Periodic state (markov chain)

For a Markov chain given by $P \in M_{N,N}(\mathbb{R})$ a state $i$ with $1 \leq i \leq N$ is said to be aperidoic if

$$gcd(\{n \in \mathbb{N} \vert P^n \mbox{ has non-zero } i'th \mbox{ diagonal value} \}) = 1$$

and is periodic otherwise.