Eigenvalues of Kronecker product of non square matrices

Question

How to prove that a Kronecker product of matrices $A \in \Bbb {R}^{m\times n}$ and $B \in \Bbb {R}^{n\times m}$ has an eigenvalue of zero?

Answer 1

Presumably $m\ne n$. If $m<n$, $Ax=0$ has a nontrivial solution $x\in\mathbb R^n$. Therefore, for every nonzero vector $y\in\mathbb R^m$, we have $(A\otimes B)(x\otimes y)=(Ax)\otimes(By)=0$ where $x\otimes y\ne0$. Hence $A\otimes B$ is singular. The case for $m>n$ is similar.