# 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

${\displaystyle |c-a/q|\leq tq^{-2},}$

for some t greater than or equal to 1, then for any positive real number ${\displaystyle \scriptstyle \varepsilon }$ one has

${\displaystyle \sum _{x=M}^{M+N}\exp(2\pi if(x))=O\left(N^{1+\varepsilon }\left({t \over q}+{1 \over N}+{t \over N^{k-1}}+{q \over N^{k}}\right)^{2^{1-k}}\right){\text{ as }}N\to \infty .}$

This inequality will only be useful when

${\displaystyle q

for otherwise estimating the modulus of the exponential sum by means of the triangle inequality as ${\displaystyle \scriptstyle \leq \,N}$ provides a better bound.

## 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 ${\displaystyle \scriptstyle M\,=\,H\,+\,P}$.

The theorem says that if M, H and P are all n by n Hermitian matrices, where M has eigenvalues

${\displaystyle \mu _{1}\geq \cdots \geq \mu _{n}}$

and H has eigenvalues

${\displaystyle \nu _{1}\geq \cdots \geq \nu _{n}}$

and P has eigenvalues

${\displaystyle \rho _{1}\geq \cdots \geq \rho _{n}}$

then the following inequalities hold for ${\displaystyle \scriptstyle i\,=\,1,\dots ,n}$:

${\displaystyle \nu _{i}+\rho _{n}\leq \mu _{i}\leq \nu _{i}+\rho _{1}}$

More generally, if ${\displaystyle \scriptstyle j+k-n\,\geq \,i\,\geq \,r+s-1,\dots ,n}$, we have

${\displaystyle \nu _{j}+\rho _{k}\leq \mu _{i}\leq \nu _{r}+\rho _{s}}$

If P is positive definite (that is, ${\displaystyle \scriptstyle \rho _{n}\,>\,0}$) then this implies

${\displaystyle \mu _{i}>\nu _{i}\quad \forall i=1,\dots ,n.}$

Note that we can order the eigenvalues because the matrices are Hermitian and therefore the eigenvalues are real.

## Applications

### 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 ${\displaystyle \|P\|_{2}\leq \epsilon }$. Then it follows that all its eigenvalues are bounded in absolute value by ${\displaystyle \epsilon }$. Applying Weyl's inequality, it follows that the spectra of M and H are close in the sense that[1]

${\displaystyle |\mu _{i}-\nu _{i}|\leq \epsilon \qquad \forall i=1,\ldots ,n.}$

### 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.[2]

This result gives the bound for the perturbation in singular values of a matrix A caused due to perturbation in A.

## References

• 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