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Weyl's inequality (Theorem)

Let $ A$ and $ E$ be two $ n \times n$ Hermitian matrices, with $ E$ positive semidefinite.

Let $ \lambda_i(A)$, $ \lambda_i(A+E)$, $ 1\leq i\leq n$ be the eigenvalues of $ A$ and $ A+E$ respectively, ordered in such a way that

$\displaystyle \vert\lambda_1\vert\leq \vert\lambda_2\vert\leq \cdots \leq \vert\lambda_n\vert. $

Then

$\displaystyle \lambda_i(A)\leq \lambda_i(A+E). $



"Weyl's inequality" is owned by Andrea Ambrosio.
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proof of Weyl's inequality (Proof) by Andrea Ambrosio
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Cross-references: eigenvalues, positive semidefinite, Hermitian matrices
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This is version 7 of Weyl's inequality, born on 2005-11-02, modified 2006-09-05.
Object id is 7463, canonical name is WeylsInequality.
Accessed 1702 times total.

Classification:
AMS MSC: 15A42 (Linear and multilinear algebra; matrix theory :: Inequalities involving eigenvalues and eigenvectors)
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