---
title: "Eigenvalue Calculator — Eigenvalues, Eigenvectors & Characteristic Polynomial"
description: "Calculate eigenvalues and eigenvectors of 2×2 and 3×3 matrices. Shows characteristic polynomial, discriminant, complex eigenvalues, and normalized eigenvectors. Essential for PCA, differential equations, and physics."
url: "https://worldcalculators.org/calculators/eigenvalues/"
canonical: "https://worldcalculators.org/calculators/eigenvalues/"
---

# Eigenvalue Calculator

Calculate eigenvalues and eigenvectors of 2×2 and 3×3 matrices. Shows characteristic polynomial, discriminant, complex eigenvalues, and normalized eigenvectors. Essential for PCA, differential equations, and physics.

## Quick answer

Every square matrix satisfies its own characteristic polynomial. If p(λ) = det(A−λI) = λ² − tr(A)λ + det(A), then p(A) = A² − tr(A)A + det(A)I = 0 (the zero matrix). This theorem is used to compute powers of matrices and matrix functions without explicitly computing each power.

## Frequently Asked Questions

Every square matrix satisfies its own characteristic polynomial. If p(λ) = det(A−λI) = λ² − tr(A)λ + det(A), then p(A) = A² − tr(A)A + det(A)I = 0 (the zero matrix). This theorem is used to compute powers of matrices and matrix functions without explicitly computing each power.
Yes — if λ has multiplicity > 1 (a repeated root of the characteristic polynomial), there may be multiple linearly independent eigenvectors for that λ. The set of all eigenvectors for λ (plus the zero vector) forms the eigenspace for λ. Its dimension is the geometric multiplicity of λ, which can be less than or equal to the algebraic multiplicity (how many times λ appears as a root).

Source: https://worldcalculators.org/calculators/eigenvalues/
Markdown mirror of the HTML page. Prefer this URL for RAG; the interactive calculator still lives on the HTML page.