# Matric equations

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

## Keywords

London Math Simple Root Characteristic Root Invariant Factor Great Common Divisor
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.

## Notes

1. 1.
Sylvester: C. R, Acad. Sci., Paris Vol. 99 (1884) pp. 117–118, 409-412, 432-436 and 527-529Google Scholar
2. 2.
Wedderburn, J. H. M.: Proc. Edinburgh Math. Soc. Vol. 22 (1904) pp. 49 to 53.Google Scholar
3. 3.
Hitchcock, F. L.: Proc. Nat. Acad. Sci. U.S.A. Vol. 8 (1922) pp. 78–83.
4. 1.
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. 2.
Cecioni, F.: Ann. Scuola norm, super. Pisa Vol. 11 (1909) pp. 1–140.Google Scholar
7. 3.
Cayley: Mess. Math. Vol. 14 (1885) pp.108–112.Google Scholar
8. 4.
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. 5.
Cecioni: Atti Accad. naz. Lincei, Rend. V Vol. 181 (1909) pp. 566–571.Google Scholar
11. 1.
Landsberg, G.: J. reine angew. Math. Vol. 116 (1896) pp. 331–349.
12. 2.
Wilson, R.: Proc. London Math. Soc. II Vol. 30 (1930) pp. 359–366.
13. Wilson, R.: Proc. London Math. Soc. II Vol. 33 (1932) pp. 517–524.
14. 4.
Cecioni: Ann. Scuola norm, super. Pisa Vol. 11 (1909) pp. 1–40.Google Scholar
15. 5.
Rutherford, D.E.: Akad. Wetensch. Amsterdam, Proc. Vol. 35 (1932) pp. 54–59.Google Scholar
16. 6.
Weitzenböck, R.: Akad. Wetensch. Amsterdam, Proc. Vol. 35 (1932) pp. 60 to 61.Google Scholar
17. 1.
Frobenius: J. reine angew. Math. Vol. 84 (1878) pp. 1–63.
18. 3.
Voss, A.: S.-B. Bayer. Akad. Wiss. Vol. 19 (1889) pp. 283–300.Google Scholar
19. 4.
Landsberg, G.: J. reine angew. Math. Vol. 116 (1896) pp. 331–349.
20. 5.
Hensel, K.: J. reine angew. Math. Vol. 127 (1904) pp. 116–166.
21. 6.
Cecioni, F.: Ann. Scuola norm, super. Pisa Vol. 11 (1909) pp. 1–140.Google Scholar
22. 7.
Amaldi, U.: Ist. Lombardo, Rend. II Vol. 45 (1912) pp. 433–445.Google Scholar
23. 9.
Sylvester: John Hopkins Circ. Vol. 3 (1884) pp. 33, 34, 57Google Scholar
24. 10.
Laurent, H.: J. Math, pures appl. V Vol. 4 (1898) pp. 75–119.Google Scholar
25. 11.
Phillips, H. B.: Amer. J. Math. Vol. 41 (1919) pp. 256–278.
26. 12.
Voss, A.: S.-B. Bayer. Akad. Wiss. II Vol. 17 (1892) pp. 235–356.Google Scholar
27. 13.
Plemelj, J.: Mh. Math. Phys. Vol. 12 (1901) pp. 82–96.
28. 14.
Taber, H.: Proc. Amer. Acad. Arts Sci. Vol. 26 (1890–1891) pp. 64–66.
29. 15.
Taber, H.: Proc. Amer. Acad. Arts Sci. Vol. 27 (1891–1892) pp. 46–56.
30. 1.
Sylvester: C. R, Acad. Sci., Paris Vol. 98 (1884) p. 471.Google Scholar
31. 2.
Frobenius: J. reine angew. Math. Vol. 84 (1878) pp. 1–63 Theorem XIII.
32. 4.
Autonne, L.: J. École polytechn. Vol. 14 (1910) pp. 83–131.Google Scholar
33. 5.
Hilton, H.: Mess. Math. Vol. 41 (1911) pp. 110–118.Google Scholar
34. 6.
Krawtchouk, M.: Rend. Circ. mat. Palermo Vol. 51 (1927) pp. 126–130.
35. 7.
Schur, I.: J. reine angew. Math. Vol. 130 (1905) pp. 66–76.Google Scholar
36. 8.
Ranum, A.: Amer. J. Math. Vol. 31 (1909) pp. 18–41.
37. 9.
Sylvester: C. R. Acad. Sci., Paris Vol. 94 (1882) pp. 396–399.Google Scholar
38. 10.
Lipschitz: Acta math. Vol. 10 (1887) pp. 137–144.
39. 11.
Baker, H. F.: Proc. London Math. Soc. Vol. 35 (1903) pp. 379–384.
40. 1.
Ranum, A.: Amer. J. Math. Vol. 31 (1909) pp. 18–41.
41. 2.
Turnbull, H. W.: J. London Math. Soc. Vol. 2 (1927) pp. 242–244.
42. 3.
Vaidyanathaswamy, R.: J. London Math. Soc. Vol. 3 (1928) pp. 268 272.Google Scholar
43. 4.
Roth, W. E.: Trans. Amer. Math. Soc. Vol. 32 (1929) pp.61–80.
44. 5.
Cayley: Philos. Trans. Roy. Soc. London Vol. 148 (1858) pp. 39–46.
45. 6.
Sylvester: C. R. Acad. Sci., Paris Vol. 99 (1884) pp. 555–558 and 621-631.Google Scholar
46. 7.
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. 1.
Buchheim, A.: Proc. London Math. Soc. Vol. 16 (1884) pp. 63–82.
51. 3.
Baker, H. F.: Proc. Cambridge Philos. Soc. Vol. 23 (1925) pp. 22–27.
52. 6.
Kreis, H.: Vjschr. naturforsch. Ges. Zürich Vol. 53 (1908) pp. 366–376.Google Scholar
53. 7.
Roth, W. F.: Trans. Amer. Math. Soc. Vol. 30 (1928) pp. 579–596.
54. 8.
Franklin, P.: J. Math. Physics, Massachusetts Inst. Technol. Vol. 10 (1932) pp. 289–314.Google Scholar
55. 9.
Rutherford, D. E.: Proc. Edinburgh Math. Soc. II Vol. 3 (1932) pp. 135 to 143.Google Scholar