# Can this block matrix be decomposed into a more succinct manner?

Let $$A_m$$ and $$B_{ij}$$ be matrices of size $$M\times N$$ where $$m\in\{1,2\}$$ and $$N \times N$$ where $$i,j\in\{1,2\}$$, respectively. Note that $$m,i,j$$ can be any number in general, but here they are specified so that the question can be clearer and easy to see.

I am wondering whether there is an alternative way to write/decompose the following block matrix more succinctly. I have had a look at Khatri-Rao product or Tracy-Singh product, but neither of them fits into the block matrix I have. I'm looking for a kind of block-wise multiplication of block matrix although I am not sure whether this type of matrix multiplication exists.

The matrix what I have is as follows:

$$\begin{bmatrix} A_1B_{11}A_1^T & A_1B_{12}A_2^T\\ A_2B_{21}A_1^T & A_2B_{22}A_2^T \end{bmatrix}$$

I would be so grateful if anyone can suggest how this matrix can be decomposed or written more succinctly.

One way is $$\begin{bmatrix} A_1B_{11}A_1^T & A_1B_{12}A_2^T\\ A_2B_{21}A_1^T & A_2B_{22}A_2^T\end{bmatrix} = \begin{bmatrix} A_1 & 0\\ 0 & A_2 \end{bmatrix} \begin{bmatrix} B_{11} & B_{12}\\ B_{21} & B_{22} \end{bmatrix} \begin{bmatrix} A_1 & 0\\ 0 & A_2 \end{bmatrix}^T\,.$$