Let $A$ be a matrix, $v$ be a unit vector, $\mu$ be a scalar, and $r = Av — \mu v$. I want to show that there is a matrix $L$ with $||L||_F=||r||_2$ such that $A + L$ has eigenvalue $\mu$ and eigenvector $x$.
The only thing that occurs to me is to start operations with the norms, but I have not reached anything
Could someone give me a hint?