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Diagonalization Process Workflow

Run the Diagonalization Workflow Fullscreen

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About This MicroSim

This interactive flowchart guides you through the complete diagonalization process, highlighting the key decision points that determine whether a matrix can be diagonalized.

Workflow Steps:

  1. Start with matrix A
  2. Compute characteristic polynomial det(A - λI)
  3. Solve for eigenvalues
  4. Check if n eigenvalues exist
  5. Find eigenvectors for each eigenvalue
  6. Compute geometric multiplicities
  7. Check m_g = m_a condition
  8. Result: A = PDP⁻¹ or NOT diagonalizable

How to Use

  1. Click "Next Step" to advance through the workflow
  2. Click "Reset" to start over
  3. Toggle "Auto-advance" for automatic progression
  4. Read the description panel for details about each step

Node Types

Shape Meaning
Rounded rectangle (gray) Start node
Rectangle (blue) Process/computation step
Diamond (yellow) Decision point
Rounded rectangle (green) Success outcome
Rounded rectangle (red) Failure outcome

The Diagonalization Algorithm

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1. Given n×n matrix A
2. Form (A - λI) and compute det(A - λI)
3. Solve characteristic equation to get λ₁, ..., λₖ
4. For each λᵢ:
   a. Solve (A - λᵢI)v = 0
   b. Find basis for null space (eigenspace)
   c. Record geometric multiplicity
5. If Σ(geometric multiplicities) = n:
   - Form P = [v₁ | v₂ | ... | vₙ]
   - Form D = diag(λ₁, ..., λₙ)
   - A = PDP⁻¹ ✓
6. Otherwise: NOT diagonalizable ✗

Embedding

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<iframe src="https://dmccreary.github.io/linear-algebra/sims/diagonalization-workflow/main.html" height="572px" scrolling="no"></iframe>

Lesson Plan

Learning Objectives

Students will be able to:

  1. Follow the systematic process for diagonalizing a matrix
  2. Identify the two key decision points in the diagonalization process
  3. Determine when a matrix is not diagonalizable and why

Suggested Activities

  1. Apply the workflow: Use the flowchart to diagonalize [[4, 2], [1, 3]]
  2. Find the failure point: Which step fails for [[2, 1], [0, 2]]?
  3. Complex eigenvalues: What happens when eigenvalues are complex?

Assessment Questions

  1. At which step do you first know the algebraic multiplicities?
  2. What is the purpose of computing geometric multiplicity?
  3. If a 3×3 matrix has only 2 linearly independent eigenvectors, what path does it take in the flowchart?

References