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$S_1$ is the state vector after one iteration of the Markov chain. Its top entry is the probability of being in state A, and its bottom entry is the probability of being in state B after one step.

Conditions: Two-state chain with states A and B.; Entries are probabilities.; In the examples shown, the entries sum to 1.

$S_1$ is the state vector after one iteration of the Markov chain. Its top entry is the probability of being in state A, and its bottom entry is the probability of being in state B after one step.

Conditions: Two-state chain with states A and B.; Entries are probabilities.; In the examples shown, the entries sum to 1.

A matrix is introduced to provide an efficient algebraic process for moving from one state vector to later state vectors, such as $S_2$ or $S_100$, rather than re-reading the diagram each time.

Conditions: The Markov chain has fixed one-step transition probabilities.; State vectors are represented in a consistent order.

A matrix is introduced to provide an efficient algebraic process for moving from one state vector to later state vectors, such as $S_2$ or $S_100$, rather than re-reading the diagram each time.

Conditions: The Markov chain has fixed one-step transition probabilities.; State vectors are represented in a consistent order.