Weyl's inequality
From Wikipedia, the free encyclopedia
In mathematics, there are at least two results known as "Weyl's inequality".
[edit] Weyl's inequality in number theory
In number theory, Weyl's inequality, named for Hermann Weyl, states that if M, N, a and q are integers, with a and q coprime, q > 0, and f is a real polynomial of degree k whose leading coefficient c satisfies
,
for some t greater than or equal to 1, then for any positive real number 
one has
This inequality will only be useful when
- q < Nk,
for otherwise estimating the modulus of the exponential sum by means of the triangle inequality as 
provides a better bound.
[edit] Weyl's inequality in matrix theory
In linear algebra, Weyl's inequality is a theorem about the changes to eigenvalues of a Hermitian matrix that is perturbed. It is useful if we wish to know the eigenvalues of the Hermitian matrix H but there is an uncertainty about the entries of H. We let H be the exact matrix and P be a perturbation matrix that represents the uncertainty. The matrix we 'measure' is M = H + P.
The theorem says that if M, H and P are all n by n Hermitian matrices, where M has eigenvalues
and H has eigenvalues
and P has eigenvalues
then the following inequalties hold for all 
:
.
If P is positive definite (e.g. ρn > 0) then this implies
.
Note that we can order the eigenvalues because the matrices are Hermitian and therefore the eigenvalues are real.
[edit] References
- "Matrix Theory", Joel N. Franklin, (Dover Publications, 1993) ISBN 0-486-41179-6
