Weyl's Inequality in Matrix Theory
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 the Hermitian matrix H but there is an uncertainty about the entries of H. We let H be the exact matrix and P be a perturbation matrix that represents the uncertainty. The matrix we 'measure' is .
The theorem says that if M, H and P are all n by n Hermitian matrices, where M has eigenvalues
and H has eigenvalues
and P has eigenvalues
then the following inequalties hold for :
If P is positive definite (e.g. ) then this implies
Note that we can order the eigenvalues because the matrices are Hermitian and therefore the eigenvalues are real.
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