2, London: MacMillan, 1911.

    MATH  Google Scholar 

  36. O. Ore, Linear equations in non-commutative fields,Ann. Math. 32 (1931), 463–77.

    Article  MathSciNet  Google Scholar 

  37. K. Hunger Parshall and D. E. Rowe,The Emergence of the American Mathematical Research Community, 1876-1900: J. J. Sylvester, Felix Klein and E. H. Moore, Providence, RI: American Mathematical Society, 1994.

    MATH  Google Scholar 

  38. J. M. Peirce, Determinants of quaternions,Bull. Amer. Math. Soc. 5 (1899), 335–337.

    Article  Google Scholar 

  39. P. Piccinni, Dieudonné determinant and invariant real polynomials on gl(n, H),Rend. Mat. (7)2 (1982), 31–45.

    MathSciNet  Google Scholar 

  40. R. S. Pierce,Associative Algebras, New York: Springer-Verlag, 1982.

    Book  MATH  Google Scholar 

  41. J. Radon, Lineare Scharen orthogonaler Matrizen,Abh. Math. Sem. Univ. Hamburg 1 (1922), 2–14.

    Google Scholar 

  42. A. R. Richardson, Hypercomplex determinants,Messenger of Math. 55 (1926), 145–152.

    Google Scholar 

  43. A. R. Richardson, Simultaneous linear equations over a division algebra,Proc. London Math. Soc. 28 (1928), 395–420.

    Article  MATH  Google Scholar 

  44. E. Study, Zur Théorie der linearen Gleichungen,Acta Math. 42 (1920), 1–61.

    Article  MathSciNet  Google Scholar 

  45. O. Teichmüller, Operatoren im Wachsschen Raum,J. Reine Angew. Math. 174 (1935), 73–124.

    MATH  Google Scholar 

  46. C. L. Tong, Symplectic Groups, honours thesis, National Univ. of Singapore, 1991.

  47. B. Leednert van der Waerden, Hamilton’s discovery of quaternions,Math. Mag. 49 (1976), 227–234.

    Article  MATH  MathSciNet  Google Scholar 

  48. B. Leednert van der Waerden,A History of Algebra, New York: Springer-Verlag, 1985.

    Book  MATH  Google Scholar 

  49. P. Van Praag, Sur les déterminants des matrices quaterniennes,Helv. Phys. Acta 62 (1989), 42–46.

    MathSciNet  Google Scholar 

  50. P. Van Praag, Sur la norme réduite du déterminant de Dieudonné des matrices quaterniennes,J. Algebra 136 (1991), 265–274.

    Article  MATH  MathSciNet  Google Scholar 

  51. L. A. Wolf, Similarity of matrices in which the elements are real quaternions,Bull. Amer. Math. Soc. 42 (1936), 737–743.

    Article  MathSciNet  Google Scholar 

  52. L. E. Zagorin, The determinants of matrices over a field (Russian),Proc. First Republican Conf. Math. Byelorussia, Izdat, Minsk: “Vysšaja Škola”, 1965, pp. 151–152.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Helmer Aslaksen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aslaksen, H. Quaternionic determinants. The Mathematical Intelligencer 18, 57–65 (1996). https://doi.org/10.1007/BF03024312

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03024312

Keywords

Navigation