Is Matrix Direct Sum Distributive over Matrix Addition?

Question

Simple question that's been bugging me - is the matrix direct sum in general distributive over matrix addition? ie is the following true? $\bigoplus_{i=1}^n (A_i+B_i)=(\bigoplus_{i=1}^n A_i) +(\bigoplus_{i=1}^n B_i) $ Thank you!

Answer 1

You can think of the direct sum using block diagonal matrices. Each block corresponds to a different orthogonal subspace. Implicitly, your matries $A_i$, $B_i$ all belong to the subspace indexed by $i$. In each subspace, you are just doing normal matrix addition. Distributivity means that you don't sum blocks from different subspaces.

Example:

\begin{align} \bigoplus_{i=1}^2 A_i+\bigoplus_{i=1}^2 B_i = & \begin{bmatrix} A_1 & 0\\ 0 & A_2 \end{bmatrix}\\ + & \begin{bmatrix} B_1 & 0\\ 0 & B_2 \end{bmatrix}\\ = & \begin{bmatrix} A_1 +B_1 & 0\\ 0 & A_2 + B_2 \end{bmatrix}\\ = & \bigoplus_{i=1}^2(A_i+B_i) \end{align}