Let T be a regular transition matrix with probability...

Let $T$ be a regular transition matrix with probability eigenvector $\mathbf{v}$. (a) Prove that $\lim _{k \rightarrow \infty} T^k=P=(\mathbf{v} \mathbf{v} \ldots \mathbf{v})$ is a matrix with every column equal to $\mathbf{v}$. (b) Explain why $(\mathbf{v} \mathbf{v} \ldots \mathbf{v}) \mathbf{v}=\mathbf{v} . \quad$ (c) Prove directly that $P$ is idempotent: $P^2=P$.

Let $T$ be a regular transition matrix with probability eigenvector $\mathbf{v}$.
(a) Prove that $\lim _{k \rightarrow \infty} T^k=P=(\mathbf{v} \mathbf{v} \ldots \mathbf{v})$ is a matrix with every column equal to $\mathbf{v}$.
(b) Explain why $(\mathbf{v} \mathbf{v} \ldots \mathbf{v}) \mathbf{v}=\mathbf{v} . \quad$ (c) Prove directly that $P$ is idempotent: $P^2=P$.

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Key Concepts

Convergence of Matrix Powers and Idempotency

When a regular transition matrix is raised to successively higher powers, it converges to a matrix whose columns are all equal to the stationary distribution. This resulting matrix behaves like a projection operator, meaning it is idempotent (P^2 = P). The idempotency reflects the fact that once the state distribution has converged, further applications of the projection (or transition) do not change it.

Regular Transition Matrix

This concept refers to a type of square matrix in which some power of the matrix has only strictly positive entries. In the context of Markov chains, it signifies that the chain is both irreducible and aperiodic, which guarantees that the state space is fully connected and that the chain will converge to a unique stationary distribution regardless of the initial state.

Stationary Distribution (Probability Eigenvector)

A stationary distribution is a probability vector that remains invariant under the action of the transition matrix. It is the eigenvector corresponding to the eigenvalue 1. In the long-term behavior of a Markov chain generated by a regular transition matrix, every column of the limit matrix converges to this stationary distribution, illustrating that the chain loses memory of its starting state.

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