How can I write a conditional expectation of finite state markov process in matrix notation NOTE: This question is related to the econometric method explored in the following two questions:


Multiplicative factorization of stochastic growth time series--solving for an eigenfunction/eigenvector
Example of the change of measure proposed in Hansen (2012)


QUESTION: Suppose that $X_t$ is an $n$-state Markov chain with transition probability matrix $\mathbb P$ and realized values given by $n$-dimensional coordinate vectors. Suppose that $\{W_{t+1} \}$ is an iid sequence of multivariate normally distributed random vectors. How would I represent an equation of the form
$$
E[\exp(D'X_t + X_t' F W_{t+1}) e(X_{t+1}) \mid X_t = x] = \exp(\eta) e(x)
$$
as an eigenvector problem for a matrix $\mathbb M$? How can I represent $\mathbb M$ in terms of the primitives of the problem?
Let the $n$ states of the finite-state markov chain be denoted
by $\{x_1,...,x_n\}$ and let $\vec e = [e(x_1), ..., e(x_n)]'$.
Now,
first note that because $X_{t+1} \mid X_t$ is independent of $W_{t+1}$, we can write
\begin{align*}
\exp(\eta) e(x) &amp;= E[\exp(D'x + x' F W_{t+1})] E[ e(X_{t+1}) \mid X_t = x] \\
  &amp;= \exp(D'x + x' F F' x)\, E[ e(X_{t+1}) \mid X_t = x] .
\end{align*}
Then, because this must hold for all values $x= x_1, ..., x_n$, we
can assemble the following vector equation,
\begin{align*}
\vec e \, \exp(\eta)
  &amp;= 
\begin{bmatrix}
\exp(D'x_1 + x_1' F F' x_1)\, E[ e(X_{t+1}) \mid X_t = x_1] \\
... \\
\exp(D'x_n + x_n' F F' x_n)\, E[ e(X_{t+1}) \mid X_t = x_n]  \\
\end{bmatrix}\\
  &amp;= 
\text{diag} \begin{bmatrix}
\exp(D'x_1 + x_1' F F' x_1) \\
... \\
\exp(D'x_n + x_n' F F' x_n) \\
\end{bmatrix}
\, \mathbb P \, \vec e \\
  &amp;= 
\text{diag}(f)\, \mathbb P \, \vec e \\
\end{align*}
where $f = [ \exp(D'x_1 + x_1' F F' x_1),  ..., \exp(D'x_n + x_n' F F' x_n) ]'$
and $\text{diag}$ is the operator that takes a vector and places
it on the diagonal of a matrix where the off-diagonal elements are zero.
So, we can express the equation as a linear operator $\mathbb M$ on $\vec e$
with
\begin{align*}
\text{diag}(f)\, \mathbb P \, \vec e
  &amp;= \vec e \, \exp(\eta) \\
\mathbb M \vec e &amp;= \vec e \, \exp(\eta).
\end{align*}