Skip to main content
SpringerLink
Log in
Book cover

Real Quaternionic Calculus Handbook pp 1–34Cite as

Birkhäuser

An Introduction to Quaternions

  • Chapter
  • First Online:

  • 2468 Accesses

Abstract

Once we start studying quaternionic analysis we take part in a wonderful experience, full of insights. This ideology is shown, for instance, when we start describing the first results and pursuing the subject, while the amazement lingers on through the elegance and smoothness of the results.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    Hamilton called pure imaginary quaternions right quaternions and real numbers scalar quaternions.

  2. 2.

    A skew field is an algebraic structure that satisfies all the properties of a field except for commutativity.

  3. 3.

    For all real angles θ and ϕ, it is easy to check that \(p =\cos \theta \cos \phi i +\cos \theta \sin \phi j +\sin \theta k\) is a square root of \(-1_{\mathbb{H}}\). This gives the set of all the points on the unit sphere in \({\mathbb{R}}^{3}\) and shows that the set of “quaternionic square roots of minus one” has infinitely many square roots.

  4. 4.

    Hamilton called this quantity the tensor of p, but nowadays this concept conflicts with modern usage.

  5. 5.

    It is also known as trigonometric representation of the quaternion p.

  6. 6.

    These matrices are used in physics to describe the angular momentum of elementary particles such as electrons.

Bibliography

  1. K. Abdel-Khalek, Quaternion analysis (1996, peprint). arXiv:hepth/9607152

    Google Scholar 

  2. G. Aeberli, Der Zusammenhang zwischen quaternären quadratischen formen und idealen in quaternionenringen. Comment. Math. Helv. 33, 212–239 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  3. V. Bargmann, Über den Zusammenhang zwischen Semivektoren und Spinoren und die Reduktion der Diracgleichungen für Semivektoren. Helv. Phys. Acta 7, 57–82 (1934)

    Google Scholar 

  4. I.Y. Baritzhack, R.R. Harman, Optimal fusion of a given quaternion with vector measurements. J. Guid. Control Dyn. 25(1), 188–190 (2002)

    Article  Google Scholar 

  5. W.E. Baylis, Why i? Am. J. Phys. 60, 788–797 (1992)

    Google Scholar 

  6. B. Beck, Sur les équations polynomiales dans les quaternions. Enseign. Math. 25(3–4), 193–201 (1979)

    MATH  MathSciNet  Google Scholar 

  7. J. Brenner, Matrices of quaternions. Pac. J. Math. 1, 329–335 (1951)

    Article  MATH  Google Scholar 

  8. E. Cartan, Sur les groupes linéaires quaternioniens. Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich 85, 191–203 (1940)

    MathSciNet  Google Scholar 

  9. G. Combebiac, Calcul des Triquaternions (GauthierVillars, Paris, 1902)

    MATH  Google Scholar 

  10. S. De Leo, P. Rotelli, Representations of U(1, q) and constructive quaternion tensor products. Nuovo Cimento B 110, 33–51 (1995)

    Article  Google Scholar 

  11. P. Du Val, Homographies, Quaternions, and Rotations. Clarendon Press, Oxford (1964)

    MATH  Google Scholar 

  12. S. Eilenberg, I. Niven, The “fundamental theorem of algebra” for quaternions. Bull. Am. Math. Soc. 50, 244–248 (1944)

    Article  MathSciNet  Google Scholar 

  13. A. Einstein, W. Mayer, Die Diracgleichungen für Semivektoren. Proc. R. Acad. Amst. 56, 497–516 (1933)

    Google Scholar 

  14. I. Gelfand, V. Retakh, R.L. Wilson, Quaternionic quasideterminants and determinants, in Lie Groups and Symmetric Spaces, ed. by F.I. Karpelevich, S.G. Gindikin. American Mathematical Society Translations Series 2, vol. 210 (American Mathematical Society, Providence, 2003), pp. 111–123

    Google Scholar 

  15. K. Gürlebeck, W. Sprößig, Quaternionic Analysis and Elliptic Boundary Value Problems (Birkhäuser, Basel, 1990)

    Book  MATH  Google Scholar 

  16. K. Gürlebeck, W. Sprößig, Quaternionic and Clifford Calculus for Physicists and Engineers (Wiley, New York, 1997)

    MATH  Google Scholar 

  17. K. Gürlebeck, K. Habetha, W. Sprößig, Holomorphic Functions in the Plane and n-Dimensional Space (Birkhäuser, Basel, 2008)

    MATH  Google Scholar 

  18. F. Gürsey, Applications of Quaternions to Field Equations. Ph.D. thesis, University of London, 1950

    Google Scholar 

  19. W.R. Hamilton, On quaternions and the rotation of a solid body. Proc. R. Ir. Acad. 4, 38–56 (1850)

    Google Scholar 

  20. W.R. Hamilton, Elements of Quaternions, Vol. I et II (First edition 1866; second edition edited and expanded by C.J. Joly 1899–1901; reprinted by Chelsea Publishing, New York, 1969)

    Google Scholar 

  21. T. Haslwanter, Mathematics of the three dimensional eye rotations. Vis. Res. 35, 1727–1739 (1995)

    Article  Google Scholar 

  22. A.S. Hathaway, Quaternion space. Trans. Am. Math. Soc. 3, 46–59 (1902)

    Article  MATH  MathSciNet  Google Scholar 

  23. D. Hestenes, A unified language for mathematics and physics, in Clifford Algebras and Their Applications in Mathematical Physics, ed. by J.S.R. Chisholmand, A.K.Common (Reidel, Dordrecht, 1986), pp. 1–23

    Chapter  Google Scholar 

  24. D. Janovská, G. Opfer, The classification and the computation of the zeros of quaternionic, two-sided polynomials. Numer. Math. 115, 81–100 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  25. D. Janovská, G. Opfer, A note on the computation of all zeros of simple quaternionic polynomials. SIAM J. Numer. Anal. 48(1), 244–256 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  26. T. Jiang, Cramer rule for quaternionic linear equations in quaternionic quantum theory. Rep. Math. Phys. 57(3), 463–468 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  27. T. Jiang, M. Wei, On a solution of the quaternion matrix equation X − AXB = C and its application. Acta Math. Sin. 21, 483–490 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. W.J. Johnston, A quaternion substitute for the theory of tensors. Proc. R. Ir. Acad. A 37, 13–27 (1926)

    Google Scholar 

  29. C.J. Joly, Quaternion invariants of linear vector functions and quaternions determinants. Proc. R. Ir. Acad. 4, 1–15 (1896)

    Google Scholar 

  30. V. Kravchenko, M. Shapiro, Integral Representations for Spatial Models of Mathematical Physics. Research Notes in Mathematics (Pitman Advanced Publishing Program, London, 1996)

    Google Scholar 

  31. T.Y. Lam, A First Course in Noncommutative Rings. Graduate Texts in Mathematics, vol. 131 (Springer, New York, 1991)

    Google Scholar 

  32. J. Lambek, If Hamilton had prevailed: quaternions in physics. Math. Intell. 17, 7–15 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  33. H.C. Lee, Eigenvalues and canonical forms of matrices with quaternion coefficients. Proc. R. Ir. Acad. Sect. A 52, 253–260 (1949)

    Google Scholar 

  34. M. Markic, Transformantes nouveau véhicule mathématique. Synth\(\grave{e}\) se des triquaternions de Combebiac et du syst\(\grave{e}\) me géométrique de Grassmann Calcul des quadriquaternions. Ann. Fac. Sci. Toulouse, 28, 103–148 (1936)

    Google Scholar 

  35. M. Markic, Transformantes nouveau véhicule mathématique. Synth\(\grave{e}\) se des triquaternions de Combebiac et du syst\(\grave{e}\) me géométrique de Grassmann Calcul des quadriquaternions. Ann. Fac. Sci. Toulouse 1, 201–248 (1937)

    Google Scholar 

  36. I. Niven, Equations in quaternions. Am. Math. Mon. 48, 654–661 (1941)

    Article  MATH  MathSciNet  Google Scholar 

  37. R.E. O’Connor, G. Pall, The quaternion congruence \(\overline{t}at = b(mod\,g)\). Am. J. Math. 61(2), 487–508 (1939)

    Article  MathSciNet  Google Scholar 

  38. S. O’Donnel, William Rowan Hamilton, Portrait of a Prodigy (Boole Press, Dublin, 1983)

    Google Scholar 

  39. H.L. Olson, Doubly divisible quaternions. Ann. Math. 31(3), 371–374 (1930)

    Article  MATH  MathSciNet  Google Scholar 

  40. G.B. Price, An Introduction to Multicomplex Spaces and Functions (Marcel Dekker, New York, 1991)

    MATH  Google Scholar 

  41. V. Retakh, R.L. Wilson, Advanced Course on Quasideterminants and Universal Localization. Notes of the Course (Centre de Recerca Matematica (CRM), Bellaterra, 2007)

    Google Scholar 

  42. M. Riesz, Clifford Numbers and Spinors. Lecture Series No. 38 (Institute for Fluid Dynamics and Applied Mathematics/University of Maryland, Baltimore, 1958). Reprinted in Clifford Numbers and Spinors, ed. by M. Riesz, E.F. Bolinder, P. Lounesto (Kluwer, Dordrecht, 1993)

    Google Scholar 

  43. W. Rindler, I. Robinson, A plainmans guide to bivectors, biquaternions, and the algebra and geometry of Lorentz transformations, in On Einsteins Path Essays in Honour of Engelbert Schucking, ed. by A. Harvey (Springer, New York, 1999), pp. 407–433

    Google Scholar 

  44. E. Sarrau, Notions sur la théorie des quaternions, Paris, Gauthier-Villars, 1889, 46 p.

    Google Scholar 

  45. R. Serôdio, L.-S. Siu, Zeros of quaternion polynomials. Appl. Math. Lett. 14, 237–239 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  46. A. Sudbery, Quaternion analysis. Math. Proc. Camb. Philos. Soc. 85, 199–225 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  47. P.G. Tait, On the linear and vector function. Proc. R. Soc. Edinb. 23, 424–426 (1899)

    Google Scholar 

  48. P.G. Tait, On the claim recently made for Gauss to the invention (not the discovery) of quaternions. Proc. R. Soc. Edinb. 23, 17–23 (1899/1900)

    Google Scholar 

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

    MATH  Google Scholar 

  50. J.P. Ward, Quaternions and Cayley Numbers (Kluwer, Dordrecht, 1997)

    Book  MATH  Google Scholar 

  51. W.H. Watson, On a system of functional dynamics and optics. Philos. Trans. R. Soc. A 236, 155–190 (1937)

    Article  Google Scholar 

  52. D. Weingarten, Complex symmetries of electrodynamics. Ann. Phys. 76(2), 510–548 (1973)

    Article  MathSciNet  Google Scholar 

  53. P. Weiss, On some applications of quaternions to restricted relativity and classical radiation theory. Proc. R. Ir. Acad. A 46, 129–168 (1941)

    MATH  Google Scholar 

  54. P. Weiss, An extension of Cauchy’s integral formula bymeans of a Maxwell’s stress tensor. J. Lond. Math. Soc. 21(3), 210–218 (1946)

    Article  MATH  Google Scholar 

  55. H. Weyl, Quantenmechanik und Gruppentheorie. Weyl, H. Publication: Zeitschrift für Physik 46(1–2), 1–46 (1927)

    Google Scholar 

  56. N.A. Wiegmann, Some theorems on matrices with real quaternion elements. Can. J. Math. 7, 191–201 (1955)

    Article  MATH  MathSciNet  Google Scholar 

  57. D.R. Wilkins, William Rowan Hamilton: mathematical genius. Phys. World 18, 33–36 (2005)

    Google Scholar 

  58. J.G. Winans, Quaternion physical quantities. Found. Phys. 7, 341–349 (1977)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  60. R.M.W. Wood, Quaternionic eigenvalues. Bull. Lond. Math. Soc. 17, 137–138 (1985)

    Article  MATH  Google Scholar 

  61. F. Zhang, Permanent Inequalities and Quaternion Matrices. Ph.D. dissertation, University of California at Santa Barbara, 1993

    Google Scholar 

  62. F. Zhang, On numerical range of normal matrices of quaternions. J. Math. Phys. Sci. 29(6), 235–251 (1995)

    MATH  MathSciNet  Google Scholar 

  63. F. Zhang, Quaternions and matrices of quaternions. Linear Algebra Appl. 251, 21–57 (1997)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CIDMA, University of Aveiro, Aveiro, Portugal

    João Pedro Morais

  2. Department of Differential Equations, University of Sofia St Kliment Ohridski Faculty of Mathematics and Informatics, Sofia, Bulgaria

    Svetlin Georgiev

  3. Institut für Angewandte Analysis, TU Bergakademie Freiberg, Freiberg, Germany

    Wolfgang Sprößig

Authors
  1. João Pedro Morais
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Svetlin Georgiev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wolfgang Sprößig
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Basel

About this chapter

Cite this chapter

Morais, J.P., Georgiev, S., Sprößig, W. (2014). An Introduction to Quaternions. In: Real Quaternionic Calculus Handbook. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0622-0_1

Download citation

Publish with us

Policies and ethics

Search

Navigation