# Do singular values of a point set determine its shape?

Suppose I have $k$ points in $d$ dimensions. Let A be a $k\times d$ matrix with $i$th row giving the coordinates of $i$th point. Do singular values of this matrix have an interpretation as some kind of geometric invariant?

I tried a set of convex point sets formed by centroids of vertices of 8 dimensional 7-simplex, and there were 49 distinct singular value sets, which is the same as the number of such point sets not equivalent under coordinate permutation, obtained by Peter Shor in earlier post, I'm wondering if it's a coincidence