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Suppose that there are some $n$ matrices, $A_1, A_2, ..., A_n$ One wants to form a square matrix $B$ that contains all aforementioned $A$ matrices as entries (this does not mean that all components of the matrix $B$ must be $A$ matrices, though.). One wants to do the following: 1) When multiplying $B$ matrix by itself, one wants to see the result of multiplication that has entries of the form $A_i \times A_j$. How would I achieve this?

2) not related to 1): Suppose that we multiply $B$ by some matrix $C$. Again, one wants to see the result of multiplication that has entries of the form $A_i \times A_j$. How would I achieve this?

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By component, do you mean an element (or sub-matrix)? If so, you may just want conformable partitioning to build a Gram (like) matrix. And $C$ is any random matrix, so not much can be said about that (2). – adam W Oct 14 '12 at 17:52
entries, oops...... – rrqq Oct 14 '12 at 18:23

For example you could take the matrix $$B=\begin{bmatrix}0&A_1&\dots&A_n\\A_1&0&\dots&0\\\vdots&\vdots&&\vdots\\A_n&0&\dots&0\end{bmatrix}.$$ Then we have $$B^2=\begin{bmatrix}\sum A_i^2&0&0&\dots&0\\0&A_1^2&A_1A_2&\dots&A_1A_n\\\vdots&\vdots&\vdots&&\vdots\\0&A_nA_1&A_nA_2&\dots&A_n^2\end{bmatrix}.$$ Thus you have all the entries $A_iA_j$ in your matrix.

For your part (2) I'm not sure what you ask for. That's certainly not always possible, e.g. if $C=0$.

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As far as I understand you, you ask about how to multiply block-matrices. Actually there is no difference to normal matrix multiplication with scalar entries

The following is valid for both scalar values of $A_i$, as well as sub-matrices $A_i$:

$\begin{bmatrix}A_1&A_2\\ A_3&A_4 \end{bmatrix} \begin{bmatrix} B_1&B_2 \\B_3& B_4 \end{bmatrix} = \begin{bmatrix}A_1 B_1 + A_2 B_3 & A_1 B_2 + A_2 B_4 \\ ... & ... \end{bmatrix}$.

For the block-matrix version you may not change the order of mutiplication in the entries on the right hand side and make sure that all mutiplications are well formed, i.e. the dimensionality fits.

You may also want to check out

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