Suppose $\{X_n\}$ is an ergodic Markov chain with state space $\mathcal{X}$, transition operator $P$, and stationary distribution $\pi$. Let $A:\mathcal{X} \to \mathbb{R}^{m \times n}$ be a matrix valued function defined on $\mathcal{X}$. For any $x \in \mathcal{X}$, the Poisson's equation for $A$ is given by
$$\hat{A}(x) - P \hat{A}(x) = A(x) - \pi(A)$$
where $\hat{A}: \mathcal{X} \to \mathbb{R}^{m \times n}$ is the solution to Poisson's equation, $P\hat{A}(x) = \int_{y \in \mathcal{X}} \hat{A}(y) P(dy \mid x)$, and $\pi(A) = \int_{x \in \mathcal{X}} A(x) \pi(dx)$ is the expected value of $A$ under the stationary distribution $\pi$. I understand that in general it is very difficult, or impossible, to obtain a closed form expression for the solution to the Poisson's equation. However, in this case, given that we are only dealing with linear functions, is there a closed form expression for $\hat{A}$?