Let $A$ be a real valued $4n \times 4n$ matrix, $B$ is a real valued $4\times4$ matrix, and $C$ is a non-negative $n\times n$ matrix. I have $A$ and $B$ and I am trying to get $C$ through this equation: $$ A = C \otimes B $$
The elements of $C$ have a certain structure:
Consider the following transition probability matrix $Q$ where $$ Q_{i,j} = \left\{ \begin{array}{cl} 1-p-q & \text{if } j=i-1 \\ p & \text{if } j=i\\ q & \text{if } j=i+1\\ \end{array} \right. $$
Then the elements of $C$ will be $(C_{k,l}) = [Q^l]_{1,k+1}$ where $Q^l$ is the $l$-step transition matrix.
I was thinking two things: (1) get $C$ by doing some sort of inverse kronecker product or (2) get $C$ by minimizing some norm $||A-C\otimes B||$ with respect to $C$. But more specifically, I want to get $p$ and $q$. Does someone have any citations I can read to help me figure it out or have any suggestions for me? Thanks.