I have the following system of equations in matrix form. It represents a series of agents, each spending a proportion of time in two states $p_1$ and $p_2$.
$ C(P-0.5) = d $
$P$ is a vector of proportions of time in $p_1$ for each agent, and $C$ is a square matrix (agent x agent). It represents how the agents interact.
I know how to solve this system with two states. However, suppose I have 3 $(p_1,p_2,p_3)$. I can no longer represent these proportions as single vector. The only solution I found was to make a new system.
$$ \begin{cases} C(P_1-P_2)=d \\ C(P_2-P_3)=e \\ P_1+P_2+P_3=1 \end{cases} $$
I want to represent this new system in matrix form also, but I'm not sure there is a way to do so. My intuition would be
$ C*M*P = R $
where M is matrix of the form:
$$ M= \begin{bmatrix} 1 & -1 & 0 \\ 0 & 1 & -1 \\ 1 & 1 & 1 \end{bmatrix} $$
The problem is that the multiplication of $M*P$ is not conformal when there is more agents than states. Is there any way to represent this sytem in a matrix form?
I don't see the problem. I assume that you seek the vectors $P_1,P_2,P_3$ and that $C$ is invertible. Then, you can calculate $P_1-P_2,P_2-P_3$ and you know $P_1+P_2+P_3$; then you can deduce easily the $3$ unknowns.