# Simplification for Kronecker product between block matrix and identity matrix (Khatri-Rao product)

Is there a short matrix operation (like matrix multiplication, Kronecker product $\otimes$ or sum $\oplus$, ...) that I can apply to a block matrix $$M_2=\left(\begin{array}{@{}cc@{}} A & B \\ C & D \end{array}\right)$$ to get the double sized matrix $$M_4 = \left(\begin{array}{@{}cccc@{}} A & 0 & B & 0 \\ 0 & A & 0 & B \\ C & 0 & D & 0 \\ 0 & C & 0 & D \end{array}\right) \;,$$ where I do not have to explicitly use the sub-matrices $A$, $B$, $C$, $D$? ($A$, $B$, $C$, $D$ are of same size, $0$ is a zero matrix of same size.)

If $A$, $B$, $C$, $D$ were scalars, $M_4=M_2\otimes I_2$ would do the job, where $I_2$ is the $2\times2$ identity matrix.

And the first block in $M_4$ can be calculated by $$\left(\begin{array}{@{}cc@{}} A & 0 \\ 0 & A \end{array}\right) = I_2\otimes A = A\oplus A \;.$$ So one way to do it would be to extract the block matrices from $M_2$ with matrix multiplications, apply the above, and put everything together again. But that would be a rather cumbersome expression, I hope to find something more direct.

I saw the Khatri–Rao product that basically is the Kronecker product applied to general sub-matrix structures, but maybe there is something more elementary for this simple block structure.

I need that for a recursion formula to construct big matrices from a $4\times4$ matrix, where I hope things simplify if I have elementary mathematical expressions for the matrix operations I do.

• If $M_2\in{\mathbb R}^{2m\times 2n}\,$ and $\,e_k\in{\mathbb R}^{2\times 1}\,$ are the standard basis vectors then $$M_4 = \sum_{i=1}^2\sum_{j=1}^2\,\,e_ie_j^T\otimes I_2 \otimes\Big((e_i\otimes I_m)^TM_2(e_j\otimes I_n)\Big)$$ – greg Jan 31 '19 at 5:46

Define the selection matrix $$Z = \left(\begin{array}{@{}cccc@{}} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$$ and let $n$ be the numbers of rows (or columns) of the square matrices A, B, C, D that define $M_2$ above. Let further $E$ be the matrix $$E = \left(\begin{array}{@{}cccc@{}} 1 & 1 \\ 1 & 1 \\ \end{array}\right) \otimes I_2 = \left(\begin{array}{@{}cccc@{}} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{array}\right)$$ where $I_2$ is the $2\times 2$ identity matrix.
The desired matrix $M_4$ then follows from $$M_4 = (Z\otimes I_n)^T \cdot (E\otimes M_2) \cdot (Z\otimes I_n)$$