Tensor Product Properties I am currently working my way through some notes and have got stuck proving a couple of tensor product properties. I have A,B,C,D as matrices and u,x,y as vectors, with a & b being constants.
I have managed to prove most of the results that arise using vectors:
(x+y)$\otimes$u = x$\otimes$u + y$\otimes$u, 
u$\otimes$(x+y) = u$\otimes$x + u$\otimes$y, &
ax$\otimes$by = ab(x$\otimes$y)
However, I am struggling to prove the following properties: 
(A$\otimes$B)(C$\otimes$D) = AC$\otimes$BD,
(A$\otimes$B)(x$\otimes$y) = Ax$\otimes$By, &
(A$\otimes$B)$^\dagger$ = A$^\dagger$$\otimes$B$^\dagger$
 A: For the first one, simply go and verify that
\begin{align}
(A \otimes B)(C \otimes D) &= \begin{bmatrix} 
a_{11}B &\ldots&a_{1n}B  \\
\vdots & \ddots & \vdots\\
a_{m1}B & \ldots & a_{mn}B \\
\end{bmatrix}
\begin{bmatrix} 
c_{11}D &\ldots&c_{1p}D  \\
\vdots & \ddots & \vdots\\
c_{n1}D & \ldots & c_{np}D \\
\end{bmatrix}\\
&= \begin{bmatrix} 
\sum_{k=1}^na_{1k}c_{k1}BD  &\sum_{k=1}^na_{1k}c_{kp}BD  \\
\vdots & \ddots & \vdots\\
\sum_{k=1}^na_{mk}c_{k1}BD & \sum_{k=1}^na_{mk}c_{kp}BD \\
\end{bmatrix}
\end{align}
And the result then follows.
A: For (A$\otimes$B)(x$\otimes$y), using the same approach as above:
\begin{align}
(A \otimes B)(x \otimes y) &= \begin{bmatrix} 
a_{11}B &\ldots&a_{1n}B  \\
\vdots & \ddots & \vdots\\
a_{m1}B & \ldots & a_{mn}B \\
\end{bmatrix}
\begin{bmatrix} 
x_{1}y \\
\vdots \\
x_{n}y \\
\end{bmatrix}\\
&= \begin{bmatrix} 
\sum_{j=1}^na_{1j}x_{j}By \\
\vdots \\
\sum_{j=1}^na_{mj}x_{j}By \\
\end{bmatrix} \\
&= \begin{bmatrix} 
\sum_{j=1}^na_{1j}x_{j} \\
\vdots \\
\sum_{j=1}^na_{mj}x_{j} \\
\end{bmatrix} \otimes By \\
&= Ax \otimes By
\end{align}
A: This is my answer for (A$\otimes$B)$^\dagger$ = A$^\dagger$$\otimes$B$^\dagger$:
\begin{align}
(A \otimes B)^\dagger &= \begin{bmatrix} 
a_{11}B &\ldots&a_{1n}B  \\
\vdots & \ddots & \vdots\\
a_{m1}B & \ldots & a_{mn}B \\
\end{bmatrix} ^\dagger \\
&= \begin{bmatrix} 
\overline{a_{11}}B^\dagger &\ldots&\overline{a_{m1}}B^\dagger  \\
\vdots & \ddots & \vdots\\
\overline{a_{1n}}B^\dagger & \ldots & \overline{a_{mn}}B^\dagger \\
\end{bmatrix} \\
&= \begin{bmatrix} 
\overline{a_{11}} &\ldots&\overline{a_{m1}}  \\
\vdots & \ddots & \vdots\\
\overline{a_{1n}} & \ldots & \overline{a_{mn}} \\
\end{bmatrix} \otimes B^\dagger \\
&= A^\dagger \otimes B^\dagger
\end{align}
I found this paper by Bobbi Jo Broxson, which confirms my results, and has several more properties on Tensor product properties included.
