# Diagonalisation

If A has enough independent eigenvectors, then A = P D P^(-1). The columns of P are the eigenvectors and D has the eigenvalues on its diagonal.

Maths (university) · Part 10: Eigenvalues and eigenvectors · Card 5 of 9 · Last updated 2026-10-19

## Analogy

Viewing a problem from the right angle, so the matrix just stretches each axis separately.

## Example

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

P = [[1,1],[1,-2]], D = [[5,0],[0,2]].
P inverse = [[2/3,1/3],[1/3,-1/3]].
P D P inverse = [[4,1],[2,3]].
Check first row: 4 and 1.

## In practice

Makes matrix powers and differential equations easy to solve.

## Practice

### Flashcards

- **Write the diagonalisation formula.** A = P D P^(-1).
- **How do you compute A^n using diagonalisation?** A^n = P D^n P^(-1).

### Interview questions

**Explain how diagonalisation helps compute high powers of a matrix.**

- A = P D P^(-1)
- A^n = P D^n P^(-1)
- D^n only needs each diagonal entry raised to n
- Largest eigenvalue dominates the long run

## Questions people ask

### Which matrices can be diagonalised?

Those with a full set of independent eigenvectors, meaning n of them for an n by n matrix. A matrix with n different eigenvalues always works. A matrix like [[1,1],[0,1]] has only one eigenvector direction, so it cannot be diagonalised.

### Why is the diagonal form useful?

A diagonal matrix is easy to work with: to square it, just square each diagonal entry. After changing basis to the eigenvectors, a complicated matrix acts like a simple diagonal one. You then switch back to the original basis at the end.

## Related concepts

- [Matrix powers](https://studyvisually.com/learn/science/maths-university/maths-university-part-10/maths-university-part-10-concept-6)
- [Symmetric matrix](https://studyvisually.com/learn/science/maths-university/maths-university-part-10/maths-university-part-10-concept-8)
- [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)

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