Advertisement

The characteristic equation

  • C. C. Mac Duffee
Chapter
  • 365 Downloads
Part of the Ergebnisse der Mathematik und Ihrer Grenƶgebiete book series (MATHE1, volume 5)

Abstract

The minimum equation. If A is a matrix of order n over a field p, the matrices I, A, A 2,..., A n 2 constitute n 2 + 1 sets of n 2 numbers each, and hence are linearly dependent in p. Thus A satisfies some equation
$$m{\text{} }(\lambda){\text{} } = {\text{} }\lambda ^u {\text{} } + {\text{} }m_1 \lambda ^{u - 1} {\text{} } + {\text{} }...{\text{} } + {\text{} }m_u {\text{} } = {\text{} }0$$
with coefficients in p of minimum degree μ. We shall call μ the index of A. The index of a scalar matrix is 1. Every matrix except 0 has an index.

Keywords

Characteristic Equation Characteristic Root Hermitian Matrix Irreducible Factor Elementary Symmetric Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

  1. 1.
    Frobenius: J. reine angew. Math. Vol. 84 (1878) p. 1–63.CrossRefGoogle Scholar
  2. 1.
    Phillips, H. B.: Amer. J. Math. Vol. 41 (1919) pp. 266–278.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Pasch: Math. Ann. Vol. 38 (1891) pp. 24–49.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 5.
    Cayley, A.: Philos. Trans. Roy. Soc, London Vol. 148 (1858) pp. 17–37.CrossRefGoogle Scholar
  5. 6.
    Laguerre, E.: J. École polytechn. Vol. 25 (1867) pp. 215–264.Google Scholar
  6. 7.
    Forsyth, A. R.: Mess. Math. Vol. 13 (1884) pp. 139–142.Google Scholar
  7. 8.
    Frobenius: Mess. Math. Vol. 13 (1884) pp. 62–66.Google Scholar
  8. 1.
    Buchheim: Proc. London Math. Soc. Vol. 16 (1884) pp. 63–82.MathSciNetCrossRefGoogle Scholar
  9. 2.
    Cauchy: Exercises d’analyse et de physique mathématique Vol. 1 (1840) p. 53.Google Scholar
  10. 3.
    Günther, S.: Z. Math. Vol. 21 (1876) pp. 178–191.zbMATHGoogle Scholar
  11. Laisant, G. A.: Bull. SOC, Math. France Vol. 17 (1889) pp. 104–107.MathSciNetGoogle Scholar
  12. Rados, G.: Math. Ann. Vol. 48 (1897) pp. 417–424.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 1.
    Loewy, A.: S.-B. Heidelberg. Akad. Wiss. Vol. 5 (1918) p. 3.Google Scholar
  14. Loewy, A.: Math. Z. Vol. 7 (1920) pp. 58–125.MathSciNetCrossRefGoogle Scholar
  15. 3.
    Frobenius: J. reine angew. Math. Vol. 84 (1878) pp. 1–63.CrossRefGoogle Scholar
  16. 4.
    Proof by O. Perron: Math. Ann. Vol. 64 (1906) pp. 248–263.MathSciNetCrossRefGoogle Scholar
  17. 2.
    Sylvester: C. R. Acad. Sci., Paris Vol. 98 I (1884) pp. 471–475.Google Scholar
  18. 3.
    Laguerre: J. École polytechn. Vol. 25 (1867) pp. 215–264.Google Scholar
  19. 4.
    Hensel, K.: J. reine angew. Math. Vol. 127 (1904) pp. 116–166.zbMATHGoogle Scholar
  20. 2.
    Borchardt, C. W.: J. reine angew. Math. Vol. 30 (1846) pp. 38–46.zbMATHCrossRefGoogle Scholar
  21. Borchardt, C. W.: J. Math, pures appl. I Vol. 12 (1847) pp. 50–67.Google Scholar
  22. 3.
    Sylvester, J. J.: Nouv. Ann. math. Vol. 11 (1852) pp. 439–440.Google Scholar
  23. 4.
    Spottiswoode, W.: J. reine angew. Math. Vol. 51 (1856) pp. 209–271 and 328-381.zbMATHCrossRefGoogle Scholar
  24. 5.
    Frobenius, G.: J. reine angew. Math. Vol. 84 (1878) pp. 1–63.CrossRefGoogle Scholar
  25. 6.
    Frobenius, G.: J. reine angew. Math. Vol. 84 (1878) pp. 1–63.CrossRefGoogle Scholar
  26. 7.
    Spottiswoode: C. R, Acad. Sci., Paris Vol. 94 (1882) pp. 55–59.Google Scholar
  27. 8.
    Frobenius, G.: Philos. Mag. V Vol. 16 (1883) pp. 267–269.Google Scholar
  28. 9.
    Bromwich, T.J. I’A.: Proc. Cambridge Philos. Soc. Vol. 11 (1901) pp.75 to 89.Google Scholar
  29. 10.
    Sylvester: Philos. Mag. V Vol. 16 (1883) pp. 267–269.CrossRefGoogle Scholar
  30. 1.
    Thurston, H. S.: Amer. Math. Monthly Vol. 38 (1931) pp. 322–324.MathSciNetzbMATHCrossRefGoogle Scholar
  31. 3.
    Châtelet, A.: Ann. École norm. III Vol. 28 (1911) pp. 105–202.Google Scholar
  32. 4.
    Lipschitz, R.: Acta math. Vol. 10 (1887) pp. 137–144.MathSciNetzbMATHCrossRefGoogle Scholar
  33. 6.
    Ranum, A.: Bull. Amer. Math. Soc. II Vol. 17 (1911) pp. 457–461.MathSciNetzbMATHCrossRefGoogle Scholar
  34. 8.
    Taber, H.: Amer. J. Math. Vol. 13 (1891) pp. 159–172.MathSciNetCrossRefGoogle Scholar
  35. 9.
    Bennett, A.A.: Ann. of Math. II Vol. 23 (1923) pp. 91–96.CrossRefGoogle Scholar
  36. 1.
    Franklin, P.: Ann. of Math. II Vol. 23 (1923) pp. 97–100.MathSciNetCrossRefGoogle Scholar
  37. 2.
    Franklin, P.: J. Math. Physics, Massachusetts Inst. Technol. Vol. 10 (1932) pp. 289–314.Google Scholar
  38. 3.
    Williamson, J.: Amer. Math. Monthly Vol. 39 (1932) pp. 280–285.MathSciNetCrossRefGoogle Scholar
  39. 4.
    Pierce, T. A.: Bull. Amer. Math. Soc. Vol. 36 (1930) pp. 262–264.MathSciNetzbMATHCrossRefGoogle Scholar
  40. 5.
    Hermite, C.: C. R, Acad. Sci., Paris Vol. 41 (1855) PP-181–183.Google Scholar
  41. 6.
    Loewy, A.: J. reine angew. Math. Vol. 122 (1900) pp. 53–72.zbMATHGoogle Scholar
  42. 7.
    Autonne, L.: Rend. Circ. mat. Palermo Vol. 16 (1902) pp. 104–128.zbMATHCrossRefGoogle Scholar
  43. 8.
    Voss, A.: Math. Ann. Vol. 13 (1878) pp. 320–374.MathSciNetzbMATHCrossRefGoogle Scholar
  44. Prym, F.: Abh. Ges. Wiss. Göttingen Vol. 38 (1892) pp. 1–42.Google Scholar
  45. 9.
    Autonne: Bull. Soc. Math. France Vol. 30 (1902) pp. 121–134.MathSciNetzbMATHGoogle Scholar
  46. 1.
    Theorem and proof by A. Hirsch: Acta math. Vol. 25 (1901) pp. 367–370.Google Scholar
  47. 3.
    Hermite: C. R, Acad. Sci., Paris Vol. 41 (1855) pp 181–183.Google Scholar
  48. 5.
    Clebsch, A.: J. reine angew. Math. Vol. 62 (1863) pp. 232–245.zbMATHCrossRefGoogle Scholar
  49. 1.
    This theorem and proof are by A. Hirsch: J. reine angew. Math. Vol. 62 (1863) pp. 232–245.Google Scholar
  50. 2.
    Bendixson, I.: Acta math. Vol. 25 (1901) pp. 359–365.MathSciNetCrossRefGoogle Scholar
  51. 3.
    The real case by Bendixson, the complex case by Hirsch: Acta math. Vol. 25 (1901) pp. 359–365.Google Scholar
  52. 4.
    Bromwich: Acta math. Vol. 30 (1906) pp. 295–304.MathSciNetCrossRefGoogle Scholar
  53. 5.
    Schur, I.: Math. Ann. Vol. 66 (1909) pp. 488–510.MathSciNetzbMATHCrossRefGoogle Scholar
  54. 1.
    Browne, E, T.: Bull. Amer. Math. Soc. Vol. 34 (1928) pp. 363–368.MathSciNetzbMATHCrossRefGoogle Scholar
  55. 3.
    Wedderburn, J. H. M.: Ann. of Math. II Vol. 27 (1926) pp. 245–248.MathSciNetzbMATHCrossRefGoogle Scholar
  56. 4.
    Aramata, H.: Tôhoku Math. J. Vol. 28 (1927) p. 281.zbMATHGoogle Scholar
  57. 5.
    Brauer, R.: Tôhoku Math. J. Vol. 30 (1928) p. 72.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1933

Authors and Affiliations

  • C. C. Mac Duffee

There are no affiliations available

Personalised recommendations