Can $(AA^T)^{-1}A$ be simplified? Given an $m\times n$ matrix $A$, where $m<n$, can the expression $(AA^T)^{-1}A$ be simplified?

In general no, but you could express it in different terms. One instance of what I say is QR decomposition, so doing the decomposition for $A^T$ $$A^T = QR$$
You will get
$$(AA^T)^{-1}A = (R^TQ^TQR)^{-1}R^TQ^T = (R^TR)^{-1}R^TQ^T$$
