Let $A \in \mathbb{R}^{n\times n}$ and let $u,v \in \mathbb{R}^{n\times 1}$. Suppose we know the following: $$ A\text{ has an eigenvalue at $1$},\quad \rho(A) = 1, \quad\text{and}\quad \rho(A + uv^\top) = \eta < 1. $$ In the case where $n=1$, we have $A=1$ and $u$ and $v$ are scalars, which implies that $uv = \pm\eta-1$.
I'm trying to find a relationship between $u,v$ and $\eta$ for general $n$. I'm not sure what it would look like exactly, but my hope is that $v^\top u$ should be related to $\eta$ somehow.
In general, under mild conditions, adding an arbitrary rank-1 matrix to $A$ can move its eigenvalues anywhere we choose (see Two matrices that are not similar have (almost) same eigenvalues and Eigenvalues of rank-$1$ update), so simply knowing the eigenvalues of $A$ doesn't tell us anything about the eigenvalues of the perturbed $A$. However, this case is different: I know something about the eigenvalues of $A$ and the perturbed $A$, and I'd like to deduce something about the size of the perturbation.
My first effort to simplify the problem was to assume a diagonal $A$ (which can be done without loss of generality if $A$ is diagonalizable), but I wasn't able to make much progress.