Follow up to this question I asked here: Is Matrix Direct Sum Distributive over Matrix Addition?. I am now wondering whether it is distributive over matrix multiplication as well? By that I mean whether the following statement is true or not:

$\bigoplus_{i=1}^n A_iB_i = \left(\bigoplus_{i=1}^n A_i\right)\left(\bigoplus_{i=1}^n B_i\right)$

By the argument given in the previous question I would take that the statement is true as in each subspace $i$ we simply have $A_iB_i$ and whether you take the matrix multiplication of the direct sum first should not matter as either way we won't have $A_iB_j$ for $i\neq j$. Can anyone confirm this result/my reasoning? Thank you!


I believe yes - taking the $2\times2$ case we get:

\begin{align*}&&\left(\bigoplus_{i=1}^n A_i\right)\left(\bigoplus_{i=1}^n B_i\right)&=\left(\begin{array}{cc} A_1 & 0 \\ 0 & A_2 \end{array}\right)\left(\begin{array}{cc} B_1 & 0 \\ 0 & B_2 \end{array}\right)&\\ &&&=\left(\begin{array}{cc} A_1B_1 & 0 \\ 0 & A_2B_2 \end{array}\right)&\\ &&&=\bigoplus_{i=1}^2 A_iB_i \end{align*} I'm almost certain this is true now - sorry for the dumb question. :P


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