MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Hello, I know that given two matrices A and B, estimating the eigenvalues of A + B = C as a function of the eigenvalues of A and of the eigenvalues of B is generally a non-easy problem. I was wondering if the solution is known in the case where A is symmetric and B is diagonal.


share|cite|improve this question
Read Fulton's survey, – Misha Mar 11 '12 at 1:56
Assuming that by symmetric you mean real-symmetric, then this case would seem to be at least as hard as the case where A and B are Hermitian (since once can always conjugate A and B by a common matrix which diagonalizes B) – Yemon Choi Mar 11 '12 at 2:07
I voted to close as an exact duplicate of the older question:, where the answers address the issues that the OP probably has in mind. – Suvrit Mar 11 '12 at 3:26
One of the works for which Terry Tao was given a Fields medal is precisely solving this problem. More precisely, he (with collaborator Knutson), proved Alfred Horn's conjecture. Well documented in Fulton's paper mentionned above. – Denis Serre Mar 11 '12 at 16:44
People mostly know the eigenvalue problem for Hermitian matrices. However, the answer in the symmetric case is given by exactly the same inequalities (Klyachko's inequalities in non-recursive form and Horn's inequalities in the recursive form), see Fulton's survey article. – Misha Mar 11 '12 at 17:15

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.