Mathematical Reflections 2014, Issue 5 - Problem U315

Problem:
Let $X$ and $Y$ be complex matrices of the same order with $XY^2-Y^2X=Y$. Prove that $Y$ is nilpotent.

Proposed by Radouan Boukharfane.

Solution:
We prove by induction on $k \in \mathbb{Z}^+$ that $\textrm{tr}(Y^k)=0$ for all $k \in \mathbb{Z}^+$, so that the conclusion will follow. We have
$$\textrm{tr}(Y)=\textrm{tr}(XY^2-Y^2X)=\textrm{tr}(XY^2)-\textrm{tr}(Y^2X)=\textrm{tr}(Y^2X)-\textrm{tr}(Y^2X)=0.$$
Assume that $\textrm{tr}(Y^k)=0$ for some $k \in \mathbb{Z}^+$. Then,
$$\begin{array}{lll}\textrm{tr}(Y^{k+1})&=&\textrm{tr}(Y^k\cdot Y)\\&=&\textrm{tr}(Y^k(XY^2-Y^2X))\\&=&\textrm{tr}((Y^kX)Y^2))-\textrm{tr}(Y^{k+2}X)\\&=&\textrm{tr}(Y^{k+2}X)-\textrm{tr}(Y^{k+2}X)\\&=&0, \end{array}$$ as desired.