Assume all matrices I discuss about are complex $N\times N$ matrices. I have a negative definite hermitian matrix $A$ whose eigen values are $(-c\lambda ,-\lambda,\ldots-\lambda) $ where $c$ and $\lambda$ are positive constants. I also have a rank one positive semi-definite hermitian Matrix $xx^{H}$ (whose eigen values clearly are $(||x||^{2},0,0...0)$. I am familiar with weyl's inequalities. I was wondering if one can find a exact formula for the eigen values of the matrix $B=A+xx^{H}$.
EDIT---
(thanks to users @adamW and @David) I have been able to reduce the problem to a seemingly simple one. There is a diagonal matrix $D$ such that all its entries are zero except for the one in the $(1,1)^{th}$ position, say $D(1,1)=c$ (hence eigen values are $(c,0,\ldots,0)$. I have another rank one positive-semi definite matrix $xx^H$ (again, whose eigen values clearly are $(||x||^{2},0,0...0)$. Even now I can't seem to find a clear solution for this seemingly simple problem. To put everything in perspective, I have the following matrix
\begin{align} B &=D+xx^H \\ D &= \left[ \begin{array}{cccc} c & 0 & \ldots & 0 \\ 0 & 0 & \ldots & 0 \\ \ldots & 0 & 0 & 0 \\ 0 & \dots & 0 & 0 \end{array} \right] \end{align}
what are the eigen values of $B$