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!
$\begingroup$
$\endgroup$
4
-
4$\begingroup$ Read Fulton's survey, arxiv.org/abs/math/9908012 $\endgroup$– MishaCommented Mar 11, 2012 at 1:56
-
3$\begingroup$ 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) $\endgroup$– Yemon ChoiCommented Mar 11, 2012 at 2:07
-
$\begingroup$ 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. $\endgroup$– Denis SerreCommented Mar 11, 2012 at 16:44
-
1$\begingroup$ 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. $\endgroup$– MishaCommented Mar 11, 2012 at 17:15
Add a comment
|