Firstly, what I want are the eigenvalues of a sum of matrices $(A + C)$.
I am not asking how to express them in terms of the eigenvalues of the summands*.
What I am hoping for is that there may be a relationship between the eigenvalues of another matrix, $(A+B)$ and those of $(A+C)$. Having read around this in the last few days I'm not optimistic about the possibility, but I would appreciate any comments or results.
$A$, $B$ and $C$ are Hermitian.
$[A,B]\ne 0$ and $[A,C]\ne 0$, however
So, the matrices $A$ and $B$ are generated, the sum is diagonalized, then B is diagonalized and C is produced from the eigenvectors of B and a function of its eigenvalues. The sum of $A$ and $C$ is then diagonalized, but I would like to be sure that there are no alternatives to this final step, as there is not just one matrix $(A+C)$ to be diagonalized but very many of them.
* I looked through Knutson and Tao (http://www.ams.org/notices/200102/fea-knutson.pdf), but it's beyond me.