Let $A \in \mathbb{R}^{d \times n}$ and let $B \in \mathbb{R}^{d \times n}$ where $d > n$.
Let $C = A \times B^{\top}$.
Let $U \Sigma V^{\top}$ be the SVD of $C$.
Can we say anything special about the matrix:
$A A^{\top} (U \Sigma^{-1} V^{\top}) B$ ?
I am especially interested in finding out about the sum of the values in this matrix.
Edit: I am pretty sure that $A^{\top} (U \Sigma^{-1} V^{\top}) B = I$. (I mocked around with that in Matlab. No real proof - but I also have a "feeling" that a proof would involve the pseudo-inverse of $AB^{\top}$ because $V \Sigma^{-1} U ^{\top}$ is the pseudo-inverse of $AB^{\top}$.