What is the name of the following matrix "product" in which elements are not multiplied? Assume that I am given the following matrices:
$A = \begin{bmatrix}
a_{1,1} & \dots & a_{1,n_a} \\
\vdots & \ddots & \vdots \\
a_{m_a,1} & \dots & a_{m_a,n_a}
\end{bmatrix}$ and $B =
\begin{bmatrix}
b_{1,1} & \dots & b_{1,n_b} \\
\vdots & \ddots & \vdots \\
b_{m_b,1} & \dots & b_{m_b,n_b}
\end{bmatrix}$. I need to consider the following matrix and I am looking for a way to relate it to $A$ and $B$.
\begin{bmatrix}
a_{1,1} & a_{1,1} & \dots & a_{1,1} & a_{1,2} & \dots & a_{1,2} & \dots & a_{1,n_a} & a_{1,n_a} & \dots & a_{1,n_a} \\
a_{2,1} & a_{2,1} & \dots & a_{2,1} & a_{2,2} & \dots & a_{2,2} & \dots & a_{2,n_a} & a_{2,n_a} & \dots & a_{2,n_a} \\
\vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\
a_{m_a,1} & a_{m_a,1} & \dots & a_{m_a,1} & a_{m_a,2} & \dots & a_{m_a,2} & \dots & a_{m_a,n_a} & a_{m_a,n_a} & \dots & a_{m_a,n_a}\\
b_{1,1} & b_{1,2} & \dots & b_{1,n_b} & b_{1,1} & \dots & b_{1,n_b} & \dots & b_{1,1} & b_{1,2} & \dots & b_{1,n_b} \\
b_{2,1} & b_{2,2} & \dots & b_{2,n_b} & b_{2,1} & \dots & b_{2,n_b} & \dots & b_{2,1} & b_{2,2} & \dots & b_{2,n_b} \\
\vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\
b_{m_b,1} & b_{m_b,2} & \dots & b_{m_b,n_b} & b_{m_b,1} & \dots & b_{m_b,n_b} & \dots & b_{m_b,1} & b_{m_b,2} & \dots & b_{m_b,n_b}
\end{bmatrix}
What would be good name and symbol for such an operation? I am thinking about cartesian product.

In terms of operations that already exist, we can write this as the block matrix
$$
\pmatrix{A \otimes X\\ X \otimes B}
$$
Where $X$ is the row-vector $(1,\dots,1)$ and $\otimes$ denotes the Kronecker product.
A nice shorthand for this operation (if the above is not sufficiently compact) could be $\frac{A}{B}$, which has no common second meaning in this context.
