I am trying to prove this equality between matrices: $(A^T A + \mu I_n)A^T=A^T(AA^T+\mu I_m)$ where $A \in \mathbb{R}^{m\times n}, \mu \in \mathbb{R}, \mu > 0$. I was given a hint that I should use the inverse of $(A^T A + \mu I_n)$ which I have proven that always exists.
I have tried few things but I just cant seem to find the right direction and I feel that it is going to be some easy trick but I cannot come up with it.
Thank you
Multiply out: $$(A^T A+ \mu I)A^T = (A^TA)A^T + \mu I A^T$$ and $$A^T(AA^T+\mu I) = A^T(AA^T) +\mu A^TI.$$
Since $(A^TA)A^T=A^T(AA^T)$ and $IA^T=A^TI$, your two expressions are equal.