# Appropriate name for matrix operation

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?

• It's not really clear to me whether, for example, the upper left hand block is all $a$'s or only $a$'s on the leading diagonal and the rest $0$'s. Feb 5 '20 at 1:52
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 $$A\otimes {\rm Id}_N$$.
This is the Kronecker product of $$A$$ with $$I_2$$ or $$I_3$$.