# Eigenvalue Calculator - Eigenvalues and Eigenvectors of 2×2 and 3×3 Matrices

> Free online eigenvalue calculator: 2×2 and 3×3 matrices — characteristic polynomial, eigenvalues (complex too), eigenvectors, trace and diagonalizability.

- URL: https://qilingtools.com/en/math/eigenvalue
- Category: Math Tools (https://qilingtools.com/en/math)
- Tags: [Math](https://qilingtools.com/en/tags/math), [Education](https://qilingtools.com/en/tags/education), [Algorithms](https://qilingtools.com/en/tags/algorithm)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Type in a matrix and get the characteristic polynomial and all of its eigenvalues. Every real eigenvalue comes with its eigenvector and an Av = λv check, alongside the trace, the determinant and a verdict on diagonalizability. 2×2 uses the quadratic formula, 3×3 solves the cubic, and complex eigenvalues are reported as such.

## How to use

1. Choose the size (2×2 or 3×3)
2. Fill in the entries of the matrix
3. Read the characteristic polynomial and the eigenvalues
4. Check each eigenvector and its Av = λv verification

## Features

- Characteristic polynomial written as λ² − tr(A)λ + |A| or λ³ − tr(A)λ² + Mλ − |A|
- Quadratic formula for 2×2 and a numerically stable trigonometric cubic solver for 3×3
- Complex eigenvalues are reported separately in the form a ± bi
- Eigenvectors come from the null space of (A − λI), normalised so the largest component is 1
- Every eigenvector is shown next to Av and λv so you can see they agree
- Trace, determinant, symmetry and diagonalizability are all reported

## FAQ

### What is an eigenvalue really?

The λ satisfying Av = λv. The matrix acting on that eigenvector only stretches it by λ without turning it; a negative λ flips the direction and |λ| is the stretch factor.

### Why is an eigenvector not unique?

If v is an eigenvector then so is any non-zero multiple kv, because A(kv) = λ(kv). An eigenvector really denotes a direction, so the tool normalises it to have largest component 1 for easy comparison.

### Can eigenvalues be complex?

Yes. The rotation matrix [[0,−1],[1,0]] has eigenvalues ±i because no direction in the plane is merely stretched by a rotation. Complex eigenvalues are listed as a ± bi.

### Which matrices are diagonalizable?

Those with n linearly independent eigenvectors. Distinct eigenvalues always suffice; with a repeated eigenvalue it depends on the dimension of its eigenspace. A symmetric matrix is always diagonalizable and its eigenvectors are orthogonal.

### How do the trace and determinant relate to the eigenvalues?

The eigenvalues sum to the trace and multiply to the determinant. Both make quick sanity checks on a computed set of eigenvalues.

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