If $A^TA \ge B^TB$ does this imply $AA^T \ge BB^T$?  

If not, is there a counter example?



If $A^TA \ge B^TB$ does this imply $AA^T \ge BB^T$?
Take
$$A = \begin{pmatrix}0 & 0 \\ 1 & 0\end{pmatrix}, \quad B = \begin{pmatrix}1 & 0 \\ 0& 0 \end{pmatrix}.$$


