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?
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