Inverse of $UAA^TU^T$

Question

Let $U\in \mathbb{R}^{n\times m}$, $n<m$ be a matrix with orthogonal rows, $UU^T=I$, and $A\in\mathbb{R}^{m\times k}$, $m<k$ be any general real matrix. What can I say on $(UAA^TU^T)^{-1}$ as a function of $U$ and $A$?

Answer 1

Well, $$(UAA^TU^T)^n=U(AA^T)^nU^T,$$ so $$(UAA^TU^T)^{-1}=U(AA^T)^{-1}U^T,$$