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Positive Definiteness Visualizer

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About This MicroSim

This visualization demonstrates positive definiteness for 2×2 symmetric matrices by plotting the quadratic form:

\[f(x, y) = \mathbf{x}^T A \mathbf{x} = a_{11}x^2 + 2a_{12}xy + a_{22}y^2\]

The 3D surface shows how the function value changes with input, revealing the geometric meaning of:

  • Positive Definite: Bowl shape (minimum at origin), all eigenvalues positive
  • Negative Definite: Inverted bowl (maximum at origin), all eigenvalues negative
  • Indefinite: Saddle shape, eigenvalues have opposite signs
  • Semi-Definite: Trough shape, one eigenvalue is zero

Key Features

  • 3D Surface Plot: Visualize the quadratic form as a surface
  • Eigenvalue Display: See eigenvalues with color coding (green=positive, red=negative)
  • Contour Lines: Level curves showing where f(x,y) = constant
  • Eigenvector Directions: Principal axes shown on the base plane
  • Interactive Matrix: Adjust matrix entries with sliders
  • Presets: Common examples of each classification

How to Use

  1. Select a preset to see classic examples of each type
  2. Adjust sliders to modify individual matrix entries
  3. Drag to rotate the 3D view
  4. Toggle contours to see level curves
  5. Observe how eigenvalues change with matrix entries

Learning Objectives

After using this MicroSim, students will be able to:

  • Connect eigenvalue signs to the shape of quadratic forms
  • Identify positive definite matrices visually
  • Understand why positive definite matrices arise in optimization
  • Recognize the relationship between contour shapes and eigenvalues

Mathematical Background

A symmetric matrix A is:

Classification Condition Surface Shape
Positive Definite All λ > 0 Bowl (upward)
Negative Definite All λ < 0 Bowl (downward)
Indefinite Mixed signs Saddle
Positive Semi-Def All λ ≥ 0, some = 0 Trough

Lesson Plan

Introduction (5 minutes)

Ask students: "What does it mean for a function to have a minimum at the origin?"

Connect this to the quadratic form: f(x,y) > 0 for all (x,y) ≠ (0,0) means the surface is a bowl opening upward.

Exploration (10 minutes)

  1. Positive Definite: Start with preset, note bowl shape and green eigenvalues
  2. Negative Definite: Switch preset, observe inverted bowl and red eigenvalues
  3. Indefinite: See the saddle point and mixed-color eigenvalues
  4. Semi-Definite: Watch how the surface flattens when one eigenvalue is zero

Key Insight

The eigenvectors are the principal axes of the contour ellipses. The eigenvalues determine how "stretched" the ellipses are along each axis.

Hands-on Activity

Have students predict what happens when:

  1. a₁₁ and a₂₂ are both positive but a₁₂ is large
  2. a₁₂ = 0 (diagonal matrix)
  3. a₁₁ = a₂₂ and a₁₂ = 0 (scalar matrix)

Assessment Questions

  1. What is the geometric meaning of xᵀAx > 0?
  2. Why must all eigenvalues be positive for positive definiteness?
  3. What shape do the contour lines form?

References