Acta Mathematica

, Volume 30, Issue 1, pp 297–304 | Cite as

On the roots of the characteristic equation of a linear substitution

  • T. J. I’a Bromwich
Article

Keywords

Characteristic Equation Linear Substitution 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literatur

  1. 1.
    Cauchy, 1829.Google Scholar
  2. 2.
    Brioschi, 1854.Google Scholar
  3. 3.
    Weierstrass, 1879.Google Scholar
  4. 1.
    Öfversigt af K. Vet. Akad. Förh. Stockholm, 1900, Bd. 57, p. 1099; Acta Mathematica, t. 25, 1902, p. 359.Google Scholar
  5. 2.
    Acta Mathematica, l. c., t. 25, 1902, p. 367.Google Scholar
  6. 3.
    Berliner Monatsberichte, 1858; Ges. Werke, Bd. 1, p. 243.Google Scholar
  7. 4.
    Rang, according toFrobenius.Google Scholar
  8. 1.
    Weierstrass, Berliner Monatsberichte, 1870; Ges. Werke, Bd. 3, p. 139.Google Scholar
  9. 1.
    That such a reduction is possible is contained implicitly inKronecker’s work on the reduction of a single bilinear form. For an explicit treatment, see my papers, Proc. Lond. Math. Soc., vol. 32, 1900, p. 321, § 4; vol. 33, 1901, p. 197, § 3; American Journal of Mathematics, vol. 23, 1901, p. 235.Google Scholar
  10. 1.
    There are onlyn(n−1) non-zero coefficients inC, becausec r,r=o.Google Scholar
  11. 2.
    Christoffel, Crelle’s Journal, Bd. 63, 1864, p. 252.Google Scholar
  12. 1.
    See for example § 6 of the first, or § 5 of the last, of my papers quoted above.Google Scholar
  13. 1.
    If it happens that the coefficients inC are pure imaginaries, so thatc r,r=0,c r,s=−c s,r, it can be proved (as in § 2) that\(\left| \beta \right| \leqq g_2 \left[ {\frac{I}{2}n(n - I)} \right]^{\frac{1}{2}}\).Google Scholar
  14. 1.
    It is obviously hopeless to use the invariant-factors of |B−λE| and |C−λE|, because these are alwayslinear; while |A−λE| may have invariant-factors of any degree up ton. In this paragraph the a’s are supposed real, so thatB andC are deduced fromA according to § 2 (not § 3).Google Scholar

Copyright information

© Beijers Bokförlagsaktiebolag 1906

Authors and Affiliations

  • T. J. I’a Bromwich
    • 1
  1. 1.GalwayIreland

Personalised recommendations