# Steady state

For a Markov chain (columns of probabilities summing to 1), the long-run distribution is the eigenvector with eigenvalue 1.

Maths (university) · Part 10: Eigenvalues and eigenvectors · Card 7 of 9 · Last updated 2026-10-19

## Analogy

A busy shop whose customers keep moving between rooms, but the crowd in each room stays the same.

## Example

**P = [[0.9,0.5],[0.1,0.5]]**

Solve P v = v: 0.5y = 0.1x, so x = 5y.
Scale to total 1: x = 5/6, y = 1/6.
Steady state (0.833, 0.167).
Trace 1.4 gives other eigenvalue 0.4.

## In practice

Used for weather models, web page ranking and population movements.

## Practice

### Flashcards

- **Which eigenvalue gives the steady state of a Markov chain?** Eigenvalue 1.

## Questions people ask

### Why is the steady state an eigenvector with eigenvalue 1?

A steady state does not change when you apply the transition matrix once more, so P v equals v. That is precisely the eigenvector condition with lambda equal to 1. You then scale the vector so its entries add up to 1.

### Does every Markov chain have a steady state?

Every one has an eigenvalue of 1, so a steady distribution exists. If every state can reach every other, it is unique and the chain settles into it. The other eigenvalues are smaller in size and fade as time passes.

## Related concepts

- [Eigenvector and eigenvalue](https://studyvisually.com/learn/science/maths-university/maths-university-part-10/maths-university-part-10-concept-1)
- [Characteristic equation](https://studyvisually.com/learn/science/maths-university/maths-university-part-10/maths-university-part-10-concept-2)
- [Finding eigenvectors](https://studyvisually.com/learn/science/maths-university/maths-university-part-10/maths-university-part-10-concept-3)
- [Trace and determinant check](https://studyvisually.com/learn/science/maths-university/maths-university-part-10/maths-university-part-10-concept-4)

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