I would like to know if there are any special analytic expressions or fast numerical methods to get the spectrum for the transition matrix corresponding to a Markovian binary process (Bernoulli process with history dependence). For example, a process that depends only on the past 3 time steps, the transition matrix has the following form:
$$ \begin{bmatrix} 1-\omega_{000} & 0 & 0 & 0 & 1-\omega_{100} & 0 & 0 & 0 \\ \omega_{000} & 0 & 0 & 0 & \omega_{100} & 0 & 0 & 0 \\ 0 & 1-\omega_{001} & 0 & 0 & 0 & 1-\omega_{101} & 0 & 0 \\ 0 & \omega_{001} & 0 & 0 & 0 & \omega_{101} & 0 & 0 \\ 0 & 0 & 1-\omega_{010} & 0 & 0 & 0 & 1-\omega_{110} & 0 \\ 0 & 0 & \omega_{010} & 0 & 0 & 0 & \omega_{110} & 0 \\ 0 & 0 & 0 & 1-\omega_{011} & 0 & 0 & 0 & 1-\omega_{111} \\ 0 & 0 & 0 & \omega_{011} & 0 & 0 & 0 & \omega_{111} \end{bmatrix} \cdot \begin{bmatrix} p_{000}\\ p_{001}\\ p_{010}\\ p_{011}\\ p_{100}\\ p_{101}\\ p_{110}\\ p_{111} \end{bmatrix} = \begin{bmatrix} p_{000}\\ p_{001}\\ p_{010}\\ p_{011}\\ p_{100}\\ p_{101}\\ p_{110}\\ p_{111} \end{bmatrix} $$ where $\omega$ is the probability of 1 in the next time step, and $p$ are probability mass over all possible histories (of length 3 in this case).
There's a nice structure to the matrix, so I am hoping that it is a known form with solutions.
