# 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 an Hermitian matrix that is perturbed. It can be used to estimate the eigenvalues of a perturbed Hermitian matrix.

Let ${\displaystyle M=N+R,\,N,}$  and ${\displaystyle R}$  be n×n Hermitian matrices, with their respective eigenvalues ${\displaystyle \mu _{i},\,\nu _{i},\,\rho _{i}}$  ordered as follows:

${\displaystyle M:\quad \mu _{1}\geq \cdots \geq \mu _{n},}$
${\displaystyle N:\quad \nu _{1}\geq \cdots \geq \nu _{n},}$
${\displaystyle R:\quad \rho _{1}\geq \cdots \geq \rho _{n}.}$

Then the following inequalities hold:

${\displaystyle \nu _{i}+\rho _{n}\leq \mu _{i}\leq \nu _{i}+\rho _{1},\quad i=1,\dots ,n,}$

and, more generally,

${\displaystyle \nu _{j}+\rho _{k}\leq \mu _{i}\leq \nu _{r}+\rho _{s},\quad j+k-n\geq i\geq r+s-1.}$

In particular, if ${\displaystyle R}$  is positive definite then plugging ${\displaystyle \rho _{n}>0}$  into the above inequalities leads to

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

Note that these eigenvalues can be ordered, because they are real (as eigenvalues of Hermitian matrices).

### Weyl's inequality between eigenvalues and singular values

Let ${\displaystyle A\in \mathbb {C} ^{n\times n}}$  have singular values ${\displaystyle \sigma _{1}(A)\geq \cdots \geq \sigma _{n}(A)\geq 0}$  and eigenvalues ordered so that ${\displaystyle |\lambda _{1}(A)|\geq \cdots \geq |\lambda _{n}(A)|}$ . Then

${\displaystyle |\lambda _{1}(A)\cdots \lambda _{k}(A)|\leq \sigma _{1}(A)\cdots \sigma _{k}(A)}$

For ${\displaystyle k=1,\ldots ,n}$ , with equality for ${\displaystyle k=n}$ . [1]

## Applications

### Estimating perturbations of the spectrum

Assume that we have a bound on R in the sense that we know that its spectral norm (or, indeed, any consistent matrix norm) satisfies ${\displaystyle \|R\|_{2}\leq \epsilon }$ . 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 N are close in the sense that[2]

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

### 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.[3]

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

## Notes

1. ^ Toger A. Horn, and Charles R. Johnson Topics in Matrix Analysis. Cambridge, 1st Edition, 1991. p.171
2. ^ Weyl, Hermann. "Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)." Mathematische Annalen 71, no. 4 (1912): 441-479.
3. ^ Tao, Terence (2010-01-13). "254A, Notes 3a: Eigenvalues and sums of Hermitian matrices". Terence Tao's blog. Retrieved 25 May 2015.

## 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