This question is a generalisation of Eigenvalues of $AB$ and $BA$ where $A$ and $B$ are square matrices.
Let $A$ and $B$ be $m\times n$ and $n\times m$ complex matrices, respectively, with $m < n$. If the eigenvalues of $AB$ are $\lambda_1, \ldots, \lambda_m$, what are the eigenvalues of $BA$?
If the matrices were square, then the conclusion would follow from the fact that $AB$ and $BA$ have the same characteristic polynomial. With rectangular matrices this is not going to happen; how to proceed then?