Weyl's inequality

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

Weyl's inequality about perturbation

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).

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

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 Let have singular values and eigenvalues ordered so that . Then

For , with equality for . Roger A. Horn, and Charles R. Johnson Topics in Matrix Analysis. Cambridge, 1st Edition, 1991. p.171

Applications

Estimating perturbations of the spectrum

Let Hermitian matrices and differ by a matrix . 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 thatWeyl, 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.

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 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. 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

References

Matrix Theory, Joel N. Franklin, (Dover Publications, 1993) "Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen", H. Weyl, Math. Ann., 71 (1912), 441–479