TITLE: tensor product and matrix multiplication distributive properties
QUESTION [3 upvotes]: I am trying to find partial trace of some matrix of the form
$M = (A \otimes B)\times (A^{T*} \otimes B^{T*})$
in which $\otimes$ is tensor product, $\times$ is matrix multiplication, $T*$ is conjugate transpose of the matrix. $A$ and $B$ are vectors (but can be matrices too).
I was wondering if I could use the following relation?
$(A \otimes B)\times (A^{T*} \otimes B^{T*}) = (A \times A^{T*})\otimes (B \times B^{T*})$
It checks out for $A=\left(\begin{matrix} a\\b \end{matrix}\right)$ and $B=\left(\begin{matrix} c\\d \end{matrix}\right)$.
if this relation is general, what about this one (for an arbitrary vectors (or matrices) C and D)
$(A \otimes B)\times (C \otimes D) = (A \times C)\otimes (B \times D)$
although I don't know how to trace this matrix. I would appreciate any help. thank you in advance

REPLY [2 votes]: Consider matrices $A,B,C,D$ of sizes such that the products $AC$ and $AD$ can be formed. We can use block matrix multiplication to show that $(A\otimes B)\,(C\otimes D)=(AC)\otimes(BD)$.
We will use the notation $A\otimes B = (a_{ij} B)_{ij}$ to denote block matrices, where indices are always supposed to range approriately. Then
\begin{align*}
(A\otimes B)\,(C\otimes D) &= (a_{ij} B)_{ij}\, (c_{ij} D)_{ij} \\
&= \left(\sum_k (a_{ik} B)(c_{kj} D)\right)_{ij} \\
&= \left( \left(\sum_k a_{ik} c_{kj}\right) BD\right)_{ij.}
\end{align*}
Note that $\sum_k a_{ik} c_{kj}$ is the $i,j$-th entry of $AC$ so the result is indeed equal to $(AC)\otimes (BD)$.
Since traces of Kronecker products are given as $\operatorname{Tr}(A\otimes B)=\operatorname{Tr}(A) \operatorname{Tr}(B)$, this yields
$$
\operatorname{Tr}\left((A\otimes B)\,(C\otimes D)\right)
= \operatorname{Tr}(AC) \operatorname{Tr}(BD).
$$
In your case that gives
$$
\operatorname{Tr}\left((A\otimes B)\,(\overline{A}^T\otimes \overline{B}^T)\right) = \operatorname{Tr}(A\overline{A}^T) \operatorname{Tr}(B\overline{B}^T) = \|A\|_F^2\, \|B\|_F^2,
$$
where $\|\cdot\|_F$ denotes the Frobenius norm.
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