Suppose two matrices have the same number of rows. I want to perform an operation of element-wise product between all possible column pairs between the two matrices. For example, if
$A = \left[{\begin{array}{ccc} 1 & 2 \\ 3 & 4 \end{array}}\right], B=\left[{\begin{array}{ccc} 5 & 6 & 7 \\ 8 & 9 & 10 \end{array}}\right], $
then the operation leads to a matrix of 6 columns,
$C = \left[{\begin{array}{ccc} 5 & 6 & 7 & 10 & 12 & 14\\ 24 & 27 & 30 & 32 & 36 & 40 \end{array}}\right]. $
My question is: Is there a name (and notation) for this kind of matrix operation, similar to Hadamard product?
The background for the question is that such operation seems to be involved in constructing the interaction columns between two dummy-coded factors.