What does it imply that all of the singular values are 1s and 0s?

Question

These were the two links I looked at:

What does it mean/imply that all my singular values are ones?

If the singular values of an $n{\times}n$ matrix $A$ are all $1$, is $A$ necessarily orthogonal?

and I understood how such matrices are necessarily orthogonal. However, what if the singular values are 0's and 1's of a square, real matrix? What does that imply?

Answer 1

A matrix $A$ will have singular values all equal to $0$ or $1$ if and only if it is a partial isometry.

Another characterization is that $A$ has singular values $1$ with multiplicity $r$ and $0$ with multiplicity $n-r$ if and only if there exist matrices $U$ and $V$ with mutually orthonormal columns such that $A = UV^T$.