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Suppose we have two positive definite matrices A and B. Is it correct to claim that all the eigenvalues of A+B are greater or equal to those of A?
Please note that: 1- I need to compare all the eigenvalues and not only the largest ones. 2- A and B are not necessarily diagonal.

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Yes. Weyl's inequality for matrices shows that what you say is true.

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The Courant minimax principle will work also: en.wikipedia.org/wiki/Courant_minimax_principle –  Terry Tao Jul 29 '11 at 5:52
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Oh, Terry! You are so kind to answer this one, after your work on A. Horn's conjecture, now Tao-Knutson's theorem. –  Denis Serre Jul 29 '11 at 8:41
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