Is it true that $\sigma(AB) =\sigma(BA) $. Denote by $\sigma$ the spectral radius.
Is it true that $\sigma(AB) =\sigma(BA) $?
Edit: I am interested in the general case, i.e. $A$ is $n \times k$ and $B$ is $k \times n$.

If $A$ and $B$ are square matrices of the same size, then the products $AB$ and $BA$ have the same eigenvalues.
As was noted in comments by @Algebraic Pavel, the result still holds for rectangular matrices $A$ and $B$ (if the products $AB$ and $BA$ make sense). The non-zero eigenvalues of these products are the same.
