Transition Matrix

Topics: Markov Chain - Transition Probabilities in a Markov Chain - n Step Transition Probability

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(definition)

Given a Markov chain with $m$ total states, we can handily write its n step transition probabilities $p_{ij}^{(n)}$ in a transition matrix, which is an $m\times m$ matrix of the form:

$$P^{(n)}=\left(\begin{array}{cccc}{p}_{00}^{(n)}& p_{01}^{(n)}& \dots & p_{0m}^{(n)}\\ p_{10}^{(n)}& p_{11}^{(n)}& \dots & p_{1m}^{(n)}\\ ⋮& ⋮& \ddots & ⋮\\ p_{m0}^{(n)}& p_{m1}^{(n)}& \dots & p_{mm}^{(n)}\end{array}\right)$$

That is, the $ij$th element of the transition matrix is the transition probability from $i$ to $j$ in $n$ steps.

$n$ step Transition Matrix §

(theorem)

From the Chapman-Kolmogorov equation it follows that:

$$P^{(n)}={\left(P^{(1)}\right)}^n$$

That is, matrix exponentiation.