Acta Mathematica

, Volume 30, Issue 1, pp 297–304

# On the roots of the characteristic equation of a linear substitution

• T. J. I’a Bromwich
Article

### Keywords

Characteristic Equation Linear Substitution

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### Literatur

1. 1.
2. 2.
3. 3.
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.
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