I am trying to find the average first passage cost of transition from $i$ to $j$ in a Markov Chain. If we consider costs of transitions as C[i,j], with the obvious property that $C[j,j]=0$, then we can calculate the costs as the expected value of the cost as a random variable for two cases, either a direct transition or an indirect transition, resulting in the following recurrence equation:
$C[i,j]=P[i,j](C[i,j]+C[j,j])+\sum_{k\in[0,n)|k\neq j}{P[i,k](C[i,k]+C[k,j])}$
where $P[i,j]$ is the transition probability from the Markov chain in question and the $C[i,j]$ is the average first passage cost that we are after. Then, considering the similarity of the two terms, and that $C[j,j]=0$, we unite the some as follows:
$C[i,j]=\sum_{k\in[0,n)}{P[i,k](C[i,k]+C[k,j])}=\sum_{k\in[0,n)}{(P[i,k]C[i,k]+P[i,k]C[k,j])}$
Which in matrix form can be vectorized as:
$\mathbf{C}=(\mathbf{P}\circ\mathbf{C})\mathbf{E}+\mathbf{P}\mathbf{C}$
I would like to find a closed-form solution by solving this matrix equation but do not know how to go further to bring all unknowns to one side.