$$\begin{bmatrix} p(A \rightarrow A) & p(A \rightarrow B) \\ p(B \rightarrow A) & p(B \rightarrow B) \end{bmatrix}$$

Multiply the transition matrix T by itself 'n' times, then multiply by the initial state vector X: $$T^n * X$$

Multiply the inverse of the transition matrix by the current state vector: $$T^{-1} * X$$

The sum of entries in each column must equal 1, and all entries must be non-negative.

Multiply the transition matrix (T) by the current state vector (X): $$T * X$$

Repeatedly multiply the transition matrix by the state vector until the state vector converges. Alternatively, solve TX = X.

Multiply the transition matrix by itself, then multiply the result by the initial state vector: $$T^2 * X$$

Multiply the transition matrix by the current state vector: $$T * X$$

Multiply the inverse of the transition matrix by the current state vector: $$T^{-1} * X$$

$$p(A \rightarrow A)$$

It shows the probabilities of moving from one state to another at each time step, allowing us to track how the system evolves.

It's the long-term distribution where the probabilities of being in each state remain constant, even after multiple transitions.

By repeatedly multiplying the transition matrix by the state vector, you can project the probabilities of being in each state at future time intervals.

Because each column represents the probabilities of transitioning from a single state to all possible states, and these probabilities must add up to 100%.

It means you can predict past states by multiplying the inverse of the transition matrix by the current state vector.

Each entry represents the probability of transitioning from one state to another in a single time step.

Because matrix multiplication is not commutative, so changing the order can lead to incorrect results.

Markov chains can model systems that transition between states based on probabilities, such as customer behavior, weather patterns, or population dynamics.

To determine the long-term distribution of states, which can help predict the eventual outcome of a system.

The initial state vector determines the starting point, and the transition matrix then dictates how the probabilities evolve over time from that starting point.

A matrix representing probabilities of moving between states in a system.

A column vector showing probabilities of being in each state at a specific time.

A model that describes transitions between states, where the probability of each transition depends only on the current state.

The distribution of states in a Markov chain that doesn't change over time.

The probability of moving from state A to state B.

A matrix that, when multiplied by the original matrix, results in the identity matrix.

A square matrix with 1s on the main diagonal and 0s elsewhere.

Each column represents the probabilities of transitioning from a specific state to all other possible states.

To represent and analyze transitions between different states over time.

A transition matrix describes how probabilities change, while a state vector describes the probabilities at a specific time.