how to show $|r(q)-\lambda|=O(\|q-x\|_2^2)$
$r(q)=q^*Aq/(q^*q)$ and $\lambda$ is an eigenvalue, $A$ is a Hermitian matrix. $x$ is the unit eigenvector corresponding to $\lambda$. and $q$ is a unit vector. vectors are allowed to have complex entries.
also I know how to show:
$|r(q)-\lambda|\le2.\|A\|_2.\|q-x\|_2$
also ,for $A$ hermitaian, $\|A\|_2$ is the largest singular value(=largest eigenvalue)