A question given at lie group theory course:
$P$ and $Q$ are $n \times n$ matrices, and some $k \in \mathbb{R}$. Assume that $$\displaystyle\forall_{x,y \in \mathbb{R}} e^{xP}e^{yQ} = e^{xyk}e^{yQ}e^{xP}$$ Show that $k=0$ and $PQ = QP$.
It seems that the trick is to differentiate both sides and substitute $x=0$ or/and $y = 0$. When differentiating first by $x$ and then by $y$ or vice versa and substituting $x,y=0$, I, indeed, get $PQ = k + QP$. Am I right? And what about $k=0$?
Thanks in advance.
You are right. Now take the trace in the equality $PQ = k + QP$ to deduce that $k = 0$.