Lecture 1: Eigenvalues and Eigenvectors

Overview

• Focus on eigenvalues and eigenvectors, crucial topics for the course.
• Distinction between eigenvectors and eigenvalues:
  • Eigenvector (x): A vector that stays in the same direction after transformation by a matrix (A).
  • Eigenvalue (λ): A scalar that describes the factor by which the eigenvector is stretched or compressed during this transformation.

Definitions

• Matrix A: Acts on a vector x, producing a new vector Ax.
• Eigenvectors are characterized by the property:
  Ax = λx (where λ is a scalar).
• Eigenvalue Zero:
  • If λ = 0, then Ax = 0, indicating that x is in the null space of A.
  • If A is singular (non-invertible), there exist eigenvalues of zero.

Finding Eigenvalues and Eigenvectors

• To find eigenvalues and eigenvectors, solve the equation: Ax = λx.
• This can be rewritten as:
  (A - λI)x = 0, where I is the identity matrix.
  • The matrix (A - λI) must be singular for non-zero solutions (x).
  • This leads to the characteristic equation:
    det(A - λI) = 0.

Example: Projection Matrix (P)

• Projection Matrix P: Projects vectors onto a specific subspace (e.g., a plane).
  • Eigenvectors are those in the plane, with λ = 1 (unchanged).
  • Eigenvectors perpendicular to the plane will result in λ = 0 (projected to zero).

Example: Permutation Matrix

• Matrix A: [ \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} ]
  • Eigenvalue λ = 1: Eigenvector (1, 1) stays unchanged.
  • Eigenvalue λ = -1: Eigenvector (-1, 1) flips sign.

Important Theorems

• Trace: Sum of eigenvalues equals the trace of the matrix (sum of diagonal elements).
• Determinant: Product of eigenvalues equals the determinant of the matrix.
• If you add a scalar multiple of the identity matrix (3I) to a matrix A, the eigenvectors remain unchanged, and the eigenvalues increase by that scalar.

Complex Eigenvalues

• Non-symmetric matrices can have complex eigenvalues. For example, a matrix representing a 90-degree rotation results in:
  • Eigenvalues: λ = i and λ = -i (complex numbers).
• Complex conjugate pairs are observed for such cases.

Example: Triangular Matrix

• For a triangular matrix, eigenvalues can be easily read off the diagonal.
• Repeated eigenvalues (e.g., λ = 3) can result in fewer independent eigenvectors than dimensions (dependency issue).

Conclusion

• Eigenvalues and eigenvectors provide insights into matrix behavior and transformations.
• The next lectures will explore further applications and implications of these concepts in linear algebra.