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The initial state vector $S_0$ is a column vector that encodes the starting probabilities of the Markov chain. In this specific example, $S_0 = (1, 0)^T$, meaning the system starts in state A with probability 1 and in state B with probability 0.

Conditions: The system has two states, A and B.; The components are ordered consistently with the states, here A then B.; The entries are probabilities describing the initial distribution.

The initial state vector $S_0$ is a column vector that encodes the starting probabilities of the Markov chain. In this specific example, $S_0 = (1, 0)^T$, meaning the system starts in state A with probability 1 and in state B with probability 0.

Conditions: The system has two states, A and B.; The components are ordered consistently with the states, here A then B.; The entries are probabilities describing the initial distribution.

The 0.75 label on the self-loop from state A to state A represents a 75% probability that a system currently in state A will remain in state A in the next iteration.

Conditions: The system is in state A at the current step.; The arrow is a self-loop on state A in the transition diagram.

The 0.75 label on the self-loop from state A to state A represents a 75% probability that a system currently in state A will remain in state A in the next iteration.

Conditions: The system is in state A at the current step.; The arrow is a self-loop on state A in the transition diagram.

Matrices and vectors are introduced as a linear algebra formalism to make manipulations and calculations simpler when studying a Markov chain, especially for iterating states efficiently over many steps.

Conditions: The problem involves a Markov chain.; The goal is to compute future states or manipulate transition probabilities.

Matrices and vectors are introduced as a linear algebra formalism to make manipulations and calculations simpler when studying a Markov chain, especially for iterating states efficiently over many steps.

Conditions: The problem involves a Markov chain.; The goal is to compute future states or manipulate transition probabilities.