In mathematics, there are at least two results known as "Weyl's inequality".
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
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 .
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 :
If P is positive definite (e.g. ) then this implies
Note that we can order the eigenvalues because the matrices are Hermitian and therefore the eigenvalues are real.
- Matrix Theory, Joel N. Franklin, (Dover Publications, 1993) ISBN 0-486-41179-6
- "Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen", H. Weyl, Math. Ann., 71 (1912), 441–479