Suppose that I have two positive matrices, $A$, and $B$, and I know their Frobenius-Perron eigenvalues ($\lambda_A$, $\lambda_B$) and eigenvectors ($v_A$, $v_B$). I'm interested in what I can say about the Frobenius-Perron eigenvalue and eigenvector of $C=A+B$. I'm sure that this question must have been considered before, but I haven't found a good source.
Here are two particular questions:
(1) One clear bound on $\lambda_C$ is that $\max(\lambda_A,\lambda_B) \le \lambda_C$. Is there a nice upper bound? Also, if $v_A=v_B$, then $\lambda_C = \lambda_A + \lambda_B$. But can we do better than this if we know $v_A$ and $v_B$?
(2) What can we say about $v_C$? I expect it to be "somewhere between" $v_A$ and $v_B$, but that's just vague intuition. (Assume all the eigenvectors are normalized.) Is there a simple bound on the elements of $v_C$ given knowledge of $\lambda_A$, $\lambda_B$, $v_A$, and $v_B$?
I'd prefer not to assume that $A$ and $B$ are symmetric, but if the question has only been solved for the symmetric case, that's fine.
Edit: I realized that I should clarify that all of the $v$'s are right eigenvectors. If it helps to know the left Frobenius-Perron eigenvectors $u_A$ and $u_B$, then feel free to use this too.
Edit 2: In light of David's answer, then I've corrected my "intuitive" bound on $\lambda_C$ which was completely wrong.