Is there a name for matrices with singular values all equal to $1$?
I think the name for this type of matrix is called isometries, that is, all of its singular values equal to $1$. Say $A$ is an $m\times n$ matrix with $m>n$. Equivalently:
- $A^\intercal A = I_n$, $\|Ax\|_2=\|x\|_2$ for all $x\in \mathbb{R}^n$.
- All singular values of $A$ are equal to 1.