Non-zero eigenvalues of $AA^T$ and $A^TA$
Let $\lambda$ be an eigenvalue of $A^TA$, i.e. $$A^T A x = \lambda x$$ for some $x \neq 0$. We can multiply $A$ from the left and get $$A A^T (Ax) = \lambda (Ax).$$
What can you conclude from this?
in fact, nonzero eigenvalues $AB$ and $BA$ are the same for any rectangular matrices $A$ and $B$. this follows from the fact that $trace((AB)^k) = trace((BA)^k)$ and the coefficients of the characteristic polynomials of a square matrix $A$ are a function of $trace(A^k).$