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Eigenvalues and Eigenvectors - Overview

Introduction​

Overview of eigenvalue theory, diagonalization, and applications to difference and differential equations.

Key Concepts​

Eigenvalues and Eigenvectors​

  • Definition: Av = λv
  • Characteristic polynomial
  • Computing eigenvalues and eigenvectors

Diagonalization​

  • When is a matrix diagonalizable?
  • A = PDP⁻¹
  • Applications to computing Aᵏ

Applications​

  • Powers of matrices and difference equations
  • Matrix exponentials eᴬᵗ
  • Solving differential equations
  • Complex eigenvalues and matrices

Similarity Transformations​

  • Similar matrices have same eigenvalues
  • Change of basis interpretation

References​