Eigenvalues and Eigenvectors Lecture

Jun 17, 2024

Eigenvalues and Eigenvectors Lecture

Introduction

Presenter: Dr. Gajendra Purohit

  • Focus: Engineering Mathematics & BSc
  • Useful for competitive exams involving higher mathematics
  • Previous uploads: Eigenvalues and Eigenvectors
  • Today's Topic: Further concepts on Eigenvalues & Eigenvectors

Key Concepts

Characteristic Equation

  • Formed from the determinant of the matrix
  • Solving gives eigenvalue (λ)
  • Example: When λ is multiplied inside the matrix, the determinant becomes the characteristic polynomial
  • Set the polynomial to 0 to solve for λ
  • Watch videos on homogeneous equations for more context (i tab)

Homogeneous Equations

  • Always consistent: solution is either trivial (zero) or non-trivial (non-zero)
  • Form matrix, subtract λ on diagonal, find determinant to get characteristic polynomial
  • Equate polynomial to 0, solve for λ (eigenvalues)
  • Short tricks available for quick calculations (i tab videos)

Eigenvectors Calculation

  • Given eigenvalue (λ=1), substitute into the matrix
  • Non-zero rank, rank reduction by 1 when λ is inserted
  • Infinite solutions represent eigenvectors; e.g., x1=k, x2=-3k corresponds to eigenvalues

Consistency Check

  • Sum of eigenvalues should equal the sum of the diagonal elements
  • Example given for a 2x2 and 3x3 matrix

Special Cases and Definitions

Linearly Independent and Dependent Eigenvectors

  • Different eigenvalues yield linearly independent eigenvectors
  • Same eigenvalues case is linked to diagonalization (discussed in next class)

Terminology

  • Eigenvalues: characteristic values, roots, latent roots
  • Spectrum: set of eigenvalues of matrix A
  • Spectral radius: largest eigenvalue

Algebraic and Geometric Multiplicity

  • Example: Eigenvalues 1, 2, 3 vs. repeating eigenvalues and their multiplicities
  • Algebraic multiplicity: power of eigenvalue in equation
  • Geometric multiplicity: number of linearly independent eigenvectors
  • Algebraic multiplicity >= Geometric multiplicity

Questions and Examples

  • Example given with determinants, sums, and product of eigenvalues

Important Metrics

  • Trace of the matrix: sum of diagonal elements, equal to sum of eigenvalues
  • Eigenvalues product equals the determinant

Next Classes

  • More examples and questions on eigenvalues and eigenvectors
  • Topic: Diagonalization

Additional Resources

  • Playlists on CSIR NET General Aptitude, Real Algebra
  • Information on the new and old YouTube channels