# Eigenvector and eigenvalue

An eigenvector of A is a non-zero vector v with A v = lambda v. The matrix only stretches it, and the stretch factor lambda is the eigenvalue.

Maths (university) · Part 10: Eigenvalues and eigenvectors · Card 1 of 9 · Last updated 2026-10-19

## Analogy

Pulling a rubber sheet: some lines of fabric only get longer or shorter and do not turn.

## Example

**A = [[4,1],[2,3]], v = (1,1)**

A v = (4+1, 2+3) = (5,5).
That equals 5 x (1,1).
So v is an eigenvector.
Its eigenvalue is 5.

## In practice

Eigenvectors reveal the main directions of a transformation and make hard problems simple.

## Practice

### Flashcards

- **What equation defines an eigenvector?** A v = lambda v, with v not zero.

## Questions people ask

### Can the zero vector be an eigenvector?

No. The zero vector satisfies A v equals lambda v for every lambda, so it would tell you nothing. Eigenvectors must be non-zero. The eigenvalue itself can be zero, though, which happens when the matrix squashes that direction to nothing.

### Does an eigenvector keep its direction?

Yes, up to a flip. A positive eigenvalue stretches it along the same line, while a negative eigenvalue also reverses it. A matrix with eigenvalue -2 turns the vector into twice its length, pointing the opposite way along the same line.

## Related concepts

- [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)
- [Diagonalisation](https://studyvisually.com/learn/science/maths-university/maths-university-part-10/maths-university-part-10-concept-5)

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