Let $K$ be a field, $V$ a finite-dimensional vector space and $\alpha, \beta \in \mathrm{End}(V)$. Show that there exists a $T \in K[x]$ such that $\textrm{Min}_{\alpha * \beta} \cdot T = x \cdot \textrm{Min}_{\beta * \alpha}$. ($\operatorname{Min}$ denotes the minimal polynomial of the endomorphismus).
Guess I can choose for $T$ the minimal polynomial of $\beta$, but than I have no idea how to prove that $\textrm{Min}_{\alpha * \beta} \cdot T = x \cdot \textrm{Min}_{\beta * \alpha}$ because I don't see what assumptions about $\alpha$ and $\beta$ can I use? Do you have any hints for me?