Suppose we have $N$ states$q_i$, $0\leq i<N$Suppose we have $N\times N$ probability matrix$p_{ij}$, $0\leq i<N$, $0\leq j<N$, And suppose a machine, which is being in state $q_i$ then goes to state $q_j$ with probability $p_{ij}$.Question #1: is this what is called probabilistic automaton?Next.We can introduce another interpretation: represent each state of a machine as a column of values$w_i$, $0\leq i<N$each representing a weight of previously defined state $q_i$.If machine is completely in state $q_0$ then we have$w_i=1, i=0$$w_i=0, i\neq0$and so on.In this interpretation each state change is just changing of the weights of a state column$w_i \rightarrow p_{ij} w_i$assumming summation by.Question #2: it this a precise interpretation of previous PA? If each first interpretation corresponds with second and vice versa?
Vector representation of probabilistic automaton
automata theory;probabilistic automata