# Eigenvalue and Eigenvector Calculator

URL: https://speedycalculator.com/calculators/math/eigenvalue-eigenvector/
Last updated: 2026-10-30

> The eigenvalue and eigenvector calculator finds the eigenvalues (real or complex) of a 2 × 2, 3 × 3 or 4 × 4 matrix, a basis of eigenvectors for each real eigenvalue, the characteristic polynomial, trace and determinant.

## Inputs

- Matrix size
- a₁
- b₁
- c₁
- d₁
- a₂
- b₂
- c₂
- d₂
- a₃
- b₃
- c₃
- d₃
- a₄
- b₄
- c₄
- d₄

## Results

- Eigenvalues
- Eigenvectors
- Characteristic polynomial
- Trace
- Determinant

## Formula

```
det(A − λI) = 0,  (A − λI)v = 0
2 × 2: λ = tr/2 ± √(tr²/4 − det)
```

## Explanation

Eigenvectors are directions a matrix only stretches, and eigenvalues are the stretch factors. They describe vibration modes, stability of systems, Google's PageRank, quantum states and the principal components of data.

### How to use the Eigenvalue and Eigenvector Calculator

- Choose the **matrix size** – the matching grid of fields appears.
- Enter the entries **row by row** (a₁ b₁ c₁ is the first row, a₂ b₂ c₂ the second, and so on).
- Read the result; integer matrices give exact fractions.

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

The characteristic polynomial is solved numerically for the eigenvalues; the null space of A − λI gives the eigenvectors. Multiplicities are reported. The sum of the eigenvalues equals the trace and their product the determinant.

## Worked example

Inputs — Matrix size: 2 × 2; a₁: 2; b₁: 1; a₂: 1; b₂: 2.

Results — **Eigenvalues = λ = 1
λ = 3**; Eigenvectors = λ = 1: (−1, 1)
λ = 3: (1, 1); Characteristic polynomial = det(A − λI) = λ² − 4λ + 3; Trace = 4; Determinant = 3.

## Key terms

- **Eigenvalue**: A scalar λ with Av = λv.
- **Multiplicity**: How many times an eigenvalue is repeated.

## FAQ

**What is an eigenvalue?**
A number λ such that Av = λv for some non-zero vector v.

**Can eigenvalues be complex?**
Yes – rotation matrices have complex eigenvalues.

**How do eigenvalues relate to the trace and determinant?**
Sum = trace, product = determinant.

**Are eigenvectors unique?**
Only up to scaling – any non-zero multiple is also an eigenvector.


## Sources

- [Wolfram MathWorld – Eigenvalue](https://mathworld.wolfram.com/Eigenvalue.html)
- [MIT OpenCourseWare – 18.06 Linear Algebra (Gilbert Strang)](https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/)
- [Jim Hefferon – Linear Algebra (free textbook)](https://hefferon.net/linearalgebra/)