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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.

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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.)

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  • $\begingroup$ No, you don't need square root. Your $B$ is positive definite. So it has a Cholesky decomposition $LL^T$. Then $tr(BVAV^T)=tr(L^TVAV^TL)\ge0$, and it is zero iff $L^TV=0$, i.e. iff $V=0$. $\endgroup$ – user1551 May 2 '17 at 19:49
  • $\begingroup$ @user1551 oh yeah cholesky. Thx. $\endgroup$ – user251257 May 2 '17 at 19:50

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