A non-square matrix with orthonormal columns

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!

Solution 1:

The $(i,j)$ entry of $U^T U$ is the dot product of the $i$'th and $j$'th columns of $U$, so the matrix has orthonormal columns if and only if $U^T U = I$ (the $n \times n$ identity matrix, that is). If $U$ is $m \times n$, this requires $m \ge n$, because the rank of $U^T U$ is at most $\min(m,n)$. On the other hand, $U U^T$ is $m \times m$, and this again has rank at most $\min(m,n)$, so if $m > n$ it can't be the $m \times m$ identity matrix.