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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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marked as duplicate by Marc van Leeuwen, user127.0.0.1, Claude Leibovici, Davide Giraudo, Sami Ben Romdhane Feb 2 at 14:50

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Not completely sure what you are asking for, since you say you know that eigenvalues of $B$, and that you want to know about the eigenvalues of $B$. –  Martin Argerami Dec 21 '12 at 3:09
    
Sorry, maybe I've not been clear enough. I know the eigenvectors of $B$ not the eigenvalues. I also know the matrix A since it's rank 1 and I know the eigenvector. –  Stan Dec 21 '12 at 9:49
    
My bad, I misread the question. –  Martin Argerami Dec 21 '12 at 11:28

1 Answer 1

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 $B$.

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