Eigenvalue of rank 1 perturbation $\lim_{n \to \infty}(A-n u v^*)$

Question

Suppose $A \in \mathcal M(n \times n)$, $u, v \in \mathbb C^n$ are fixed and $u$ is not an eigenvector of $A$. What will be the eigenvalue of following limit \begin{align*} \lim_{n \to \infty} (A-nuv^*). \end{align*} Is it minus infinity?