I know these 2 statements to be true:
1) An $n$ x $n$ matrix U has orthonormal columns iff. $U^TU=I=UU^T$.
2) An $m$ x $n$ matrix U has orthonormal columns iff. $U^TU=I$.
But can (2) be generalised to become "An $m$ x $n$ matrix U has orthonormal columns iff. $U^TU=I=UU^T$" ? Why or why not?
Thanks!