Known Results on eigenvalues of $A$, $B$, $A+B$? Consider the matrices $A$, $B$ and $A+B$. What are the best known results on relations between eigenvalues of these matrices? Please provide a reference. 
 A: Not an answer, adding resources from the comments so that it will be helpful for others who happen across this in future. 


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*https://mathoverflow.net/questions/90861/eigenvalues-of-the-sum-of-two-matrices

*https://mathoverflow.net/questions/4224/eigenvalues-of-matrix-sums
A: There is plenty of results on similar problem called one-rank update of eigenvalues formulated as follows: given eigendecomposition of $A$, find eigenvalues of matrix $C=A+uv^t$ where $uv^t$ is a one-rank matrix. There is special case of $u=v$ which is often covered in literature, for example, by Golub https://pdfs.semanticscholar.org/d2c3/bad41634b72d1677c2bef77f5220c85a05dc.pdf There is also a lot of results for symmetric real matrices or diagonal matrices, so I think it would be better to know something about $A$ and $B$ properties before searching more references.
Having suitable method for finding one-rank updates of eigenvalues, you can basically split summation of $A$ and $B$ on sequence of such one-rank updates, for example, by making each $uv^t$ contain just one element equal to $b_{ij}-a_{ij}$. There is also similar problems with two-rank updates and multiple-rank updates which may be useful, for example, https://arxiv.org/ftp/arxiv/papers/1706/1706.00773.pdf
