Suppose we have an orthogonal projection $P$ of rank $r$, which has an eigendecomposition $P=Q\begin{bmatrix}I_r & 0 \\ 0 & 0\end{bmatrix}Q^T$. Here $Q$ is an orthogonal matrix.
Is it true that $P$ and $QPQ^T$ have different eigenvectors? What is the relation between the new and old non-zero eigenvectors $Q_r$?
Since $Q$ is orthogonal, $Q^T=Q^{-1}$, the eigenvector of $QPQ^T$ are the images of the eigenvector of $P$ by $Q$.
If $P(x) =cx$, $QPQ^T(Q(x)) =QP(x)) =Q(cx) =cQ(x) $.
It is possible to choose $Q$ such that the eigenvectors are different.