Calculator
Example Data Table
| State | Transition Row | Example Equilibrium Probability |
|---|---|---|
| State 1 | 0.70, 0.20, 0.10 | 0.456522 |
| State 2 | 0.30, 0.40, 0.30 | 0.282609 |
| State 3 | 0.20, 0.30, 0.50 | 0.260870 |
This sample shows a valid stochastic matrix. Each row sums to 1. The listed equilibrium vector satisfies πP = π and totals 1.
Formula Used
The equilibrium vector is the steady state probability row vector. It is written as π. The main equation is πP = π, where P is the transition matrix.
The vector must also satisfy π1 + π2 + ... + πn = 1. This condition makes the probabilities add to one.
The calculator solves the linear system formed by PT − I and then replaces one equation with the normalization rule. That creates a solvable system for the stationary probabilities.
It also iterates vk+1 = vkP from the chosen starting vector. Those repeated multiplications show whether the distribution moves toward the equilibrium vector.
For regular stochastic matrices, the equilibrium vector is unique and the iteration history approaches the same long run distribution from many starting points.
How to Use This Calculator
- Enter a square transition matrix with one row per line.
- Make sure every row sums to 1.
- Enter an optional initial probability vector.
- Choose the number of iteration steps.
- Choose the number of decimal places.
- Press the calculate button.
- Review the equilibrium vector, summary table, graph, and iteration table.
- Use the CSV or PDF buttons to save the output.
Understanding Equilibrium Vectors in Transition Matrices
What the equilibrium vector means
An equilibrium vector describes the long run behavior of a Markov process. Each entry represents the probability of being in one state after many transitions. When the distribution reaches equilibrium, applying the same transition matrix again does not change it. That is why the steady state vector satisfies the condition πP = π.
Why the transition matrix matters
A transition matrix stores the movement probabilities between states. Every row must sum to 1 because each row represents a full set of possible next outcomes. If a row does not total 1, the matrix is not a valid stochastic matrix and the results would not represent probability behavior correctly.
Why an initial vector is useful
The initial vector gives the starting distribution. It shows where the process begins before any transitions occur. Repeated multiplication by the transition matrix creates a sequence of distributions. That sequence helps you study convergence and compare the current distribution with the equilibrium vector at each step.
Why iteration tables help
The iteration table makes the calculation more transparent. Instead of only showing the final answer, it reveals how the probabilities evolve from step 0 to the selected number of transitions. This is helpful in teaching, verification, and practical modelling.
Where this calculator can help
This type of calculator is useful in population movement models, customer retention analysis, web navigation paths, queueing systems, game transitions, and many other probabilistic systems. By combining a solver, a convergence table, exports, and a graph, the page gives both a final answer and a clear explanation of how the answer behaves over time.
FAQs
1. What is an equilibrium vector?
An equilibrium vector is a probability vector that remains unchanged after multiplication by the transition matrix. It represents the long run distribution of the system.
2. Does every transition matrix have one unique equilibrium vector?
No. A valid stochastic matrix always has at least one stationary solution, but uniqueness depends on the matrix structure. Regular chains usually give a unique equilibrium vector.
3. Why must each row sum to 1?
Each row represents all possible next states from one state. Since those outcomes cover every possibility, their probabilities must add to 1.
4. What happens if I leave the initial vector blank?
The calculator uses a uniform starting vector automatically. That means it begins with equal probability in every state.
5. Why does the calculator normalize my initial vector?
A probability vector must total 1. If your values are proportional but do not add to 1, the calculator rescales them so they become a valid distribution.
6. What does the residual value mean?
The residual measures how closely the computed equilibrium vector satisfies πP = π. A very small residual indicates a strong numerical solution.
7. Why do I need the iteration graph?
The graph shows how each state probability changes across steps. It helps you see convergence speed and whether the distribution stabilizes.
8. Can I use decimals and different matrix sizes?
Yes. You can enter decimal probabilities and any square matrix size, as long as all rows have equal length and each row sums to 1.