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Characteristic Polynomial Explorer

Run the Characteristic Polynomial Explorer Fullscreen

Edit the MicroSim with the p5.js editor

About This MicroSim

This interactive tool helps you understand how eigenvalues are found by computing the characteristic polynomial det(A - λI) and finding its roots. The visualization shows both the algebraic computation and the graphical representation of the polynomial.

Key Features:

  • 2×2 and 3×3 matrices: Toggle between matrix sizes
  • Editable matrix entries: Click cells to enter custom values
  • Step-by-step calculation: See the polynomial derivation
  • Polynomial graph: Visualize p(λ) and see eigenvalues as x-intercepts
  • Trace slider: Explore points along the polynomial curve
  • Preset examples: Identity, random, and symmetric matrices

How to Use

  1. Select matrix size using the 2×2/3×3 toggle button
  2. Click matrix cells to edit values
  3. View the characteristic polynomial in the left panel
  4. See eigenvalues where the curve crosses the x-axis (red dots)
  5. Use the λ trace slider to explore values along the polynomial

Mathematical Background

The characteristic equation is: det(A - λI) = 0

For a 2×2 matrix [[a, b], [c, d]]:

  • p(λ) = λ² - (a+d)λ + (ad-bc)
  • = λ² - trace(A)λ + det(A)

The eigenvalues are the roots of this polynomial, found using the quadratic formula:

λ = (trace ± √(trace² - 4·det)) / 2

Embedding

1
<iframe src="https://dmccreary.github.io/linear-algebra/sims/characteristic-polynomial/main.html" height="532px" scrolling="no"></iframe>

Lesson Plan

Learning Objectives

Students will be able to:

  1. Compute the characteristic polynomial for 2×2 and 3×3 matrices
  2. Find eigenvalues as roots of the characteristic polynomial
  3. Connect the algebraic formula to the graphical representation

Suggested Activities

  1. Verify by hand: Compute the characteristic polynomial for [[4, 2], [1, 3]] and verify it matches the display
  2. Trace exploration: For eigenvalue λ = 5, verify that p(5) = 0 by using the slider
  3. Complex eigenvalues: Try [[0, -1], [1, 0]] - what happens to the graph?
  4. Triple root: Can you create a 3×3 matrix where all three eigenvalues are equal?

Assessment Questions

  1. What is the relationship between the trace of A and the coefficient of λ in the characteristic polynomial?
  2. If the characteristic polynomial never crosses the x-axis, what does this mean about the eigenvalues?
  3. For a triangular matrix, how do eigenvalues relate to diagonal entries?

References