# Is there a name for matrices with singular values all equal to $1$?

I at first thought these are just the orthogonal matrices, since the SVD of $$X$$ is $$U \Sigma V^T$$ and if $$\Sigma$$ is the identity matrix then this implies X is orthogonal.

However, singular values equal to 1 doesnt imply $$\Sigma$$ is identity, since $$X$$ may be non-square.

So is there a name for this more general class, including nonsquare matrices? Also, can we characterize that class in a more similar way that the orthogonal matrices are characterized?