In trying to find an MLE, I have run into the following matrix equation that if feasible would allow me to use least squares to optimize $A$. Suppose that matrices in the set $\{B_k\}$ are square, and not necessarily commutative with $A$. Let $\{x_k\}$ be a set of vectors. Can I always find $C$ such that:
$$\sum_{k=1}^NB_kA\mathbf{x_k}=AC$$
I'm not terribly familiar with matrix algebra so even some starting directions would be helpful.
No, not even in the case $N=1$. $AC$ would be a linear combination of the columns of $A$, but $B_k A x_k$ might not be.