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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. Thanks!

Rodrigo de Azevedo
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Michele
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    Read Fulton's survey, http://arxiv.org/abs/math/9908012 – Misha Mar 11 '12 at 01:56
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    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 02:07
  • 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

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