Name for this matrix operation? 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.
 A: I am not sure what that matrix operator is called, but it is very close to being a  Kronecker product (i.e., $A\otimes B$ ) so maybe there is something similar to that?  In either case here is some fast code for getting it out of R using Kronecker products as well.
> kronecker(A,B)
     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    5    6    7   10   12   14
[2,]    8    9   10   16   18   20
[3,]   15   18   21   20   24   28
[4,]   24   27   30   32   36   40

> kronecker(A,B)[c(1,4),]
     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    5    6    7   10   12   14
[2,]   24   27   30   32   36   40

A: No there is not a defined and studied compact matrix operation that does this - "this" being a combination of column-wise Hadamard multiplication and matrix augmentation, which uses the symbol "|".
Your operation requires conformable row dimensions. Using small letters to denote the columns of a matrix we have
$$A_{k\times n} = \begin{matrix} [a_1 & ...& a_n] \end{matrix}, \qquad B_{k\times m}=\begin{matrix}[b_1 & ...& b_m]\end{matrix}$$
$$ C_{k\times (n\times m)} = \Big [a_1*b_1|...|a_1*b_m|...|a_n*b_1|...|a_n*b_m\Big]$$
A: I don't know what you would call this, but here's a quick way you can do this in R:
a = t(matrix(c(1,2,3,4), nrow=2))
b = t(matrix(c(5,6,7,8,9,10), ncol=2))

matrix(apply(a,2, function(x) x * b)
       ,ncol(a)
       ,ncol(a)*ncol(b))

