Suppose we've got a matrix $P$ and a perturbed version $\hat{P}=P+E.$ Given that $v$ is an eigenvector of $P$ with $Pv=0,$ I'd like to get as sharp a bound as possible on $\hat{P}v$ (in terms of whatever perturbation-measure is most appropriate.
I could write $\hat{P}v = \hat{P}\hat{v} + \hat{P}(v-\hat{v})$ for the eigenvector $\hat{v}$ of $\hat{P}$ corresponding to $v$ (i.e. the eigenvector whose associated eigenvalue is smallest) and then apply classical perturbation theorems to both expressions, but this seems crude and also to miss the point. I feel like working with a zero-eigenvalue should allow me to find a better bound, somehow.
Any help would be tremendously appreciated, guys!