There is a two-state discrete-time Markov chain with a random variable $ y_t = yx_t $, where $ y = \left[\array{1 \\ 5}\right] $. It is known that $ E\left(y_{t+1} \mid x_t\right) = \left[\array{1.8 \\ 3.4}\right] $ and that $ E\left(y_{t+1}^2 \mid x_t \right) = \left[\array{5.8 \\ 15.4}\right] $. How to find a transition matrix consistent with these conditional expectations? Is it possible? Is the transition matrix unique (i.e., can you find another one that is consistent with these conditional expectations)?
$ Ey_t = \left(\pi_0^{\intercal}P^t\right)y $, where $ P $ is a transistion matrix, $ \pi_0 $ is an $ \left(n \times 1\right) $ vector whose $ i $th element is the probability of being in state $ i $ at time $ 0 $. As I understood, $ E\left(y_{t+1} \mid x_t\right) = Py $, where $ P $ is the transistion matrix, and $ E\left(y^2_{t+1} \mid x_t \right) = P^2y$. Could $ P $ be found using linear algebra?