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Weyl's inequality

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In linear algebra, Weyl's inequality is a theorem about the changes to eigenvalues of an Hermitian matrix that is perturbed. It can be used to estimate the eigenvalues of a perturbed Hermitian matrix.

Weyl's inequality about perturbation

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Let be Hermitian on inner product space with dimension , with spectrum ordered in descending order . Note that these eigenvalues can be ordered, because they are real (as eigenvalues of Hermitian matrices).[1]

Weyl inequality — 

Proof

By the min-max theorem, it suffices to show that any with dimension , there exists a unit vector such that .

By the min-max principle, there exists some with codimension , such that Similarly, there exists such a with codimension . Now has codimension , so it has nontrivial intersection with . Let , and we have the desired vector.

The second one is a corollary of the first, by taking the negative.

Weyl's inequality states that the spectrum of Hermitian matrices is stable under perturbation. Specifically, we have:[1]

Corollary (Spectral stability) —  where
is the operator norm.

In jargon, it says that is Lipschitz-continuous on the space of Hermitian matrices with operator norm.

Weyl's inequality between eigenvalues and singular values

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Let have singular values and eigenvalues ordered so that . Then

For , with equality for . [2]

Applications

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Estimating perturbations of the spectrum

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Assume that is small in the sense that its spectral norm satisfies for some small . Then it follows that all the eigenvalues of are bounded in absolute value by . Applying Weyl's inequality, it follows that the spectra of the Hermitian matrices M and N are close in the sense that[3]

Note, however, that this eigenvalue perturbation bound is generally false for non-Hermitian matrices (or more accurately, for non-normal matrices). For a counterexample, let be arbitrarily small, and consider

whose eigenvalues and do not satisfy .

Weyl's inequality for singular values

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Let be a matrix with . Its singular values are the positive eigenvalues of the Hermitian augmented matrix

Therefore, Weyl's eigenvalue perturbation inequality for Hermitian matrices extends naturally to perturbation of singular values.[1] This result gives the bound for the perturbation in the singular values of a matrix due to an additive perturbation :

where we note that the largest singular value coincides with the spectral norm .

Notes

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  1. ^ a b c Tao, Terence (2010-01-13). "254A, Notes 3a: Eigenvalues and sums of Hermitian matrices". Terence Tao's blog. Retrieved 25 May 2015.
  2. ^ Roger A. Horn, and Charles R. Johnson Topics in Matrix Analysis. Cambridge, 1st Edition, 1991. p.171
  3. ^ 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.

References

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  • 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