Question

Suppose that $A$ is a positive matrix and that we let $R$ be the diagonal matrix of $A$'s row-sums. What can be said about the spectrum of $R-A$? I am particularly interested in the largest eigenvalue of $R-A$.

Answer 1

It is just a point of view. But it is more long for writing it as a comment.

If $A$ be the adjacency matrix of graph $G$, then $R-A$ is its Laplacian matrix and there are some good bounds for the radius (or maximum eigenvalues of this matrix) of matrix. For example, if we denote the maximum eigenvalue of this matrix by $\mu$, we have: $$\mu\leq \sqrt(2)\times max(d(v)^2+\sum_{uv\in E(G)}{d(u)})^\frac{1}{2}$$ where $v$ changes in the vertices of the graph $G$. In most cases, we can extend such relations to the positive definite matrices. For more such bounds you can see the paper:

Bounds on the (Laplacian) spectral radius of graphs, written by $Lingsheng$ $Shi$ and published in Linear Algebra and Its Application.

But in general, as you want, I think there is not good bound.