<s>[INST] Does such a operator exist? I have been looking for a matrix multiplier that is similar to a tensor product. The best way I can define the product is with the following example:
Suppose $A=\left[\begin{array}{cc}
1 & 2\\
3 & 4
\end{array}\right]$, and $B=\left[\begin{array}{cc}
0 & 5\\
6 & 7
\end{array}\right]$. 
Does their exist a multiplier (call it $\#$) such that
$A\#B=\left[\begin{array}{cc}
1\left[\begin{array}{cc}
0 & 5\end{array}\right] & 2\left[\begin{array}{cc}
0 & 5\end{array}\right]\\
3\left[\begin{array}{cc}
6 & 7\end{array}\right] & 4\left[\begin{array}{cc}
6 & 7\end{array}\right]
\end{array}\right]=\left[\begin{array}{cccc}
0 & 5 & 0 & 10\\
18 & 21 & 24 & 28
\end{array}\right]$. 
Does such a multiplier exist? If not, how could I use existing operators to attain my desired outcome?
 [/INST] Can someone verify?
Suppose $A=\left[\begin{array}{cc}
1 & 2\\
3 & 4
\end{array}\right],$and $B=\left[\begin{array}{cc}
0 & 5\\
6 & 7
\end{array}\right]$. 
By definintion of the Khatri-Rao product: $A*B=\left(A_{ij}\otimes B_{ij}\right)_{ij}$.
$A^{T}=\left[\begin{array}{cc}
1 & 3\\
2 & 4
\end{array}\right],$and $B^{T}=\left[\begin{array}{cc}
0 & 6\\
5 & 7
\end{array}\right]$
\begin{eqnarray*}
\left(A^{T}*B^{T}\right)^{T} & = & \left[A_{1}\otimes B_{1}|A_{2}\otimes B_{2}\right]^{T}\\
 & = & \left[\begin{array}{cc}
1\cdot0 & 3\cdot6\\
1\cdot5 & 3\cdot7\\
2\cdot0 & 4\cdot5\\
2\cdot5 & 4\cdot7
\end{array}\right]^{T}\\
 & = & \left[\begin{array}{cc}
1\cdot0 & 3\cdot6\\
1\cdot5 & 3\cdot7\\
2\cdot0 & 4\cdot6\\
2\cdot5 & 4\cdot7
\end{array}\right]^{T}\\
 & = & \left[\begin{array}{cc}
0 & 18\\
5 & 21\\
0 & 24\\
10 & 28
\end{array}\right]^{T}\\
 & = & \left[\begin{array}{cccc}
0 & 5 & 0 & 10\\
18 & 21 & 24 & 28
\end{array}\right]
\end{eqnarray*}
This seems related: Macedo, Hugo Daniel, and José Nuno Oliveira. 2015.
A linear algebra approach to OLAP. Formal Aspects of Computing 27,
(2) (03): 283-307
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