Markov Chain Steady State Calculator
The steady-state distribution of a Markov chain tells you the long-run share of time spent in each state, no matter where you start. Enter the row-stochastic transition matrix for a three-state chain, where each row gives the probabilities of moving to states 1, 2, and 3. The calculator iterates the chain to convergence and returns the steady-state probabilities.
Steady-state method
Each row of P is normalised to sum to 1
Start: pi = (1/3, 1/3, 1/3)
Iterate: pi_new = pi * P
Repeat until max change < 1e-12
Result satisfies pi * P = pi and pi_1 + pi_2 + pi_3 = 1
The steady state is the left eigenvector of P with eigenvalue 1, normalised to sum to one. Power iteration converges to it for any regular chain.
Worked example
For the default matrix, iterating pi times P to convergence gives the steady state approximately (0.2245, 0.4286, 0.3469). Each value is the long-run fraction of time the chain spends in that state, and the three sum to 1.00.
Markov steady state: frequently asked questions
What is a steady-state distribution?
It is the long-run fraction of time a Markov chain spends in each state. For a regular chain it is the unique probability vector that does not change when multiplied by the transition matrix: the row vector pi where pi times P equals pi, with the entries summing to one.
How is it computed here?
The calculator uses power iteration: it starts from a uniform distribution and repeatedly multiplies by the transition matrix until the vector stops changing. For a regular chain this converges to the steady state regardless of the starting point.
What is a row-stochastic matrix?
It is a transition matrix where each row holds the probabilities of moving from one state to every state, so each row sums to one. Enter probabilities as decimals between 0 and 1. The calculator normalises each row so it sums to one before iterating.
When does a steady state exist?
A unique steady state exists when the chain is irreducible and aperiodic, called regular. Chains with absorbing or periodic structure may converge to a vector that depends on the start or may not settle. The result here assumes a regular chain.
Sources
- NIST Digital Library of Mathematical Functions: DLMF (linear algebra and stochastic matrices).
Reviewed by the CalculatorHub team, edited by James Graham, 19 June 2026. See our methodology.