Explore transition probabilities, matrix powers, and long-run trends. Validate sums, normalize inputs, and compare paths. See each state evolve through clear tables and charts.
Enter a square transition matrix, define an initial distribution, and choose how many steps to project. The solver supports 2 to 5 states.
This example models three customer loyalty states. Start fully in State A with a five-step projection.
| From \ To | State A | State B | State C | Row sum |
|---|---|---|---|---|
| State A | 0.70 | 0.20 | 0.10 | 1.00 |
| State B | 0.25 | 0.50 | 0.25 | 1.00 |
| State C | 0.15 | 0.25 | 0.60 | 1.00 |
Next-state distribution: pt+1 = ptT
n-step distribution: pn = p0Tn
Row constraint: Σj Tij = 1 for every row i
A transition matrix T stores probabilities of moving from one state to another in one step. The current probability vector pt is multiplied by T to obtain the next distribution.
To project farther ahead, the solver computes Tn with repeated squaring. This method is efficient and accurate for larger step counts.
The steady-state estimate is found by repeatedly applying the transition matrix until the change between iterations falls below a very small tolerance.
It represents one-step movement probabilities among states in a Markov process. Each row corresponds to a current state, and each column shows the chance of moving to a future state.
Each row is a full probability distribution for the next step. Since one of the listed outcomes must occur, the probabilities across that row must total exactly 1.
It describes where the system starts before any transitions happen. A single-state start uses 1 for one state and 0 for others, while mixed starts use fractional probabilities.
Tn is the transition matrix multiplied by itself n times. It gives direct n-step movement probabilities without manually simulating every intermediate step.
Steady state is a long-run probability distribution that stays unchanged after further transitions. Some chains converge to it quickly, while others may oscillate or converge slowly.
Enable it when your inputs are close to correct but slightly off because of rounding or draft estimates. It rescales rows and the initial vector to valid probability totals.
Yes. This page supports two to five states. That range keeps the interface readable while still covering many educational, operational, and analytical Markov-chain use cases.
The graph shows how each state probability changes across time steps. It is useful for spotting dominant states, stable behavior, convergence patterns, and slow-moving transitions.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.