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Matric equations

  • Chapter
The Theory of Matrices

Part of the book series: Ergebnisse der Mathematik und Ihrer Grenƶgebiete ((MATHE1,volume 5))

Abstract

The general linear equation. If A 1 A 2,..., A h , B 1, B 2,..., B h , C are matrices of order n with elements in a field p, the general linear equation is of the form

$$A_1 XB_1 {\text{} } + {\text{} }A_2 XB_2 {\text{} } + {\text{} }...{\text{} } + {\text{} }A_h XB_h {\text{} } = {\text{} }C$$
((46.1))

where X is a matrix of order n, with elements in F, to be found. By replacing C by 0 we obtain the corresponding auxiliary equation. It is evident that if X 1 and X 2 are solutions of (46.1), their difference is a solution of the auxiliary equation. Hence the sum of a particular solution of (46.1) and the general solution of the corresponding auxiliary equation gives the general solution of (46.1).

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Notes

  1. Sylvester: C. R, Acad. Sci., Paris Vol. 99 (1884) pp. 117–118, 409-412, 432-436 and 527-529

    Google Scholar 

  2. Wedderburn, J. H. M.: Proc. Edinburgh Math. Soc. Vol. 22 (1904) pp. 49 to 53.

    Google Scholar 

  3. Hitchcock, F. L.: Proc. Nat. Acad. Sci. U.S.A. Vol. 8 (1922) pp. 78–83.

    Article  Google Scholar 

  4. Frobenius: J. reine angew. Math. Vol. 84 (1878) p. 8.

    Google Scholar 

  5. Autonne, L.: Ann. Univ. Lyon II Vol. 25 (1909) pp. 1–79.

    Google Scholar 

  6. Cecioni, F.: Ann. Scuola norm, super. Pisa Vol. 11 (1909) pp. 1–140.

    Google Scholar 

  7. Cayley: Mess. Math. Vol. 14 (1885) pp.108–112.

    Google Scholar 

  8. Sylvester: C. R, Acad. Sci., Paris Vol. 99 (1884) pp. 67–71 and 115-116.

    Google Scholar 

  9. Cecioni: Ann. Scuola norm, super. Pisa Vol. 11 (1909) pp. 1–40.

    Google Scholar 

  10. Cecioni: Atti Accad. naz. Lincei, Rend. V Vol. 181 (1909) pp. 566–571.

    Google Scholar 

  11. Landsberg, G.: J. reine angew. Math. Vol. 116 (1896) pp. 331–349.

    MATH  Google Scholar 

  12. Wilson, R.: Proc. London Math. Soc. II Vol. 30 (1930) pp. 359–366.

    Article  MATH  Google Scholar 

  13. Wilson, R.: Proc. London Math. Soc. II Vol. 33 (1932) pp. 517–524.

    Article  Google Scholar 

  14. Cecioni: Ann. Scuola norm, super. Pisa Vol. 11 (1909) pp. 1–40.

    Google Scholar 

  15. Rutherford, D.E.: Akad. Wetensch. Amsterdam, Proc. Vol. 35 (1932) pp. 54–59.

    Google Scholar 

  16. Weitzenböck, R.: Akad. Wetensch. Amsterdam, Proc. Vol. 35 (1932) pp. 60 to 61.

    Google Scholar 

  17. Frobenius: J. reine angew. Math. Vol. 84 (1878) pp. 1–63.

    Article  Google Scholar 

  18. Voss, A.: S.-B. Bayer. Akad. Wiss. Vol. 19 (1889) pp. 283–300.

    Google Scholar 

  19. Landsberg, G.: J. reine angew. Math. Vol. 116 (1896) pp. 331–349.

    MATH  Google Scholar 

  20. Hensel, K.: J. reine angew. Math. Vol. 127 (1904) pp. 116–166.

    MATH  Google Scholar 

  21. Cecioni, F.: Ann. Scuola norm, super. Pisa Vol. 11 (1909) pp. 1–140.

    Google Scholar 

  22. Amaldi, U.: Ist. Lombardo, Rend. II Vol. 45 (1912) pp. 433–445.

    Google Scholar 

  23. Sylvester: John Hopkins Circ. Vol. 3 (1884) pp. 33, 34, 57

    Google Scholar 

  24. Laurent, H.: J. Math, pures appl. V Vol. 4 (1898) pp. 75–119.

    Google Scholar 

  25. Phillips, H. B.: Amer. J. Math. Vol. 41 (1919) pp. 256–278.

    Article  Google Scholar 

  26. Voss, A.: S.-B. Bayer. Akad. Wiss. II Vol. 17 (1892) pp. 235–356.

    Google Scholar 

  27. Plemelj, J.: Mh. Math. Phys. Vol. 12 (1901) pp. 82–96.

    MathSciNet  MATH  Google Scholar 

  28. Taber, H.: Proc. Amer. Acad. Arts Sci. Vol. 26 (1890–1891) pp. 64–66.

    Article  Google Scholar 

  29. Taber, H.: Proc. Amer. Acad. Arts Sci. Vol. 27 (1891–1892) pp. 46–56.

    Article  Google Scholar 

  30. Sylvester: C. R, Acad. Sci., Paris Vol. 98 (1884) p. 471.

    Google Scholar 

  31. Frobenius: J. reine angew. Math. Vol. 84 (1878) pp. 1–63 Theorem XIII.

    Article  Google Scholar 

  32. Autonne, L.: J. École polytechn. Vol. 14 (1910) pp. 83–131.

    Google Scholar 

  33. Hilton, H.: Mess. Math. Vol. 41 (1911) pp. 110–118.

    Google Scholar 

  34. Krawtchouk, M.: Rend. Circ. mat. Palermo Vol. 51 (1927) pp. 126–130.

    Article  MATH  Google Scholar 

  35. Schur, I.: J. reine angew. Math. Vol. 130 (1905) pp. 66–76.

    Google Scholar 

  36. Ranum, A.: Amer. J. Math. Vol. 31 (1909) pp. 18–41.

    Article  MathSciNet  MATH  Google Scholar 

  37. Sylvester: C. R. Acad. Sci., Paris Vol. 94 (1882) pp. 396–399.

    Google Scholar 

  38. Lipschitz: Acta math. Vol. 10 (1887) pp. 137–144.

    Article  MathSciNet  MATH  Google Scholar 

  39. Baker, H. F.: Proc. London Math. Soc. Vol. 35 (1903) pp. 379–384.

    Article  MATH  Google Scholar 

  40. Ranum, A.: Amer. J. Math. Vol. 31 (1909) pp. 18–41.

    Article  MathSciNet  MATH  Google Scholar 

  41. Turnbull, H. W.: J. London Math. Soc. Vol. 2 (1927) pp. 242–244.

    Article  MathSciNet  MATH  Google Scholar 

  42. Vaidyanathaswamy, R.: J. London Math. Soc. Vol. 3 (1928) pp. 268 272.

    Google Scholar 

  43. Roth, W. E.: Trans. Amer. Math. Soc. Vol. 32 (1929) pp.61–80.

    Article  Google Scholar 

  44. Cayley: Philos. Trans. Roy. Soc. London Vol. 148 (1858) pp. 39–46.

    Article  Google Scholar 

  45. Sylvester: C. R. Acad. Sci., Paris Vol. 99 (1884) pp. 555–558 and 621-631.

    Google Scholar 

  46. Sylvester: Johns Hopkins Circ. Vol. 3 (1884) p. 122.

    Google Scholar 

  47. Sylvester: Philos. Mag. Vol. 18 (1884) pp. 454–458.

    Google Scholar 

  48. Sylvester: Quart. J. Math. Vol. 20 (1885) pp. 305–312.

    Google Scholar 

  49. Sylvester: C. R. Acad. Sci., Paris Vol. 99 (1884) pp. 13–15.

    Google Scholar 

  50. Buchheim, A.: Proc. London Math. Soc. Vol. 16 (1884) pp. 63–82.

    Article  MathSciNet  Google Scholar 

  51. Baker, H. F.: Proc. Cambridge Philos. Soc. Vol. 23 (1925) pp. 22–27.

    Article  Google Scholar 

  52. Kreis, H.: Vjschr. naturforsch. Ges. Zürich Vol. 53 (1908) pp. 366–376.

    Google Scholar 

  53. Roth, W. F.: Trans. Amer. Math. Soc. Vol. 30 (1928) pp. 579–596.

    MathSciNet  MATH  Google Scholar 

  54. Franklin, P.: J. Math. Physics, Massachusetts Inst. Technol. Vol. 10 (1932) pp. 289–314.

    Google Scholar 

  55. Rutherford, D. E.: Proc. Edinburgh Math. Soc. II Vol. 3 (1932) pp. 135 to 143.

    Google Scholar 

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Authors
  1. C. C. Mac Duffee
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Mac Duffee, C.C. (1933). Matric equations. In: The Theory of Matrices. Ergebnisse der Mathematik und Ihrer Grenƶgebiete, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-99234-6_8

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  • DOI: https://doi.org/10.1007/978-3-642-99234-6_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-98421-1

  • Online ISBN: 978-3-642-99234-6

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