Enter Matrix Data
Use decimal probabilities. Choose row-stochastic when each row sums to one, or column-stochastic when each column sums to one.
Example Data Table
This sample row-stochastic matrix models three-state movement. Every row totals 1, so each row can be interpreted as a full transition distribution.
| State | S1 | S2 | S3 | Row Sum |
|---|---|---|---|---|
| S1 | 0.50 | 0.30 | 0.20 | 1.00 |
| S2 | 0.20 | 0.60 | 0.20 | 1.00 |
| S3 | 0.10 | 0.30 | 0.60 | 1.00 |
Formula Used
Row-stochastic condition: ∑ pij = 1 for every row i.
Column-stochastic condition: ∑ pij = 1 for every column j.
k-step transition matrix: Pk, where repeated multiplication estimates movement after k steps.
Row-vector propagation: xk = x0Pk.
Column-vector propagation: xk = Pkx0.
Stationary distribution: π = πP for row-stochastic form, or Pπ = π for column-stochastic form.
Normalization: each selected row or column is divided by its own total so the corrected probabilities sum to 1.
How to Use This Calculator
- Select the matrix dimension from 2 to 8.
- Choose whether your probabilities are row-stochastic or column-stochastic.
- Enter all matrix values in decimal form.
- Enter the starting distribution vector for the states.
- Set the power exponent to inspect multi-step transitions.
- Enable auto-normalization when your probabilities need correction.
- Click Calculate Matrix to see validation, steady-state estimates, and transition behavior.
- Use the CSV and PDF buttons to export the results.
Frequently Asked Questions
1. What does a stochastic matrix represent?
It represents transition probabilities between states. Each valid row or column forms a complete probability distribution, making the matrix useful for Markov chains and long-run state analysis.
2. What is the difference between row-stochastic and column-stochastic?
In a row-stochastic matrix, every row sums to one. In a column-stochastic matrix, every column sums to one. The propagation formula changes with that choice.
3. Why would I use the power matrix?
The power matrix shows probabilities after several transitions. For example, P5 reveals the five-step transition pattern instead of only one-step movement.
4. What is a stationary distribution?
It is a probability vector that remains unchanged after another transition. It often describes the long-run behavior of an irreducible and regular Markov process.
5. What is an absorbing state?
An absorbing state keeps probability mass once entered. Its self-transition probability is 1, while transitions from that state to different states are 0.
6. What does irreducible mean here?
Irreducible means every state can reach every other state through some sequence of transitions. This is important when studying unique long-run distributions.
7. What is a doubly stochastic matrix?
A doubly stochastic matrix has all rows and all columns summing to one. In many cases, the uniform distribution becomes a stationary distribution.
8. When should I auto-normalize the entries?
Use auto-normalization when your entries are nonnegative but the chosen rows or columns do not sum exactly to one because of rounding or manual entry errors.