A bizarre matrix product -- does it have a name? Let $\mathbb{F}$ be a field, let $A\in\mathbb{F}^{n\times \ell m}$, and let $B \in\mathbb{F}^{m\times p}$. Let $A_j$ be the submatrix of $A$ consisting of columns $(j-1)m+1$ through $jm$ and define
$$
A*B = \begin{bmatrix}A_1B \ | \ \dots \ | \ A_\ell B\end{bmatrix} \in \mathbb{F}^{n\times \ell p}
$$
For example,
$$
\begin{bmatrix}
1 & 2 & 3 & 4\\ 5 & 6 & 7 & 8
\end{bmatrix}*\begin{bmatrix}0 & 1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix}\begin{bmatrix}
1 & 2 \\ 5 & 6
\end{bmatrix}\begin{bmatrix}0 & 1 \\ 1 & 0 \end{bmatrix}\ | \ \begin{bmatrix}
3 & 4\\ 7 & 8
\end{bmatrix}\begin{bmatrix}0 & 1 \\ 1 & 0 \end{bmatrix} \end{bmatrix}=\begin{bmatrix}2 & 1 & 4 & 3\\ 6 & 5 & 8 & 7 \end{bmatrix}
$$
Does this bizarre matrix product have a name? I would like to use it to write something like $XY = C*\text{vec}(X)$ where $C = [I\otimes \text{col}_1(Y)^T\dots I\otimes \text{col}_p(Y)^T]$
 A: Perhaps you aren't familiar with modules, but if you have an inkling what they are, this isn't so odd.
Let $R=M_2(F)$ You could view it as the natural action of $R$ on $R\times R$ by left multiplication.
I'm not sure how attached you are to writing the eight elements in a single matrix, but the resulting object is isomorphic to the left $R$ module $R\times R$.
So... I don't have any fancy name for it, just a simpler description.
A: I don't know of any name for this product.  That being said, the following characterization might be useful:
$$
A*B = \sum_{k=1}^\ell e_k^T \otimes (A_k B) = \sum_{k=1}^\ell (e_k^T \otimes A_k)(I_{\ell} \otimes B)
$$
where $e_k$ dentoes the $k$th column of the size $\ell$ identity matrix, $I_\ell$.  
I think that the identity $\operatorname{vec}(P \otimes Q) = \operatorname{vec}(P) \otimes \operatorname{vec}(Q)$ generally holds.  If this is correct, then we also find
$$
\operatorname{vec}(A * B) = 
 \sum_{k=1}^\ell \operatorname{vec}\left[ e_k^T \otimes (A_k B) \right] = 
\sum_{k=1}^\ell \operatorname{vec}[ e_k^T] \otimes \operatorname{vec}[(A_k B) ]\\
= \sum_{k=1}^\ell e_k \otimes [(I \otimes A_k)\operatorname{vec}(B)]\\
%= \left[\sum_{k=1}^\ell I_{\ell^2} \otimes (I \otimes A_k)\right][\operatorname{vec}(I_{\ell})\otimes \operatorname{vec}(B)]
$$
I hope you find some of this helpful.
