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The equations derived from $\pi P = \pi$ are linearly dependent because the rows of the transition matrix $P$ sum to 1. This implies that the sum of the equations is identically zero, leaving only $N-1$ independent constraints for $N$ states.

Conditions: $P$ is a stochastic matrix (rows sum to 1).; $\pi$ is a row vector of probabilities.

The equations derived from $\pi P = \pi$ are linearly dependent because the rows of the transition matrix $P$ sum to 1. This implies that the sum of the equations is identically zero, leaving only $N-1$ independent constraints for $N$ states.

Conditions: $P$ is a stochastic matrix (rows sum to 1).; $\pi$ is a row vector of probabilities.

To find a stationary distribution for a Markov chain, one solves the system of linear equations given by $\pi P = \pi$, subject to the constraint that the probabilities in $\pi$ sum to 1.

Conditions: $P$ is the transition matrix of the Markov chain.; $\pi$ is a row vector of probabilities.

To find a stationary distribution for a Markov chain, one solves the system of linear equations given by $\pi P = \pi$, subject to the constraint that the probabilities in $\pi$ sum to 1.

Conditions: $P$ is the transition matrix of the Markov chain.; $\pi$ is a row vector of probabilities.

The stationary distribution is the row vector $\pi = [4/11, 7/11]$.

Conditions: The transition matrix is $P = \begin{bmatrix} 0.3 & 0.7 \\ 0.4 & 0.6 \end{bmatrix}$.; $\pi$ is a row vector satisfying $\pi P = \pi$ and $\sum \pi_i = 1$.

The stationary distribution is the row vector $\pi = [4/11, 7/11]$.

Conditions: The transition matrix is $P = \begin{bmatrix} 0.3 & 0.7 \\ 0.4 & 0.6 \end{bmatrix}$.; $\pi$ is a row vector satisfying $\pi P = \pi$ and $\sum \pi_i = 1$.

A simple sanity check is to verify that the sum of the probabilities in each row of the transition matrix equals 1.

Conditions: The matrix is intended to be a transition matrix for a Markov chain.

A simple sanity check is to verify that the sum of the probabilities in each row of the transition matrix equals 1.

Conditions: The matrix is intended to be a transition matrix for a Markov chain.