Weyl's inequality

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

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

for some "t" greater than or equal to 1, then for any positive real number scriptstylevarepsilon one has

:sum_{x=M+1}^{M+N}exp(2pi if(x))=Oleft(N^{1+varepsilon}left({tover q}+{1over N}+{tover N^{k-1+{qover N^k} ight)^{2^{1-k ight) ext{ as }N oinfty.

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 scriptstylele, 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 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

: u_1 ge cdots ge u_n,

and "P" has eigenvalues

: ho_1 ge cdots ge ho_n,

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

: u_i + ho_n le mu_i le u_i + ho_1.,

If "P" is positive definite (e.g. scriptstyle ho_n ,>, 0) then this implies

:mu_i > u_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.

References

* "Matrix Theory", Joel N. Franklin, (Dover Publications, 1993) ISBN 0-486-41179-6

* "Das asymptotisher Verteilungsgesetz der Eigenwerte lineare partialler Differentialgleichungen", H.Weyl, Math.Ann., 71 (1912),441-479


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