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Multiplicity Comparison Chart

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About This MicroSim

This comparison chart displays three canonical cases that illustrate the relationship between algebraic and geometric multiplicity, and how this relationship determines whether a matrix is diagonalizable.

The Three Cases:

  1. Distinct Eigenvalues (Green): Each eigenvalue has multiplicity 1 → Always diagonalizable
  2. Repeated with Full Eigenspace (Blue): Repeated eigenvalue but m_g = m_a → Diagonalizable (scalar matrix example)
  3. Defective Matrix (Red): Repeated eigenvalue with m_g < m_a → NOT diagonalizable

Key Concepts

Term Definition
Algebraic Multiplicity (m_a) How many times λ appears as a root of the characteristic polynomial
Geometric Multiplicity (m_g) Dimension of the eigenspace, i.e., number of linearly independent eigenvectors
Defective Matrix A matrix where m_g < m_a for some eigenvalue

The Multiplicity Inequality

For any eigenvalue λ:

1 ≤ geometric multiplicity ≤ algebraic multiplicity

A matrix is diagonalizable if and only if geometric multiplicity equals algebraic multiplicity for ALL eigenvalues.

How to Use

  1. Compare the three cards side by side
  2. Examine the ratio bar showing m_g/m_a
  3. Hover over cards to highlight them
  4. Note the status indicator showing diagonalizability

Embedding

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

Lesson Plan

Learning Objectives

Students will be able to:

  1. Distinguish between algebraic and geometric multiplicity
  2. Identify when a matrix is diagonalizable based on multiplicities
  3. Recognize defective matrices and explain why they cannot be diagonalized

Suggested Activities

  1. Verification: Compute the eigenspaces for each example matrix and verify the geometric multiplicities
  2. Create examples: Find a 3×3 defective matrix with a different structure
  3. Borderline cases: Why is [[2, 0], [0, 2]] diagonalizable but [[2, 1], [0, 2]] is not?

Assessment Questions

  1. If a 4×4 matrix has characteristic polynomial (λ-3)⁴, what are the possible geometric multiplicities? Which would make it diagonalizable?
  2. Can a matrix with distinct eigenvalues ever be defective? Explain.
  3. What is special about the eigenvectors of a defective matrix?

References