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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.

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

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  • $\begingroup$ Indeed, this is definitely one way to decompose the matrix I post!!!!! Great! $\endgroup$ – Creatlee Jun 4 at 20:44

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