Weyl's inequality


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
for otherwise estimating the modulus of the exponential sum by means of the triangle inequality as provides a better bound.

Weyl's inequality in matrix theory

Weyl's inequality about perturbation

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 a Hermitian matrix but there is an uncertainty about its entries. 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 any two of M, H and P are n by n Hermitian matrices, where M has eigenvalues
and H has eigenvalues
and P has eigenvalues
then the following inequalities hold for :
More generally, if , we have
If P is positive definite then this implies
Note that we can order the eigenvalues because the matrices are Hermitian and therefore the eigenvalues are real.

Weyl's inequality between eigenvalues and singular values

Let have singular values and eigenvalues ordered so that. Then
For, with equality for.

Applications

Estimating perturbations of the spectrum

Assume that we have a bound on P in the sense that we know that its spectral norm satisfies. Then it follows that all its eigenvalues are bounded in absolute value by. Applying Weyl's inequality, it follows that the spectra of M and H are close in the sense that

Weyl's inequality for singular values

The singular values of a square matrix M are the square roots of eigenvalues of M*M. Since Hermitian matrices follow Weyl's inequality, if we take any matrix A then its singular values will be the square root of the eigenvalues of B=A*A which is a Hermitian matrix. Now since Weyl's inequality hold for B, therefore for the singular values of A.
This result gives the bound for the perturbation in the singular values of a matrix A due to perturbation in A.