Let $A$ and $B$ be $m\times n$ and $n\times m$ matrices respectively.
- Prove that if $\lambda$ is a non-zero eigenvalue of $AB$ then it is also an eigenvalue of $BA$
- Prove that $I_m-AB$ is invertible if and only if $I_n-BA$ is invertible.
Part (1) is easy:
By definition, $x\ne 0$, and by assumption $\lambda \ne 0$. So we have $Bx\ne 0$
Now, $$B(AB)x=(BA)Bx=\lambda Bx$$ and so $Bx$ and a non-zero vector with eigenvalue $\lambda$.
My problem is I have no idea how to use this to do (2). Any help is much appreciated. Thanks!