# Characteristic equation

The eigenvalues of A are the solutions of det(A - lambda I) = 0. This gives a polynomial in lambda.

Maths (university) · Part 10: Eigenvalues and eigenvectors · Card 2 of 9 · Last updated 2026-10-19

## Analogy

A lock whose combination is the set of stretch factors that make the matrix collapse.

## Example

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

det = (4 - L)(3 - L) - (1)(2).
= L^2 - 7L + 10.
= (L - 5)(L - 2).
Eigenvalues L = 5 and L = 2.

## In practice

The first step in finding eigenvalues by hand.

## Practice

### Flashcards

- **How do you find eigenvalues?** Solve det(A - lambda I) = 0.
- **What are the eigenvalues of [[4,1],[2,3]]?** 5 and 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 does det(A minus lambda I) equal zero?

We need A v equals lambda v for a non-zero v, which means A minus lambda I sends v to zero. A matrix that kills a non-zero vector has no inverse, so its determinant must be zero. Solving this finds each allowed lambda.

### How many eigenvalues does an n by n matrix have?

Up to n, counting repeats, and allowing complex numbers. The characteristic polynomial has degree n, so it has n roots in the complex numbers. A 2 by 2 matrix has two eigenvalues, though they may be equal or complex. For example, a rotation has two complex eigenvalues.

## Related concepts

- [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)
- [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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