Simpler than in in-review work (Wijayasari et al., 2022), it can be written in the form of
$$\tag{1} \mathbf{X}^{t + 1} = \mathbf{X}^t \ \mathbf{B} $$
with
$$\tag{2} \mathbf{B} = \left[ \begin{array}{ccc} b_{11} & b_{12} & b_{13} \newline b_{21} & b_{22} & b_{23} \newline b_{31} & b_{32} & b_{33} \end{array} \right] $$
if only three states or classes are considered and
$$\tag{3} \mathbf{X}^t = \left[ \begin{array}{ccc} x_{11}^t & x_{12}^t & x_{13}^t \newline x_{11}^t & x_{12}^t & x_{13}^t \newline x_{11}^t & x_{12}^t & x_{13}^t \newline \dots & \dots & \dots \newline x_{N1}^t & x_{N2}^t & x_{N3}^t \end{array} \right], $$
for $N$ points of data. From Eqns (1)-(3) it can be written
$$\tag{4} x_{ij}^{t+1} = x_{i1}^t b_{1j} + x_{i2}^t b_{2j} + x_{i3}^t b_{3j}, $$
where
$$\tag{5} \sum_{j = 1}^3 b_{ij} = 1 $$
in each row. Value of $b_{ij}$ means, in this case, the probability of transition from state $i$ to state $j$, where Eqn (5) describe that probability of state $i$ becames all states is equal to one.
- Winda Wijayasari, Faizal Rohmat, Sparisoma Viridi, “An R-Based End-to-End Markov Transition Matrix Extraction for Land Cover Datasets”, SSRN 4246799, 2022, url https://dx.doi.org/10.2139/ssrn.4246799.