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.


1 Answer 1


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

  • $\begingroup$ The Courant minimax principle will work also: en.wikipedia.org/wiki/Courant_minimax_principle $\endgroup$
    – Terry Tao
    Jul 29, 2011 at 5:52
  • 1
    $\begingroup$ Oh, Terry! You are so kind to answer this one, after your work on A. Horn's conjecture, now Tao-Knutson's theorem. $\endgroup$ Jul 29, 2011 at 8:41

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