If we have the QR decomposition of $A^p = Q_p R_p$, can we say anything about the matrix $Q_p^T A Q_p$?
For $p=1$, we clearly have that $Q^T A Q = RQ$. However, even at $Q_2^T A Q_2$, I don't really see any meaningful connection. The only thing I feel we can say about this matrix is that it has the same eigenvalues as $A$ due to $Q_j$ being unitary. However, I'm wondering if we can say anything beyond this basic fact, presumably by using the fact that $Q_p$ is part of the factorization of the $p$th power of $A$.