If we know the $A A^T $'s bounded , what is the $A^T A$'s bound? If I have a matrix $A\in \mathcal{R}^{m\times n}$, and it satisfies $\underline{a}^2 I\leq A A^T\leq \overline{a}^2 I$, can we conclude that the matrix $A^T A$ is bounded, too? and what's the bound? and what's relationship between new bound and the original one? 

The nonzero eigenvalues of $A A^T$ are the same as those of $A^T A$.  Of course $0$ is an eigenvalue as well if the matrix has less than full rank (which is true if $A A^T$ if $m > n$ and of $A^T A$ if $m < n$).
