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

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[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 .

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 , we have

If P is positive definite (that is, ) then this implies

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

Applications[edit]

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 . 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[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.

Notes[edit]

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

References[edit]

  • 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