What is a Markov chain's steady-state distribution?

It's the probability distribution over states that remains unchanged after one more transition (πP = π) — intuitively, the long-run fraction of time the chain spends in each state.

What does 'row-stochastic' mean?

It means every row of the transition matrix represents a valid probability distribution: all entries are non-negative and each row sums to exactly 1, since row i, column j gives the probability of moving from state i to state j.

How does power iteration work?

It starts with any probability vector and repeatedly multiplies it by the transition matrix; for well-behaved chains, this sequence of vectors converges to the unique stationary distribution regardless of the starting point.

Does every Markov chain have a unique steady state?

No — a chain needs to be irreducible (every state reachable from every other) and aperiodic (doesn't cycle in a fixed rhythm) to guarantee a unique steady state that power iteration will find; periodic or reducible chains may not converge.

What happens if the calculator doesn't converge?

It runs up to 1000 iterations and reports whether the vector stabilized within a tight tolerance; if not, the chain may be periodic or reducible, and the closest approximation found is shown instead of a false claim of convergence.

How large a matrix can this calculator handle?

This calculator supports transition matrices up to 20×20, keeping each power-iteration step instant while covering the vast majority of hand-entered chain examples.

Is this the same as an eigenvector calculation?

Conceptually yes — the steady-state distribution is the left eigenvector of the transition matrix for eigenvalue 1 — but power iteration reaches it through simple repeated matrix-vector multiplication rather than general eigenvalue decomposition.

Where are Markov chains used in real life?

They model web page ranking (PageRank), weather prediction, board game and queueing systems, genetics (DNA sequence modeling), and customer behavior/churn prediction.

Markov Chain Steady State Calculator — Find a Chain's Long-Run Distribution

Our Markov Chain Steady State Calculator finds the long-run probability distribution across a Markov chain's states — the distribution that no longer changes once the chain settles down — from a row-stochastic transition matrix, using power iteration (repeatedly multiplying a probability vector by the matrix until it stabilizes).

Quick Answer

Enter a row-stochastic transition matrix to instantly find the chain's steady-state distribution via power iteration.

Transition matrix

Enter one row per line, comma-separated probabilities that sum to 1, e.g. 0.9,0.1 then 0.5,0.5.

How to Use the Markov Chain Steady State Calculator — Stationary Distribution Online

1
Enter your transition matrix, one row per line, with comma-separated probabilities, e.g. 0.9,0.1 then 0.5,0.5.
2
Each row must be non-negative and sum to 1 (a valid probability distribution over the next state).
3
Click 'Calculate' to find the steady-state distribution via power iteration.

Why Use Markov Chain Steady State Calculator — Stationary Distribution Online?

A Markov chain moves between states with fixed transition probabilities, and for many chains, the probability of being in each state converges to a fixed 'steady-state' distribution no matter where you started — the long-run fraction of time spent in each state. Power iteration finds this distribution directly: start with any probability vector (this calculator uses a uniform starting guess), repeatedly multiply it by the transition matrix (π ← πP), and watch it converge toward the stationary distribution π that satisfies πP = π. This iterative approach is simpler and more numerically robust to implement correctly than solving the eigenvector equation directly, and works reliably for irreducible, aperiodic chains.

Frequently Asked Questions

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