Here, https://math.stackexchange.com/a/1146455, it is noted that eigenprojections are continuous, but eigenvectors are not. Are there any conditions where the eigenvalues are not simple, but the eigenvectors are continuous?
For example, assume we're using the $\infty$ norm and the eigenvectors are assumed to be normalized to some constant $c$. If $v_P$ and $v_Q$ are the first eigenvectors of $P$ and $Q$, then $||v_P−v_Q||_{\infty}$ is always less than some constant, for example, $c+1$. Can we claim continuity in this case?
EDIT:
To clarify, given that the eigenprojections of $P$ and $Q$, call them $R_P$ and $R_Q$, are continuous functions of $P$ and $Q$, doesn't this imply continuity of the eigenvectors under certain conditions? For example, if we note that $||R_P-R_Q||_{\infty} < \delta$, for some constant $\delta$, then this implies a constant bound on $||v_P-v_Q||_{\infty}$ as well. While continuity may not be the best term to use to describe their relationship with $P$ and $Q$, all the eigenvectors are "close".
