Mr. Integral

THE BIG IDEA

Eigenvalues and Dynamics

Which directions keep their identity as a system evolves?

An eigenvector keeps its direction under a transformation while its eigenvalue changes its scale. An eigenbasis separates coupled dynamics into modes; complex and symmetric cases add useful structure.

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4 lessons · No deadlines or locked lessons

SEE THE CONNECTIONS

How the ideas fit together.

A map of the main ideas. Follow a node to its lesson.

  1. Start with01Preserved directions
  2. an eigenbasis separates02Independent modes
  3. complex pairs encode03Oscillatory modes
  4. orthogonal modes clarify04Symmetric structure

YOUR LEARNING ROUTE

One idea at a time.

Read the explanation, use the visualization, then try the check before opening its reasoning.

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Lessons in this unit4 lessons
  1. 01Eigenvalues and EigenspacesFind directions preserved by a square transformation and separate eigenvalues from eigenvectors.
  2. 02Diagonalization and Discrete DynamicsUse an eigenvector basis to compute matrix powers and interpret long-term behavior.
  3. 03Complex Numbers and Oscillatory ModesUse complex arithmetic to interpret conjugate eigenvalues of a real matrix.
  4. 04Symmetric Matrices and Quadratic FormsUse an orthonormal eigenbasis to classify quadratic energy and optimization models.