Let $A\otimes B$ denote the tensor product of two matrices, $A$ and $B$. I can show the trace of it is the same as the product of the traces of $A$ and $B$, which follows from computation.
Is there some conceptual explanation for this? I believe there should be one related to the fact that $V\otimes W$ is the space of functions over $\{1,2,\dots n\}\times \{1,2,3,\dots m\}$ if $V$ is of dimension $n$ and $W$ of dimension $m$.And $A\otimes B$ acts on this space if $A$ acts on $V$ and $B$ acts on $W$.
Thanks!