IMC 1994 · Day 1 · P4

For a) using the assumptions we have $$\begin{align*} F^k \circ G - G \circ F^k &= \sum_{i=1}^{k} \left( F^{k-i+1} \circ G \circ F^{i-1} - F^{k-i} \circ G \circ F^{i} \right) = \\ &= \sum_{i=1}^{k} F^{k-i} \circ (F \circ G - G \circ F) \circ F^{i-1} = \\ &= \sum_{i=1}^{k} F^{k-i} \circ \alpha F \circ F^{i-1} = \alpha k F^k. \end{align*}$$ b) Consider the linear operator $L(F) = F \circ G - G \circ F$ acting over all $n \times n$ matrices $F$. It may have at most $n^2$ different eigenvalues. Assuming that $F^k \ne 0$ for every $k$ we get that $L$ has infinitely many different eigenvalues $\alpha k$ in view of a) – a contradiction.