TITLE: Kronecker product of positive definite matrices
QUESTION [6 upvotes]: I am looking for a reference where it is proved that given two positive definite matrices $A\in M_n$, $B \in M_m$, their Kronecker product $A\otimes B$ is positive definite.
More precisely, I am looking for a computation showing that 
$$\langle (A\otimes B)v,v\rangle \ge 0$$
for every $v\in \mathbb{C}^{mn}$.
I specifically don't want to use the argument about the eigenvalues or the mixed-product and square roots, but a very direct computation of the inner product above. I would appreciate the help.

REPLY [5 votes]: Let $V=\operatorname{mat}(v)$ such that stacking its columns yields $v$. Then, we have 
$$ \langle (A\otimes B)v, v\rangle = tr(BVAV^T). $$ 
Since $A$ is positive definite, $VAV^T$ is positive semidefinite. As $B$ is positive definite, $tr(B (VAV^T))$ is non-negative. 
In fact, if $B=R^T R$ is a Cholesky decomposition, we have
$$ tr( B VAV^T) = tr( RV A (RV)^T) \ge 0, $$
which is zero if and only if $RV=0$, that is $V=0$. (Thanks to @user1551 for tipping me of.)