# Finding eigenvectors

For each eigenvalue lambda, solve (A - lambda I) v = 0. Any non-zero solution is an eigenvector, and any scalar multiple of it is too.

Maths (university) · Part 10: Eigenvalues and eigenvectors · Card 3 of 9 · Last updated 2026-10-19

## Analogy

Once you know the stretch factor, you search for the direction that gets that stretch.

## Example

**A = [[4,1],[2,3]], lambda = 2**

A - 2I = [[2,1],[2,1]].
2x + y = 0, so y = -2x.
Take x = 1: v = (1, -2).
Check: A v = (2,-4) = 2v.

## In practice

Gives the actual directions, which you need to build the diagonal form.

## Practice

### Flashcards

- **Find an eigenvector of [[4,1],[2,3]] for lambda = 2.** (1, -2).

### Interview questions

**Find the eigenvalues and eigenvectors of A = [[4,1],[2,3]].**

- Characteristic equation L^2 - 7L + 10 = 0
- Eigenvalues 5 and 2
- For 5: (A - 5I)v = 0 gives v = (1,1)
- For 2: gives v = (1,-2)
- Check using trace 7 and determinant 10

## Questions people ask

### Why are the rows of A minus lambda I dependent?

Because the determinant is zero by design. The two equations in the example are the same line, 2x plus y equals 0, repeated. That leaves one free variable, which is why you get a whole line of eigenvectors, not a single vector. That is why eigenvectors are described as directions or lines.

### Is an eigenvector unique?

No. Any non-zero multiple of an eigenvector is also an eigenvector for the same eigenvalue. For lambda equals 2, both (1,-2) and (2,-4) work. We usually pick the simplest one, or scale it to length 1 so that comparing answers is easier.

## Related concepts

- [Characteristic equation](https://studyvisually.com/learn/science/maths-university/maths-university-part-10/maths-university-part-10-concept-2)
- [Trace and determinant check](https://studyvisually.com/learn/science/maths-university/maths-university-part-10/maths-university-part-10-concept-4)
- [Eigenvector and eigenvalue](https://studyvisually.com/learn/science/maths-university/maths-university-part-10/maths-university-part-10-concept-1)
- [Diagonalisation](https://studyvisually.com/learn/science/maths-university/maths-university-part-10/maths-university-part-10-concept-5)

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