Eigenpair residual ($r=Av-\lambda v$)

For an approximate eigenpair $(\tilde{\lambda },\tilde{v})$, the residual is

If $\Vert \tilde{v}{\Vert }_2=1$, the smallest operator-norm perturbation that makes $(\tilde{\lambda },\tilde{v})$ an exact eigenpair has norm $\Vert r{\Vert }_2$. One such rank-one perturbation is