Suppose $A$ is a square matrix over complex numbers and $u$ is an eigenvector with eigenvalue $\alpha$. Consider perturbing $A$ using $u$ to get $B = A + uv^H$ for some vector $v$. Then we would like to relate the characteristic polynomial of $A$ with that of $B$ as
$(\lambda - \alpha)$ $det(\lambda I_n - B)$ = $(\lambda - \alpha - v^Hu)$ $det(\lambda I_n - A)$
Any hints will be appreciated.