In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matric ...
, Weyl's inequality is a theorem about the changes to
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of an
Hermitian matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -t ...
that is perturbed. It can be used to estimate the eigenvalues of a perturbed Hermitian matrix.
Weyl's inequality about perturbation
Let
and
be ''n''×''n'' Hermitian matrices, with their respective eigenvalues
ordered as follows:
:
:
:
Then the following inequalities hold:
:
and, more generally,
:
In particular, if
is positive definite then plugging
into the above inequalities leads to
:
Note that these eigenvalues can be ordered, because they are real (as eigenvalues of Hermitian matrices).
Weyl's inequality between eigenvalues and singular values
Let
have singular values
and eigenvalues ordered so that
. Then
:
For
, with equality for
.
Applications
Estimating perturbations of the spectrum
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
:
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 value In mathematics, in particular functional analysis, the singular values, or ''s''-numbers of a compact operator T: X \rightarrow Y acting between Hilbert spaces X and Y, are the square roots of the (necessarily non-negative) eigenvalues of the se ...
s
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
{{DEFAULTSORT:Weyl's Inequality
Diophantine approximation
Inequalities
Linear algebra