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Let $A$ be a rank 1 positive semidefinite matrix and $B$ a Hermitian matrix. Suppose I know the eigenvectors of both $A$ and $B$ and that $A-B$ is also positive semidefinite.

Apart from Weyl's inequality is there anything that can be deduced about the eigenvalues of $B$?

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    My bad, I misread the question.2012-12-21

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I think that you should expect to be able to say very little about the eigenvalues of $B$: any negative definite $B$ will satisfy your conditions, which means that any $n$-tuple of negative numbers can appear as eigenvalues of $B$.

Some more information could be gathered from specific knowledge of the eigenvectors; for example, if $A$ and $B$ share an eigenvector, that gives you a condition on an eigenvalue of $B$.