Markov Chain Calculator
Enter the row-stochastic transition matrix P and an initial distribution p₀. The calculator left-multiplies p₀ by P at each step to track how the distribution evolves, then solves (Pᵀ − I)·π = 0 with Σ πᵢ = 1 to find the steady-state distribution.
Worked examples
Frequently asked questions
What does row-stochastic mean?
Each row of P is a probability distribution: every entry is between 0 and 1, and the row sums to 1.
Why might the steady state not be unique?
If the chain is periodic or reducible, the steady-state equation has more than one solution. The calculator flags this when it cannot solve uniquely.
How is the distribution updated?
By left-multiplication: pₖ₊₁ⱼ = Σᵢ pₖᵢ · Pᵢⱼ. The new entry in column j is the dot product of the current distribution with column j of P.