Transition matrices
A transition matrix stores the one-step transition probabilities of a finite-state DTMC.
\[P=(p_{ij}),\qquad p_{ij}=P(X_{n+1}=j\mid X_n=i).\]Probability rules
\[p_{ij}\ge0,\qquad \sum_jp_{ij}=1.\]Thus every row is a probability distribution over possible next states when the row-vector convention is used.
Multiple steps
The \(m\)-step transition matrix is
\[P^m,\]and its \((i,j)\) entry gives the probability of being in state \(j\) after \(m\) steps given initial state \(i\).
Key idea. A transition matrix collects the complete one-step probabilistic rule of a finite DTMC.