Understanding Eigenvectors and Eigenvalues

Sep 18, 2024

Lecture Notes: Eigenvectors and Eigenvalues

Introduction

  • Unintuitive Topic: Many students find eigenvectors and eigenvalues confusing and hard to understand.
  • Visualization Importance: A solid visual understanding of foundational topics is crucial.
  • Foundational Topics: Important to understand matrices as linear transformations, determinants, linear systems, and change of basis.

Linear Transformations

  • Example: 2D linear transformation with basis vector movement.
    • Matrix representation: columns (3, 0) and (1, 2).
  • Span of a Vector: Most vectors are displaced from their span by a transformation.
  • Special Vectors: Vectors that remain on their span are eigenvectors.

Eigenvectors and Eigenvalues

  • Eigenvectors: Special vectors that are only stretched or squished by a transformation.
    • Example: i-hat on x-axis, stretched by factor of 3.
    • Negative 1, 1 stretched by a factor of 2.
  • Eigenvalues: The factor by which an eigenvector is stretched or squished.
  • Example in 3D Rotation: Eigenvectors remain on axis of rotation; eigenvalue would be 1.
  • Utility: Helps understand transformations irrespective of coordinate system.

Computation of Eigenvectors and Eigenvalues

  • Symbolic Representation: A * v = λ * v
  • Matrix Representation: A - λI * v = 0
  • Determinant: Eigenvalues are found by setting determinant to zero (det(A - λI) = 0).
    • Example: Matrix A with columns (2, 1) and (2, 3), λ = 1 is an eigenvalue.
  • Quadratic Polynomial in λ: Determines possible eigenvalues.

Examples and Special Cases

  • Matrix Example: (3, 0) and (1, 2).
    • Eigenvalues λ = 2 and λ = 3.
    • Eigenvectors on diagonal line for λ = 2.
  • No Eigenvectors: 90-degree rotation, eigenvalues are imaginary numbers i and -i.
  • Shear Example: i-hat fixed, j-hat moves. Eigenvalue is 1.
  • Scale Matrix: All vectors are eigenvectors with eigenvalue 2.

Eigenbasis

  • Definition: Basis of eigenvectors; results in a diagonal matrix.
  • Benefits of Diagonal Matrices: Easier for repeated transformations (e.g., computing matrix powers).
  • Change of Basis: Transform matrix with eigenvectors as basis vectors to a diagonal form.
  • Limitations: Not all transformations have enough eigenvectors for a full eigenbasis.

Conclusion

  • Final Point: Finding an eigenbasis simplifies matrix operations.
  • Next Topic: Abstract vector spaces.

Note: Engage with the provided problems for deeper understanding. Final video will cover abstract vector spaces.