<s>[INST] Kronecker Product Interpretation The algebraic expression for a Kronecker product is simple enough.  Is there some way to understand what this product is?
The expression for matrix-vector multiplication is easy enough to understand.  But realizing that the multiplication yields a linear combination of the columns of the matrix is a useful insight.
Is there some analogous insight for the Kronecker product?
 [/INST] A nice motivating example is provided in Kronecker Products & Matrix Calculus with Applications by A. Graham.
We consider two linear transformations
\begin{align*}
\boldsymbol{x}=\boldsymbol{A}\boldsymbol{z}\qquad\text{and}\qquad \boldsymbol{y}=\boldsymbol{B}\boldsymbol{w}
\end{align*}
and the simple case
\begin{align*}
\begin{pmatrix}
x_1\\x_2
\end{pmatrix}
=
\begin{pmatrix}
a_{11}&a_{12}\\
a_{21}&a_{22}\\
\end{pmatrix}
\begin{pmatrix}
z_1\\z_2
\end{pmatrix}
\qquad\text{and}\qquad
\begin{pmatrix}
y_1\\y_2
\end{pmatrix}
=
\begin{pmatrix}
b_{11}&b_{12}\\
b_{21}&b_{22}\\
\end{pmatrix}
\begin{pmatrix}
w_1\\w_2
\end{pmatrix}
\end{align*}

The Kronecker product:
Now we consider the two transformations simultaneously by defining two vectors
\begin{align*}
\color{blue}{\boldsymbol{\mu}=\boldsymbol{x}\otimes\boldsymbol{y}}
=
\begin{pmatrix}
x_1y_1\\x_1y_2\\x_2y_1\\x_2y_2
\end{pmatrix}
\qquad\text{and}\qquad
\color{blue}{\boldsymbol{\nu}=\boldsymbol{z}\otimes\boldsymbol{w}}
=
\begin{pmatrix}
z_1w_1\\z_1w_2\\z_2w_1\\z_2w_2
\end{pmatrix}
\end{align*}

We want to find the transformation between $\boldsymbol{\mu}$ and $\boldsymbol{\nu}$. We look at the relation between the components of the two vectors and show it exemplarily for $x_1y_1$.
We obtain
\begin{align*}
x_1y_1&=\left(a_{11}z_1+a_{12}z_2\right)\left(b_{11}w_1+b_{12}w_2\right)\\
&=a_{11}b_{11}\left(z_1w_1\right)+a_{11}b_{12}\left(z_1w_2\right)+a_{12}b_{11}\left(z_2w_1\right)\\
&\qquad+a_{12}b_{12}\left(z_2w_2\right)
\end{align*}

It follows
\begin{align*}
\color{blue}{\boldsymbol{\mu}}&=
\begin{pmatrix}
a_{11}b_{11}&a_{11}b_{12}&a_{12}b_{11}&a_{12}b_{12}\\
a_{11}b_{21}&a_{11}b_{22}&a_{12}b_{21}&a_{12}b_{22}\\
a_{21}b_{11}&a_{21}b_{12}&a_{22}b_{11}&a_{22}b_{12}\\
a_{21}b_{12}&a_{21}b_{22}&a_{22}b_{21}&a_{22}b_{22}\\
\end{pmatrix}\boldsymbol{\nu}\\
&\color{blue}{=\left(\boldsymbol{A}\otimes\boldsymbol{B}\right)\boldsymbol{\nu}}
\end{align*}
We find
\begin{align*}
\color{blue}{\boldsymbol{Az}\otimes\boldsymbol{Bw}
=\left(\boldsymbol{A}\otimes \boldsymbol{B}\right)
\left(\boldsymbol{z}\otimes \boldsymbol{w}\right)}
\end{align*}

</s>