|
|
| Line 1: |
Line 1: |
| In mathematics, there are at least two results known as '''"Weyl's inequality"'''.
| | Hello and welcome. My title is Figures Wunder. My day occupation is a meter reader. One of the extremely best things in the world for me is to do aerobics and now I'm attempting to earn money with it. For many years he's been residing in North Dakota and his family enjoys it.<br><br>Also visit my website ... at home std testing ([http://vei.cuaed.unam.mx/es/node/4882 simply click the up coming internet page]) |
| | |
| ==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 number|real]] [[polynomial]] of degree ''k'' whose leading coefficient ''c'' satisfies
| |
| | |
| :<math>|c-a/q|\le tq^{-2},\,</math>
| |
| | |
| for some ''t'' greater than or equal to 1, then for any positive real number <math>\scriptstyle\varepsilon</math> one has
| |
| | |
| :<math>\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.</math>
| |
| | |
| This inequality will only be useful when
| |
| | |
| :<math>q < N^k,\,</math>
| |
| | |
| for otherwise estimating the modulus of the [[exponential sum]] by means of the [[triangle inequality]] as <math>\scriptstyle\le\, N</math> 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 <math>\scriptstyle M \,=\, H \,+\, P</math>.
| |
| | |
| The theorem says that if ''M'', ''H'' and ''P'' are all ''n'' by ''n'' Hermitian matrices, where ''M'' has eigenvalues
| |
| | |
| :<math>\mu_1 \ge \cdots \ge \mu_n\, </math> | |
| | |
| and ''H'' has eigenvalues
| |
| | |
| :<math>\nu_1 \ge \cdots \ge \nu_n\, </math>
| |
| | |
| and ''P'' has eigenvalues
| |
| | |
| :<math>\rho_1 \ge \cdots \ge \rho_n\, </math>
| |
| | |
| then the following inequalties hold for <math>\scriptstyle i \,=\, 1,\dots ,n</math>:
| |
| | |
| :<math>\nu_i + \rho_n \le \mu_i \le \nu_i + \rho_1\, </math>
| |
| | |
| More generally, if <math>\scriptstyle j+k-n \,\ge\, i \,\ge\, r+s-1,\dots ,n</math>, we have
| |
| | |
| :<math>\nu_j + \rho_k \le \mu_i \le \nu_r + \rho_s\, </math>
| |
| | |
| If ''P'' is positive definite (that is, <math>\scriptstyle\rho_n \,>\, 0</math>) then this implies
| |
| | |
| :<math>\mu_i > \nu_i \quad \forall i = 1,\dots,n.\,</math>
| |
| | |
| 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 asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen", H. Weyl, Math. Ann., 71 (1912), 441–479
| |
| | |
| {{DEFAULTSORT:Weyl's Inequality}}
| |
| [[Category:Diophantine approximation]]
| |
| [[Category:Inequalities]]
| |
| [[Category:Linear algebra]]
| |
Hello and welcome. My title is Figures Wunder. My day occupation is a meter reader. One of the extremely best things in the world for me is to do aerobics and now I'm attempting to earn money with it. For many years he's been residing in North Dakota and his family enjoys it.
Also visit my website ... at home std testing (simply click the up coming internet page)