In mathematics, there are at least two results known as Weyl's inequality.
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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
This inequality will only be useful when
for otherwise estimating the modulus of the exponential sum by means of the triangle inequality as
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 inequalities hold for
More generally, if
If P is positive definite (that is,
Note that we can order the eigenvalues because the matrices are Hermitian and therefore the eigenvalues are real.
Estimating perturbations of the spectrum
Assume that we have a bound on P in the sense that we know that its spectral norm (or, indeed, any consistent matrix norm) satisfies
Weyl's inequality for singular values
The singular values {σk} of a square matrix M are the square roots of eigenvalues of M*M (equivalently MM*). 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 singular values of a matrix A caused due to perturbation in A.