best way to show tr(AB) = tr(BA) for non square A and b matrices

Question

I have matrices $A \in \mathbb{K}^{n \times m}$ and $B \in \mathbb{K}^{m \times n}$

What is the best way to prove that tr(AB) = tr(BA). I found a prove in Matrix Analysis by Horn and Johnson, but they only prove it if $n \leq m$.

Answer 1

Either you have $n\leq m$ or $m\leq n$. If it is the latter, exchange $n$ with $m$ and $A$ with $B$, and the same proof works.