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Eigenvector Calculator

Calculate eigenvectors and eigenvalues for 2×2 and 3×3 matrices with step-by-step solutions. Enter matrix elements, detect complex roots, and share results.

Calculate Eigenvectors & Eigenvalues

Matrix Size
Matrix Elements
Enter the elements of your 2×2 matrix:
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Your Results

Eigenvalue 1 (λ₁)
3.0000
Eigenvector 1:
[1.0000, 0.0000]
Eigenvalue 2 (λ₂)
2.0000
Eigenvector 2:
[-1.0000, 1.0000]

Calculation Steps

1Matrix A = [3 1; 0 2]
2Calculate trace: tr(A) = 3 + 2 = 5
3Calculate determinant: det(A) = 3 × 2 - 1 × 0 = 6
4Discriminant = tr(A)² - 4×det(A) = 1.0000 ≥ 0
5Eigenvalues: λ₁ = 3.0000, λ₂ = 2.0000
6Eigenvector 1: [1.0000, 0.0000]
7Eigenvector 2: [-1.0000, 1.0000]

Understanding Eigenvectors & Eigenvalues

Mathematical Definition

Av = λv
Where A is a matrix, v is an eigenvector, and λ is the eigenvalue

Key Properties

🔄 Direction Preserved

Eigenvectors only change in magnitude, not direction when multiplied by the matrix

📏 Scaling Factor

Eigenvalues tell us how much the eigenvector is scaled

🎯 Characteristic Equation

Found by solving det(A - λI) = 0

Real-World Examples

🎵 Vibration Analysis

Natural frequencies and mode shapes of structures

🎞️ Image Processing

Principal component analysis for data compression

🌐 Google PageRank

Web page ranking algorithm using dominant eigenvector

Calculation Methods

2×2 Matrix Method

Step 1: Characteristic polynomial det(A - λI) = 0
Step 2: λ² - trace(A)λ + det(A) = 0
Step 3: Solve quadratic equation for eigenvalues
Step 4: Find eigenvectors from (A - λI)v = 0

2×2 matrices always have analytical solutions

3×3 Matrix Complexity

• Requires solving cubic characteristic polynomial

• No simple closed-form solutions (unlike quadratics)

• Numerical methods needed: QR algorithm, power iteration

• Our calculator provides simplified educational results

Complex Eigenvalues

When discriminant < 0, eigenvalues are complex:

  • • Often represent rotational or oscillatory behavior
  • • Common in systems with periodic motion
  • • Complex conjugate pairs in real matrices

Applications in Different Fields

🔬 Engineering
  • • Structural vibration analysis
  • • Control systems stability
  • • Circuit analysis
  • • Modal analysis
📊 Data Science
  • • Principal Component Analysis (PCA)
  • • Dimensionality reduction
  • • Data compression
  • • Feature extraction
⚛️ Physics
  • • Quantum mechanics operators
  • • Crystal lattice vibrations
  • • Normal modes
  • • Energy states
💻 Computer Science
  • • PageRank algorithm
  • • Image recognition
  • • Machine learning
  • • Network analysis

Example Matrices

Identity Matrix (2×2)

[1 0]
[0 1]
Eigenvalues: λ₁ = 1, λ₂ = 1
Eigenvectors: Any vector
Property: Every vector is an eigenvector

Diagonal Matrix Example

[3 0]
[0 2]
Eigenvalues: λ₁ = 3, λ₂ = 2
Eigenvectors: [1,0] and [0,1]
Property: Eigenvalues are diagonal elements

Rotation Matrix (90°)

[0 -1]
[1 0]
Eigenvalues: Complex (±i)
Property: Pure rotation has complex eigenvalues
Application: 2D rotational transformations

Frequently Asked Questions

What are eigenvectors and eigenvalues?
Eigenvectors are special vectors that, when multiplied by a matrix, only change in magnitude (scale) but not in direction. The scaling factor is called the eigenvalue. Mathematically, for matrix A, vector v, and scalar λ: Av = λv, where v is the eigenvector and λ is the eigenvalue.
How does this eigenvector calculator work?
This calculator finds eigenvalues by solving the characteristic equation det(A - λI) = 0. For 2x2 matrices, this results in a quadratic equation. Once eigenvalues are found, corresponding eigenvectors are calculated by solving (A - λI)v = 0 for each eigenvalue.
What are the applications of eigenvectors?
Eigenvectors have many applications including: Principal Component Analysis (PCA) in data science, vibration analysis in engineering, quantum mechanics in physics, Google's PageRank algorithm, image compression, facial recognition, and stability analysis in control systems.
Can this calculator handle complex eigenvalues?
Yes, the calculator detects when eigenvalues are complex (when discriminant < 0 for 2x2 matrices). Complex eigenvalues often occur in systems with oscillatory behavior, such as rotational transformations or underdamped oscillations.
Why are 3x3 matrices more complex to solve?
3x3 matrices require solving cubic characteristic polynomials, which don't have simple closed-form solutions like quadratics. In practice, numerical methods like QR decomposition or power iteration are used. This calculator provides simplified results for educational purposes.

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Eigenvector Calculator - 2x2 & 3x3 Eigenvalues & Vectors