I know that given two matrices $A$ and $B$, estimating the eigenvalues of $A + B$ by the eigenvalues of $A$ and $B$ is generally a non-easy problem. In particular, there are some results for matrices that commute (multiplicatively!), hermitian matrices etc.

In this case $B=\operatorname{diag}(1, 0,\dots,0)$ and the sum of the elements of every row of $A$ is $0$; each eigenvalue of $A$ is non-negative. I was wondering if the solution is known in this case, at least if one can say something about the sign of eigenvalues of $A+B$.

Thanks in advance.