Eigenvalue & Eigenvector Calculator

Linear Algebra Calculator

Calculate eigenvalues and eigenvectors for 2x2 and 3x3 matrices with detailed solutions.

Matrix Input

Eigenvalue Equation:
det(A - λI) = 0
where λ is eigenvalue, I is identity matrix

What are Eigenvalues?

Eigenvalues (λ) and eigenvectors (v) satisfy the equation: Av = λv

Eigenvalue:

A scalar that scales the eigenvector

Eigenvector:

A non-zero vector that only changes by a scalar factor when the matrix is applied

Features

  • 2×2 and 3×3 matrix support
  • Eigenvalue calculation
  • Eigenvector calculation
  • Trace calculation
  • Determinant calculation
  • Characteristic polynomial
  • Normalized eigenvectors
  • Quick example matrices
  • Copy results

Properties

Trace:

Sum of diagonal elements = sum of eigenvalues

Determinant:

Product of all eigenvalues

Identity Matrix:

All eigenvalues are 1

Diagonal Matrix:

Eigenvalues are diagonal entries

Note

2×2 Matrices: Exact calculation using quadratic formula

3×3 Matrices: Approximation using power iteration method (dominant eigenvalue)

Complex eigenvalues are not supported in this calculator

Use Cases

  • Linear algebra homework
  • Principal component analysis (PCA)
  • Stability analysis
  • Quantum mechanics
  • Vibration analysis
  • Graph theory
  • Machine learning
  • Differential equations

Applications

Real-World Applications:

Data Science:

Dimensionality reduction and feature extraction

Engineering:

Structural analysis and vibration modes

Physics:

Quantum states and energy levels

Computer Graphics:

3D transformations and rotations

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