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Complex Eigenvalue Visualizer

Run the Complex Eigenvalue Visualizer Fullscreen

Edit the MicroSim with the p5.js editor

About This MicroSim

Complex eigenvalues reveal the rotational nature of certain linear transformations. When a real matrix has complex eigenvalues, they always appear in conjugate pairs λ = a ± bi, and the transformation involves both rotation and scaling.

Key Features:

  • Dual view: Transformation plane (left) and complex plane (right)
  • Adjustable eigenvalue: Sliders for real and imaginary parts
  • Spiral animation: Watch repeated transformation create spirals
  • Conjugate pair: Toggle to show both λ and λ̄
  • Behavior indicators: Shows whether spiral expands, contracts, or circles

How to Use

  1. Adjust "Re" slider to change the real part of λ
  2. Adjust "Im" slider to change the imaginary part
  3. Click "Animate" to see repeated transformation
  4. Click "Reset" to start over from (1, 0)
  5. Toggle "Show Conjugate" to display both eigenvalues

Geometric Interpretation

For complex eigenvalue λ = a + bi:

Property Formula Meaning
Magnitude |λ| = √(a² + b²) Scaling factor per step
Angle θ = arctan(b/a) Rotation angle per step

Behavior Patterns:

  • |λ| > 1: Spiral outward (expanding)
  • |λ| < 1: Spiral inward (contracting)
  • |λ| = 1: Pure rotation (circle)
  • Im(λ) = 0: No rotation (pure scaling)

Example: Rotation Matrix

The 90° rotation matrix has eigenvalues λ = ±i: - |λ| = 1 → no scaling - θ = 90° → quarter turn each step

Try setting Re = 0, Im = 1 to see pure rotation!

Embedding

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

Lesson Plan

Learning Objectives

Students will be able to:

  1. Connect complex eigenvalues to rotation-scaling transformations
  2. Interpret eigenvalue magnitude as scaling factor
  3. Interpret eigenvalue argument (angle) as rotation amount
  4. Predict trajectory behavior from eigenvalue position in complex plane

Suggested Activities

  1. Find the circle: Adjust sliders until |λ| = 1 exactly and observe pure rotation
  2. Predict behavior: Before animating, predict if spiral will expand or contract
  3. Conjugate pairs: Why do complex eigenvalues of real matrices come in conjugate pairs?
  4. Matrix connection: For λ = 0.9 + 0.4i, what 2×2 matrix produces this behavior?

Assessment Questions

  1. If λ = 2i, what happens to points under repeated transformation?
  2. What eigenvalue produces a 60° rotation with 10% shrinkage per step?
  3. Where on the complex plane are eigenvalues of stable systems located?

References