# Symmetric matrix

A matrix equal to its own transpose. Its eigenvalues are always real, and eigenvectors for different eigenvalues are perpendicular.

Maths (university) · Part 10: Eigenvalues and eigenvectors · Card 8 of 9 · Last updated 2026-10-19

## Analogy

A perfectly balanced seesaw: its natural axes sit at right angles.

## Example

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

Trace 4, det 3, so L = 3 and 1.
For L = 3: v = (1,1).
For L = 1: v = (1,-1).
Dot product: 1 - 1 = 0.

## In practice

Symmetric matrices appear in statistics, physics and optimisation, and are the best behaved of all.

## Practice

### Flashcards

- **What is special about eigenvectors of a symmetric matrix?** Those with different eigenvalues are perpendicular.

## Questions people ask

### Why do symmetric matrices have real eigenvalues?

This is a proven theorem, the spectral theorem. Symmetry forces every eigenvalue to be a real number, so no complex surprises appear. It is also why symmetric matrices model quantities such as energy and variance, which must be real. Engineers rely on this guarantee when analysing vibrations.

### What does orthogonal eigenvectors mean in practice?

The eigenvector directions are at right angles, like the x and y axes. You can then use them as perpendicular coordinate axes, which makes calculations clean. In the example, (1,1) and (1,-1) have dot product zero. Using such perpendicular axes keeps the algebra clean and the numbers stable.

## Related concepts

- [Diagonalisation](https://studyvisually.com/learn/science/maths-university/maths-university-part-10/maths-university-part-10-concept-5)
- [Matrix powers](https://studyvisually.com/learn/science/maths-university/maths-university-part-10/maths-university-part-10-concept-6)
- [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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