# Eigenvalues of the sum of two matrices

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.

Thanks!

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Read Fulton's survey, arxiv.org/abs/math/9908012 – 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: mathoverflow.net/questions/4224/eigenvalues-of-matrix-sums, 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