IMC 1994 · Problem 4

Let $\alpha \in \mathbb{R} \setminus \{0\}$ and suppose that $F$ and $G$ are linear maps (operators) from $\mathbb{R}^n$ into $\mathbb{R}^n$ satisfying $F \circ G - G \circ F = \alpha F$.

a) Show that for all $k \in \mathbb{N}$ one has $F^k \circ G - G \circ F^k = \alpha k F^k$.

b) Show that there exists $k \ge 1$ such that $F^k = 0$.