I would like to know if there is a standard name for the matrix operation that changes a $2 \times 2$ or a $3 \times 3$ matrix into a $2N \times 2N$ or a $3N \times 3N$ matrix, such that every block diagonal $N \times N$ sub-matrix is made up of each element of the original matrix.
If that statement was not clear, here is an example for the $2 \times 2$ case:
$$\begin{bmatrix} a & b \\ c& d \end{bmatrix} \mapsto \underset{2N \times 2N}{\underbrace{\begin{bmatrix} a & & & & b & & & & \\ & \ddots & & & & \ddots & \\ & & & a & & & & b \\ c & & & & d & & & & \\ & \ddots & & & & \ddots & \\ & & & c & & & & d\end{bmatrix}}}$$
Here, the matrix on the right-hand side represents four block-diagonal matrices, each of dimension $N \times N$.
My question is: would you happen to know a standard nomenclature for such a matrix operation? If yes, what would be the correct notation for representing this?
Please help.
You're taking the Kronecker product with the identity matrix of the correct order. For instance, if $A$ is your original $2\times 2$ matrix, your example is just $A\otimes {\rm Id}_N$.