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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.

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  • $\begingroup$ 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. $\endgroup$ Feb 5 '20 at 1:52
  • $\begingroup$ @user5713492a's on the leading diagonal only $\endgroup$
    – metsburg
    Feb 5 '20 at 2:13
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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 $A\otimes {\rm Id}_N$.

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This is the Kronecker product of $A$ with $I_2$ or $I_3$.

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