I'm trying to find a solution to a system of equations, but its quite different from anything I've come across before. I believe there is a solution, but I could be wrong.
$\mathbf{A} = \mathrm{diag}(\mathbf{a})$ is an $n \times n$ diagonal matrix whose diagonal entries $\mathbf{a}_i \in [0,1]$ are the unknown quantities in my system. The equation is as follows: $$ \mathbf{p}^T [ \mathbb{I} - \mathbf{T} + \mathbf{AT}]^{-1}\mathbf{A} = \mathbf{c}^T$$ where $\mathbf{p}$ is an $n$-vector of initial state probabilities, $\mathbf{T}$ is an $n \times n$ transition matrix of probabilities whose rows sum to one, and $\mathbf{c}$ is an $n$-vector of capacities. The above equation is based on an absorbing markov chain, and the idea is that it represents a model where capacity is limited, and in equilibrium, the acceptance probabilities $a_i$ "clear the market" so that no absorbing state is overfull. The equals sign should actually be an inequality but I'm trying to solve a simpler version for now.
I need to solve for $\mathbf{A}$. I don't know a lot about linear algebra, but I don't think that systems usually look like this, and you don't usually try to solve for the matrix. However, since the matrix is diagonal there are actually only $n$ equations in $n$ unknowns. The vector $\mathbf{a}$ is actually what I'm interested in, but I couldn't see another way to make it work in matrix notation.
I have got as far as $$ \mathbf{p}^T = \mathbf{c}^T\mathbf{A}^{-1} - \mathbf{c}^T\mathbf{A}^{-1}\mathbf{T} + \mathbf{c}^T\mathbf{T} $$ but now I'm hopelessly stuck. I hope someone can help me. Thank you!
Assuming that you want to solve the last eqaution I don't see any problem here. Define $x=A^{-1}c$. Then you have $$ p-T^Tc=(I-T^T)x $$ Solve it. (Take in account that $I-T^T$ probably is singular).
Recover $A$ from $x$
The only problem I observe is to guarantee ($a\in [0,1]$. Does it follow from the nature of the problem or not?