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.
be Hermitian on inner product space
, 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 —
By the min-max theorem, it suffices to show that any
, there exists a unit vector
By the min-max principle, there exists some
Similarly, there exists such a
, so it has nontrivial intersection with
, 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) —
is the operator norm.
is Lipschitz-continuous on the space of Hermitian matrices with operator norm.
have singular values
and eigenvalues ordered so that
is small in the sense that its spectral norm satisfies
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).
be arbitrarily small, and consider whose eigenvalues
Its singular values
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