Week 12: Eigenvalues and Eigenvectors

Week 12 Overview

This week covered eigenvalues and eigenvectors and their properties.

Key Concepts

• Eigenvalues and Eigenvectors

  • Definition: $T(v) = \lambda v$ for non-zero $v$
  • $\lambda$ is the eigenvalue (scalar)
  • $v$ is the eigenvector (direction preserved under transformation)

• Characteristic Polynomial

  • $\det(A - \lambda I) = 0$ gives eigenvalues
  • Connects to determinants from last week

• Eigenspaces

  • Set of all eigenvectors for a given eigenvalue
  • Geometric multiplicity = dimension of eigenspace

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