Functions of Matrices

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

Abstract

Power series in matrices. Let \(P\left( \lambda \right) = \sum\limits_{i = 0}^\infty {{a_i}{\lambda ^i}} \) be an ordinary power series with complex coefficients in the complex variable λ. If for a matrix A of order n with complex elements every element of
$$P_m (A) = \sum\limits_{i = 0}^m {a_i \lambda ^{_i} } $$
approaches a finite limit as m → ∞
$$P(A) = \sum\limits_{i = 0}^\infty {a_i \lambda ^{_i} } $$
the matrix is said to exist and to be equal to the matrix of these limiting values.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

  1. 1.
    Weitzenböck, R.: Akad. Wetensch. Amsterdam, Proc. Vol. 35 (1932) pp. 157 to 161.MATHGoogle Scholar
  2. 2.
    Roth, W. E.: Trans. Amer. Math. Soc. Vol. 32 (1930) pp. 61–80.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Hurwitz, A.: Math. Ann. Vol. 88 (1923) PP-1–25.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Eddington, A. S.: J. London Math. Soc. Vol. 7 (1932) pp. 58–68.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Newman, M. H. A.: J. London Math. Soc. Vol. 7 (1932) pp. 93–99.CrossRefGoogle Scholar
  6. 1.
    Hensel, K.: J. reine angew. Math. Vol. 155 (1926) pp. 107–110.MATHGoogle Scholar
  7. 2.
    Weyr, E.: Bull. Sci. math. II Vol. 11 (1887) pp. 205–215.Google Scholar
  8. 1.
    Peano, G.: Math. Ann. Vol. 32 (1888) pp. 450–456.MathSciNetMATHCrossRefGoogle Scholar
  9. 2.
    Carvallo, E.: Mh. Math. Phys. Vol. 2 (1891) PP-177–216, 225-266 and 311-330.MathSciNetMATHGoogle Scholar
  10. 3.
    Taber, H.: Amer. j. Math. Vol. 12 (1890) pp. 337–396.MathSciNetCrossRefGoogle Scholar
  11. Taber, H.: Amer. j. Math. Vol. 13 (1891) pp. 159–172.MathSciNetCrossRefGoogle Scholar
  12. 4.
    Taber, H.: Proc. Amer. Acad. Arts Sci. Vol. 27 (1891–1892) pp. 163–165.CrossRefGoogle Scholar
  13. 5.
    Metzler, W, H.: Amer. J. Math. Vol. 14 (1892) pp. 326–377.MathSciNetCrossRefGoogle Scholar
  14. 6.
    Phillips, H. B.: Amer. J. Math. Vol. 41 (1919) pp. 266–278.MathSciNetMATHCrossRefGoogle Scholar
  15. 7.
    Sylvester: Philos. Mag. Vol. 16 (1883) pp. 267–269.Google Scholar
  16. 1.
    Buchheim, A.: Philos. Mag. V Vol. 22 (1886) pp. 173–174.CrossRefGoogle Scholar
  17. 2.
    Dirac, P. A. M.: Proc. Cambridge Philos. Soc. Vol. 23 (1926) pp. 412–418.MATHCrossRefGoogle Scholar
  18. 4.
    Fantappié, L.: C. R, Acad. Sci., Paris Vol. 186 (1928) pp. 619–621.MATHGoogle Scholar
  19. 5.
    Giorgi, G.: Atti Accad. naz. Lincei, Rend. VI Vol. 8 (1928) pp. 3–8.MATHGoogle Scholar
  20. 1.
    Cipolla, M.: Rend. Circ. mat. Palermo Vol. 56 (1932) pp. 144–154.CrossRefGoogle Scholar
  21. 2.
    Botasso, M.: Rend. Circ. mat. Palermo Vol. 35 (1913) pp.1–46.CrossRefGoogle Scholar
  22. 3.
    Martis-Biddau, S.: Atti Accad. naz. Lincei, Rend. VI Vol. 8 (1928) pp. 130 to 133.Google Scholar
  23. 4.
    Martis-Biddau, S.: Atti Accad. naz. Lincei, Rend. VI Vol. 8 (1928) pp. 276 to 280.Google Scholar
  24. 5.
    Martis-Biddau, S.: Atti Accad. naz. Lincei, Rend. VI Vol. 9 (1929) pp. 206 to 213.Google Scholar
  25. 6.
    Porcu-Tortrini, E.: Atti Accad. naz. Lincei, Rend. VI Vol. 7 (1928) pp. 206 to 208.Google Scholar
  26. 7.
    Amante, S.: Atti Accad. naz. Lincei, Rend. VI Vol. 12 (1930) p. 290.MATHGoogle Scholar
  27. 8.
    Wedderburn, J. H. M.: Trans. Amer. Math. SOC, Vol. 16 (1915) PP. 328 to 332.MathSciNetCrossRefGoogle Scholar
  28. 1.
    Birkhoff, G. D.: Math. Ann Vol. 74 (1913) pp. 122–133.MathSciNetCrossRefGoogle Scholar
  29. 2.
    Birkhoff: Trans. Amer. Math. Soc. Vol. 17 (1916) pp. 386–404.MathSciNetMATHCrossRefGoogle Scholar
  30. 3.
    Wedderburn, J. H. M.: Bull. Amer. Math. Soc. Vol. 31 (1925) pp. 304 to 308.Google Scholar
  31. 4.
    Volterra, V.: Atti Accad. naz. Lincei, Rend. IV Vol. 31 (1887) PP. 393–396.Google Scholar
  32. 1.
    Volterra: Rend. Circ. mat. Palermo Vol. 2 (1888) pp. 69–75.MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1933

Authors and Affiliations

  • C. C. Mac Duffee

There are no affiliations available

Personalised recommendations