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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?

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