Consider a matrix $A\in\mathbb R^{n\times m}$ with $n<m$.
Given that $\|A\|_2 = \gamma_0$ and $AA^T$ is invertible, can we find any equality/upper bound for $\|(AA^T)^{-1}\|_2$ in terms of $\gamma_0$?
For the case that $A$ is an invertible square matrix, I can simply do that. But, I couldn't find the relation for non-square $A$ matrix, if there is any.
Thank you in advance for your help.
Hint:
Use $$(AA^T)^{-1}(AA^T) = I$$ and the $l_2$ norm property $$ ||AB||_2 \le ||A||_2||B||_2$$