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Eigenvector Transformation Visualization

Run the Eigenvector Transformation Fullscreen

Edit the MicroSim with the p5.js editor

About This MicroSim

This interactive visualization demonstrates the fundamental concept of eigenvectors: special vectors that maintain their direction under a linear transformation. When matrix A transforms a vector v, most vectors change both direction and magnitude. However, eigenvectors only scale by their corresponding eigenvalue λ.

Key Features:

  • Draggable vector: Move the blue vector to explore how different vectors transform
  • Eigenvector detection: When you align with an eigenvector direction, both vectors glow green
  • Editable matrix: Click matrix cells to enter custom values
  • Animation: Watch the transformation animate smoothly
  • Real-time computation: See Av computed for any vector position

How to Use

  1. Drag the blue vector around the coordinate plane
  2. Watch the red/green vector show the transformed result Av
  3. When the vectors align (same direction), you've found an eigenvector!
  4. Click matrix cells to edit values and explore different transformations
  5. Toggle "Show Eigenvectors" to see the eigenvector directions as dashed lines
  6. Click "Animate Transform" to see the transformation animated

Mathematical Background

The eigen equation is: Av = λv

Where: - A is a 2×2 matrix - v is the eigenvector (non-zero) - λ is the eigenvalue (scalar)

This equation states that when we apply transformation A to eigenvector v, the result is simply v scaled by λ. The default matrix [[2, 1], [1, 2]] has eigenvalues λ₁ = 3 and λ₂ = 1.

Embedding

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<iframe src="https://dmccreary.github.io/linear-algebra/sims/eigenvector-transformation/main.html" height="502px" scrolling="no"></iframe>

Lesson Plan

Learning Objectives

Students will be able to:

  1. Identify eigenvectors by observing which vectors maintain their direction under transformation
  2. Understand the relationship between eigenvalues and scaling factors
  3. Explain why eigenvectors are the "natural axes" of a linear transformation

Suggested Activities

  1. Eigenvector Hunt: Find both eigenvector directions for the default matrix by dragging the vector
  2. Eigenvalue Verification: When on an eigenvector, verify that the scale factor equals the eigenvalue
  3. Matrix Exploration: Try different matrices and predict how many real eigenvectors they have
  4. Rotation Matrix: Enter [[0, -1], [1, 0]] - why does it have no real eigenvectors?

Assessment Questions

  1. For matrix [[3, 0], [0, 2]], what are the eigenvector directions? Why?
  2. If a vector is scaled by factor 5 when transformed, what can you say about the eigenvalue?
  3. Can a non-zero vector ever transform to zero? What would this imply about the eigenvalue?

References