Eigenvalue and Eigenvector Calculator

Category: Linear Algebra
- April 16, 2025
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Calculate the eigenvalues and eigenvectors of a square matrix. Eigenvalues and eigenvectors have important applications in linear transformations, differential equations, quantum mechanics, and many other fields of mathematics and physics.

Matrix Input

Display Options

About Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental concepts in linear algebra. An eigenvector of a square matrix A is a non-zero vector v that, when multiplied by A, yields a scalar multiple of itself. This scalar is called the eigenvalue.

Mathematical Definition

For a square matrix A, a non-zero vector v is an eigenvector if:

Av = λv

where λ is the eigenvalue associated with the eigenvector v.

Finding Eigenvalues and Eigenvectors
  1. Eigenvalues: Solve the characteristic equation det(A - λI) = 0
  2. Eigenvectors: For each eigenvalue λ, solve the homogeneous system (A - λI)v = 0
Applications of Eigenvalues and Eigenvectors
  • Differential Equations: Solving systems of linear differential equations
  • Principal Component Analysis: Dimensionality reduction in statistics and data science
  • Quantum Mechanics: Describing physical observables and quantum states
  • Engineering: Vibration analysis, structural stability, and control systems
  • Matrix Diagonalization: Simplifying matrix operations through transformation
  • Graph Theory: Spectral analysis of networks and connectivity
Properties
  • The determinant of a matrix equals the product of its eigenvalues
  • The trace of a matrix equals the sum of its eigenvalues
  • A matrix is diagonalizable if and only if there are n linearly independent eigenvectors for an n×n matrix
  • Symmetric matrices always have real eigenvalues and orthogonal eigenvectors
  • Similar matrices share the same eigenvalues

What Is the Eigenvalue and Eigenvector Calculator?

The Eigenvalue and Eigenvector Calculator is a powerful tool designed to compute the eigenvalues and eigenvectors of any square matrix. This type of calculation is commonly used in fields such as engineering, physics, data science, and linear algebra for understanding transformations, solving systems of equations, and performing matrix analysis.

Eigenvalue Equation:

Av = λv

Where:

  • A is a square matrix
  • v is the eigenvector
  • λ (lambda) is the eigenvalue

How to Use the Calculator

Follow these steps to calculate the eigenvalues and eigenvectors of a matrix:

  • Select the matrix size (from 2×2 up to 6×6).
  • Click "Create Matrix" to generate input fields.
  • Enter the values for your matrix.
  • Optionally adjust display settings such as decimal precision or showing steps.
  • Click "Calculate Eigenvalues & Eigenvectors."

After calculation, the tool displays:

  • The original matrix
  • All eigenvalues and their corresponding eigenvectors
  • The characteristic polynomial
  • Matrix properties like determinant and trace
  • Verification of results (Av = λv)
  • Diagonalization steps if applicable

Why This Calculator Is Useful

Eigenvalues and eigenvectors help simplify complex linear systems and reveal important properties about matrices. This calculator is especially useful for:

  • Students: Learn and verify matrix diagonalization, normalization, and eigen-analysis
  • Researchers: Quickly compute spectral data without manual calculation
  • Engineers & Data Scientists: Use in vibration analysis, PCA, stability studies, and more

This tool also complements other matrix calculators, including:

Key Features

  • Supports matrices from 2×2 to 6×6
  • Handles real and complex eigenvalues
  • Normalization of eigenvectors
  • Step-by-step display of calculations
  • Diagonalization verification with P, D, and P⁻¹ matrices

Frequently Asked Questions (FAQ)

What are eigenvalues and eigenvectors used for?

They are used in many areas such as differential equations, quantum mechanics, machine learning (PCA), and structural analysis.

What is a characteristic polynomial?

The characteristic polynomial is derived from a matrix and is used to find eigenvalues by solving the equation det(A - λI) = 0.

Can this calculator handle complex numbers?

Yes. It can display and calculate with complex eigenvalues if enabled in the options.

What does diagonalization mean?

Diagonalization rewrites a matrix in the form A = PDP⁻¹, which simplifies matrix operations. The calculator checks if the matrix is diagonalizable.

Does this help with other matrix operations?

Yes, this complements tools such as the matrix multiplication tool, matrix division calculator, matrix transpose tool, and the matrix trace calculator for a broader linear algebra workflow.

Summary

The Eigenvalue and Eigenvector Calculator simplifies matrix analysis and supports learning and problem-solving in linear algebra. Whether you're exploring eigenvalues and diagonalization, using matrix LU breakdown techniques, or comparing outputs with a matrix inverse tool, this calculator offers a clear, efficient, and educational way to work with matrices.