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

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

[math]\displaystyle{ |c-a/q|\le tq^{-2}, }[/math]

for some t greater than or equal to 1, then for any positive real number [math]\displaystyle{ \scriptstyle\varepsilon }[/math] one has

[math]\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. }[/math]

This inequality will only be useful when

[math]\displaystyle{ q \lt N^k, }[/math]

for otherwise estimating the modulus of the exponential sum by means of the triangle inequality as [math]\displaystyle{ \scriptstyle\le\, N }[/math] provides a better bound.

Weyl's inequality in matrix theory

Weyl's inequality about perturbation

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 a Hermitian matrix but there is an uncertainty about its entries. We let H be the exact matrix and P be a perturbation matrix that represents the uncertainty. The matrix we 'measure' is [math]\displaystyle{ \scriptstyle M \,=\, H \,+\, P }[/math].

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

[math]\displaystyle{ \mu_1 \ge \cdots \ge \mu_n }[/math]

and H has eigenvalues

[math]\displaystyle{ \nu_1 \ge \cdots \ge \nu_n }[/math]

and P has eigenvalues

[math]\displaystyle{ \rho_1 \ge \cdots \ge \rho_n }[/math]

then the following inequalities hold for [math]\displaystyle{ \scriptstyle i \,=\, 1,\dots ,n }[/math]:

[math]\displaystyle{ \nu_i + \rho_n \le \mu_i \le \nu_i + \rho_1 }[/math]

More generally, if [math]\displaystyle{ \scriptstyle j+k-n \,\ge\, i \,\ge\, r+s-1 }[/math], we have

[math]\displaystyle{ \nu_j + \rho_k \le \mu_i \le \nu_r + \rho_s }[/math]

If P is positive definite (that is, [math]\displaystyle{ \scriptstyle\rho_n \,\gt \, 0 }[/math]) then this implies

[math]\displaystyle{ \mu_i \gt \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.

Weyl's inequality between eigenvalues and singular values

Let [math]\displaystyle{ A \in \mathbb{C}^{n \times n} }[/math] have singular values [math]\displaystyle{ \sigma_1(A) \geq \cdots \geq \sigma_n(A) \geq 0 }[/math] and eigenvalues ordered so that [math]\displaystyle{ |\lambda_1(A)| \geq \cdots \geq |\lambda_n(A)| }[/math]. Then

[math]\displaystyle{ |\lambda_1(A) \cdots \lambda_k(A)| \leq \sigma_1(A) \cdots \sigma_k(A) }[/math]

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

Applications

Estimating perturbations of the spectrum

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 [math]\displaystyle{ \|P\|_2 \le \epsilon }[/math]. Then it follows that all its eigenvalues are bounded in absolute value by [math]\displaystyle{ \epsilon }[/math]. Applying Weyl's inequality, it follows that the spectra of M and H are close in the sense that[2]

[math]\displaystyle{ |\mu_i - \nu_i| \le \epsilon \qquad \forall i=1,\ldots,n. }[/math]

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". https://terrytao.wordpress.com/2010/01/12/254a-notes-3a-eigenvalues-and-sums-of-hermitian-matrices/. 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