Weyl's inequality

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In mathematics, there are at least two results known as "Weyl's inequality".

Weyl's inequality in number theory[edit]

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

|c-a/q|\le tq^{-2},\,

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

\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

q < N^k,\,

for otherwise estimating the modulus of the exponential sum by means of the triangle inequality as \scriptstyle\le\, N provides a better bound.

Weyl's inequality in matrix theory[edit]

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 \scriptstyle M \,=\, H \,+\, P.

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

\mu_1 \ge \cdots \ge \mu_n\,

and H has eigenvalues

\nu_1 \ge \cdots \ge \nu_n\,

and P has eigenvalues

\rho_1 \ge \cdots \ge \rho_n\,

then the following inequalities hold for \scriptstyle i \,=\, 1,\dots ,n:

\nu_i + \rho_n \le \mu_i \le \nu_i + \rho_1\,

More generally, if \scriptstyle j+k-n \,\ge\, i \,\ge\, r+s-1,\dots ,n, we have

\nu_j + \rho_k \le \mu_i \le \nu_r + \rho_s\,

If P is positive definite (that is, \scriptstyle\rho_n \,>\, 0) then this implies

\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.


Estimating perturbations of the spectrum[edit]

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 \|P\|_2 \le \epsilon. Then it follows that all its eigenvalues are bounded in absolute value by \epsilon. Applying Weyl's inequality, it follows that the spectra of M and H are close in the sense that

|\mu_i - \nu_i| \le \epsilon \qquad \forall i=1,\ldots,n.

Weyl's inequality for singular values[edit]

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.[1]

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


  1. ^ Tao, Terence. "254A, Notes 3a: Eigenvalues and sums of Hermitian matrices". Terence Tao's blog. Retrieved 25 May 2015. 


  • 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