# Eigenvalues of a sum of matrices given eigenvalues of different sum

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

$[B,C]=0$.

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.

Thanks,

Joe

* I looked through Knutson and Tao (http://www.ams.org/notices/200102/fea-knutson.pdf), but it's beyond me.

• In short, you have $B = \sum_k \lambda_ke_ke_k^*$ and $C = \sum_k f(\lambda_k)e_ke_k^*$ as well as $A + B = \sum_k\mu_kf_kf_k^*$ and you would like to get the spectral representation of $A+C$ from this information? – Friedrich Philipp Mar 26 '16 at 23:28
• Yes, I think that's a nice way to summarize it. Thanks. – holyjoly Mar 26 '16 at 23:47