Let's consider the following block diagonal matrix $A\in \mathbb{R}^{3n\times3n}$ with equal diagonal sub matrices $A_1 \in \mathbb{R}^{n\times n}$ and zero matrices $0 \in \mathbb{R}^{n\times n}$:
$$ A = \left( \begin{matrix} A_1 & 0 & 0 \\ 0 & A_1 & 0 \\ 0 & 0 & A_1 \end{matrix} \right).$$
According to the definition of the direct sum operator, the following applies:
$$A = A_1\oplus A_1 \oplus A_1 = \oplus_{i=1}^3A_1$$.
Let's consider an equation of the following form:
$$s = B\cdot A\cdot C = B\cdot \left( \begin{matrix} A_1 & 0 & 0 \\ 0 & A_1 & 0 \\ 0 & 0 & A_1 \end{matrix} \right)\cdot C = B\cdot(\oplus_{i=1}^3A_1)\cdot C $$ with $$ s \in \mathbb{R}^{n\times 1},$$ $$ B \in \mathbb{R}^{n\times 3n},$$ $$ C \in \mathbb{R}^{3n\times 1}.$$
Is there a way to solve for the diagonal elements in $A_1$ as a function of $s$, $B$, and $C$?. The notation of the direct matrix sum is somehow inconvenient in my opinion. Might there be a different notation such that the system can be solved? For $n=1$ it is obvious since $A= \left(\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}\right)A_1 $ with scalar $A_1 $. But what about $n>1$?