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Similarity

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

Abstract

Similar matrices. Two matrices A and B with elements in a principal ideal ring V are called similar (written A = B) if there exists a unimodular matrix P such that A = P I BP.5 Similarity is an instance of equivalence, and is determinative, reflexive, symmetric and transitive (§ 22). More than this, every unimodular matrix P determines an automorphism of the ring of matrices with elements in V, for if
$$A_1 {\text{} } = {\text{} }P^I B_1 P,{\text{} }A_2 {\text{} } = {\text{} }P^I B_2 P$$
than
$$A_1 {\text{} } + {\text{} }A_2 {\text{} } = {\text{} }P^I (B_1 {\text{} } + {\text{} }B_2)P,{\text{} }A_1 A_2 {\text{} } = {\text{} }P^I (B_1 B_2)P$$
A matrix may be interpreted as a linear homogeneous transformation in vector space. From this point of view similar matrices represent the same transformation referred to different bases. All the theorems of this chapter may be interpreted from this standpoint.

Keywords

Orthogonal Matrix Characteristic Root Unitary Matrice Invariant Factor Orthogonal Matrice 
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.

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Notes

  1. 1.
    Smith, P. F.: Trans. Amer. Math. Soc. Vol. 6 (1905) pp. 1–16.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Hilton, H.: Ann. of Math. II Vol. 15 (1914) pp. 195–201.zbMATHCrossRefGoogle Scholar
  3. 3.
    Autonne, L.: Ann. Univ. Lyon II Vol. 38 (1915) pp. 1–77.Google Scholar
  4. 5.
    Frobenius: J. reine angew. Math. Vol. 84 (1878) p. 21.Google Scholar
  5. 1.
    Fuchs, L.: J. reine angew. Math. Vol. 66 (1866) pp. 121–160.zbMATHCrossRefGoogle Scholar
  6. 3.
    Dickson, L.: Amer. J. Math. Vol. 22 (1900) pp. 121–137.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 4.
    Dickson, L. E.: Froc. London Math. Soc. Vol. 32 (1900) pp. 165–170.zbMATHCrossRefGoogle Scholar
  8. 2.
    Netto, E.: Acta math. Vol. 17 (1893) pp. 265–280.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 3.
    Hilton, H.: Mess, of Math. Vol. 39 (1909) pp. 24–26.Google Scholar
  10. 4.
    Voghera, G.: Boll. Un. Mat. Ital. Vol. 7 (1928) pp. 32–34.zbMATHGoogle Scholar
  11. 5.
    Frobenius: J. reine angew. Math. Vol. 86 (1879) pp.146–208.Google Scholar
  12. 6.
    Landsberg, G.: J. reine angew. Math. Vol. 116 (1896) pp. 331–349.zbMATHGoogle Scholar
  13. 7.
    Burnside, W.: Proc. London Math. Soc. Vol. 30 (1898) pp. 180–194.MathSciNetCrossRefGoogle Scholar
  14. 9.
    Lattès, S.: Ann. Fac. Sci. Univ. Toulouse Vol. 28 (1914) pp. 1–84.CrossRefGoogle Scholar
  15. 10.
    Segre, C.: Atti Accad. naz. Lincei, Mem., III Vol. 19 (1884) pp. 127–148.zbMATHGoogle Scholar
  16. 1.
    Kowalewski, G.: Ber. Verh. sächs. Akad. Leipzig Vol. 68 (1916) pp. 325 to 335.Google Scholar
  17. 3.
    Bennett, A.A.: Amer. Math. Monthly II Vol. 38 (1931) pp. 377–383.CrossRefGoogle Scholar
  18. 9.
    Ingraham, M. H.: Abstr. Bull. Amer. Math. Soc. Vol. 38 (1932) p. 814.zbMATHGoogle Scholar
  19. 1.
    Schur, I.: Trans. Amer. Math. Soc. Vol. 10 (1909) pp. 159–175.MathSciNetzbMATHGoogle Scholar
  20. 3.
    Weyr, H.: C. R, Acad. Sci., Pans Vol. 100 (1885) pp. 966 969.Google Scholar
  21. Weyr, H.: Mh. Math. Phys. Vol. 1 (1990) pp. 163–236.MathSciNetGoogle Scholar
  22. 1.
    Wedderburn, J. H. M.: Ann. of Math. II Vol. 23 (1921) p. 123.MathSciNetGoogle Scholar
  23. 5.
    Metzler, W, H.: Amer. J. Math. Vol. 14 (1892) pp. 326–377.MathSciNetCrossRefGoogle Scholar
  24. 6.
    Hensel, K.: J. reine angew. Math. Vol. 127 (1904) pp. 116–166.zbMATHGoogle Scholar
  25. 7.
    Wellstein, J.: J. reine angew. Math. Vol. 163 (1930) pp. 166–182.zbMATHGoogle Scholar
  26. 8.
    Menge, W. O.: Bull. Amer. Math. Soc. Vol. 38 (1932) pp. 88–94.MathSciNetCrossRefGoogle Scholar
  27. 9.
    Autonne, L.: Nouv. Ann. Math. IV Vol. 12 (1912) pp. 118–127.Google Scholar
  28. 2.
    Sylvester, J. J.: Mess, of Math. Vol. 19 (1890) pp. 1–5.Google Scholar
  29. 3.
    Schur, I.: Math. Ann. Vol. 66 (1909) pp. 488–510.MathSciNetzbMATHCrossRefGoogle Scholar
  30. 1.
    Toeplitz, O.: Math. Z. Vol. 2 (1918) pp. 187–197.MathSciNetCrossRefGoogle Scholar
  31. 3.
    Autonne: Rend. Circ. mat. Palermo Vol. 16 (1902) pp. 104–128.zbMATHCrossRefGoogle Scholar
  32. Autonne: Bull. Soc. Math. France Vol. 31 (1903) pp. 140–155.MathSciNetzbMATHGoogle Scholar
  33. 1.
    Autonne, L.: Bull. Soc. Math. France Vol. 31 (1903) pp. 140–155.MathSciNetzbMATHGoogle Scholar
  34. 2.
    Autonne, L.: Bull. Soc. Math. France Vol. 30 (1902) pp. 121–134.MathSciNetzbMATHGoogle Scholar
  35. Wintner, A., and F. D. Murnaghan: Proc. Nat. Acad. Sci. U.S.A. Vol. 17 (1931) pp. 676–678.CrossRefGoogle Scholar
  36. 3.
    Murnaghan, F. D., and A. Wintner: Proc. Nat. Acad. Sci. U.S.A. Vol. 17 (1931) pp. 417–420.CrossRefGoogle Scholar
  37. 4.
    Weitzenböck, R.: Akad. Wetensch. Amsterdam, Proc. Vol. 35 (1932) pp. 328–330.zbMATHGoogle Scholar
  38. 1.
    Autonne, L.: Ann. Univ. Lyon II Vol. 38 (1915) pp. 1–77.Google Scholar
  39. 2.
    Beltrami, E.: Giorn. Mat. Battaglini Vol. 11 (1873) pp. 98–106.Google Scholar
  40. 3.
    Jordan, C.: J. Math, pures appl. II Vol. 19 (1874) pp. 35–54.Google Scholar
  41. 4.
    Sylvester, J. J.: C. R. Acad. Sci., Paris Vol. 108 (1889) pp. 651–653.zbMATHGoogle Scholar
  42. Sylvester, J. J.: Mess, of Math. Vol. 19 (1890) pp. 42–46.Google Scholar
  43. 5.
    Cosserat, E.: Ann. Fac. Sci. Univ. Toulouse Vol. 3 (1889) M. 1–12.Google Scholar
  44. 6.
    Schläfli: J. reine angew. Math. Vol. 65 (1866) pp. 185–187.zbMATHCrossRefGoogle Scholar
  45. 7.
    Hilton, H.: Mess, of Math. Vol. 41 (1912) pp. 146–154.Google Scholar
  46. 2.
    Muir, T.: Proc. Roy. Soc. Edinburgh Vol. 47 (1926–1927) pp. 252–282.Google Scholar
  47. 3.
    Loewy, A.: C.R. Acad. Sci., Paris Vol. 123 (1896) pp. 168–171.zbMATHGoogle Scholar
  48. Autonne, L.: Rend. Circ. mat. Palermo Vol. 16 (1902) pp. 104–128.zbMATHCrossRefGoogle Scholar
  49. 4.
    Cayley: J. reine angew. Math. Vol. 32 (1846) pp. 119–123.zbMATHCrossRefGoogle Scholar
  50. 5.
    Metzler, W, H.: Amer. J. Math. Vol. 15 (1892) pp. 274–282.MathSciNetCrossRefGoogle Scholar
  51. Prym, F.: Abh. Ges. Wiss. Göttingen Vol. 38 (1892) pp. 1–42.Google Scholar
  52. Taber, H.: Proc. London Math. Soc. Vol. 24 (1892) pp. 290–306.MathSciNetCrossRefGoogle Scholar
  53. Taber, H.: Proc. Amer. Acad. Arts Sci. Vol. 28 (1892–1893) pp. 212–221.CrossRefGoogle Scholar
  54. Taber, H.: Amer. J. Math. Vol. 16 (1893) pp. 123–130.MathSciNetCrossRefGoogle Scholar
  55. Taber, H.: Nachr. Ges. Wiss. Göttingen Vol. 3 (1900) pp. 298–303.Google Scholar
  56. 7.
    Voss, A.: Math. Ann. Vol. 13 (1878) pp. 320–374.MathSciNetzbMATHCrossRefGoogle Scholar
  57. 8.
    Goursat, E.: Ann. École norm. III Vol. 6 (1889) pp. 1–102.MathSciNetGoogle Scholar
  58. 10.
    Autonne, L.: C. R. Acad. Sci., Paris Vol. 136 (1903) pp. 1185–1186.zbMATHGoogle Scholar
  59. Autonne, L.: Ann. Univ. Lyon II Vol. 12 (1903) pp. 1–124.Google Scholar
  60. 11.
    Vitali, G.: Boll. Un. Mat. Ital. Vol. 7 (1928) pp. I–7.Google Scholar
  61. 1.
    Stieltjes, T. J.: Acta math. Vol. 6 (1885) pp. 319–320.MathSciNetzbMATHCrossRefGoogle Scholar
  62. 2.
    Netto, E.: Acta math. Vol. 9 (1887) pp. 295–300.MathSciNetzbMATHCrossRefGoogle Scholar
  63. 4.
    Frobenius: J. reine angew. Math. Vol. 84 (1878) p. 48.Google Scholar
  64. 5.
    Toscano, L.: Rend. Roy. Inst. Lombardo IIa Vol. 61 (1928) pp. 187–195.Google Scholar
  65. 6.
    Voss, A.: Math. Ann. Vol. 13 (1878) pp. 320–374.MathSciNetzbMATHCrossRefGoogle Scholar
  66. 9.
    Stenzel, H.: Math. Z. Vol. 15 (1922) pp. 1–25.MathSciNetCrossRefGoogle Scholar
  67. 11.
    Schmidt, E.: Math. Ann. Vol. 63 (1907) pp. 433–476.MathSciNetzbMATHCrossRefGoogle Scholar
  68. 13.
    Radon, J.: Abh. math. Semin. Hamburg. Univ. Vol. 1 (1921) pp. 1–14.CrossRefGoogle Scholar

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  • C. C. Mac Duffee

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