Suppose I have a sequence of matrices, $A_1,...A_k$, each of dimension $n \times m$. Suppose I also have two $m \times m$ matrices $B$ and $C$. What are the broadest conditions under which its true that:
$$\sum_i A_i B A_i' = \sum_i A_i C A_i'$$
Implies $B=C$? And when does this implication definitely not hold?
With the vectorization operator, we find that the matrix of the linear map $X \mapsto \sum_{i} A_i X A_j'$ is given by $$ M = \sum_i \bar A_i \otimes A_i, $$ where $\otimes$ denotes the Kronecker product. It will hold that $\sum_i A_i B A_i' = \sum_i A_i C A_i'$ implies that $B = C$ if and only if the matrix $M$ is invertible.