This is a bit of an amorophic question, as I am still exploring.
Let $A$ be a $d \times d$ matrix. We perform truncated SVD to get $A = U \Sigma V^{\top}$ where $U \in \mathbb{R}^{d \times m}$ for $m < d$.
What can we say about the matrix $U U^{\top} \in \mathbb{R}^{d \times d}$?
Clearly $U^{\top} U$ which is $m \times m$ would be the identity.
But I am trying to figure out what "interesting" things (without a very high bar for "interesting") we can say about $U U^{\top}$.