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Abstract Linear Algebra II

Eigenvalues and eigenvectors are key concepts in linear algebra. They help us understand how linear transformations affect vector spaces, revealing the fundamental properties of these operations.

These tools are crucial for solving complex problems in various fields. From physics to computer science, eigenvalues and eigenvectors provide insights into system behavior, stability, and optimization techniques.

Eigenvalues and Eigenvectors

Definitions and Properties

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  • Eigenvalue represents a scalar λ where T(v) = λv for a non-zero vector v in the vector space
  • Eigenvector constitutes a non-zero vector v satisfying T(v) = λv for a given eigenvalue λ
  • Eigenspace encompasses all eigenvectors corresponding to a particular eigenvalue λ along with the zero vector
  • Linear operators possess intrinsic eigenvalues and eigenvectors regardless of chosen basis
  • Algebraic multiplicity denotes the multiplicity of an eigenvalue as a root of the characteristic polynomial
  • Geometric multiplicity signifies the dimension of the corresponding eigenspace for an eigenvalue

Multiplicities and Complex Eigenvalues

  • Algebraic and geometric multiplicities provide comprehensive characterization of the eigenstructure
  • Complex eigenvalues and eigenvectors may arise in certain cases
    • Require computations in the complex field
    • Represent rotations combined with scalings in the plane spanned by real and imaginary parts of corresponding eigenvectors
  • Eigenvalues with absolute value greater than 1 indicate expansion along eigenvector direction
  • Eigenvalues with absolute value less than 1 signify contraction along eigenvector direction

Computing Eigenvalues and Eigenvectors

Characteristic Equation and Eigenvalues

  • Construct characteristic equation det(TλI)=0det(T - λI) = 0 to find eigenvalues
    • T represents matrix of linear operator
    • I denotes identity matrix
  • Solve characteristic equation to obtain eigenvalues
  • Characteristic equation forms a polynomial equation in λ
    • Roots of equation correspond to eigenvalues of linear operator T
    • Degree of characteristic polynomial equals dimension of vector space on which T acts
  • Coefficient of highest degree term in characteristic polynomial always equals (1)n(-1)^n
    • n represents dimension of space
  • Constant term of characteristic polynomial equals (1)n(-1)^n times determinant of T

Eigenvector Computation

  • Find non-trivial solutions to equation (TλI)v=0(T - λI)v = 0 for each eigenvalue λ to obtain corresponding eigenvectors
  • Employ Gaussian elimination or other matrix methods to solve homogeneous system (TλI)v=0(T - λI)v = 0 for each eigenvalue
  • Verify computed eigenvectors satisfy equation T(v)=λvT(v) = λv
  • Sum of eigenvalues (counting multiplicity) equals trace of T
    • Trace represents sum of diagonal elements of matrix representation

Geometric Interpretation of Eigenvalues

Transformation Properties

  • Eigenvector indicates direction in which linear operator acts as scalar multiplication
  • Eigenvalue represents scaling factor applied by linear operator to eigenvector
  • For 2D or 3D linear operators, eigenvectors denote principal axes of transformation
  • Positive real eigenvalues signify stretching along corresponding eigenvector direction
  • Negative real eigenvalues indicate reflection and stretching along eigenvector direction
  • Determinant of linear operator equals product of its eigenvalues
    • Represents overall scaling factor of transformation

Visualization Examples

  • Rotation matrix with eigenvalues eiθe^{iθ} and eiθe^{-iθ} rotates vectors by angle θ
  • Shear matrix with eigenvalues 1 and k stretches space along one direction while preserving area
  • Projection matrix onto a line has eigenvalues 0 and 1
    • Eigenvector with eigenvalue 1 represents direction of projection line
    • Eigenvectors with eigenvalue 0 represent directions perpendicular to projection line

Eigenvalues, Eigenvectors, and the Characteristic Equation

Relationships and Theorems

  • Characteristic equation det(TλI)=0det(T - λI) = 0 links eigenvalues, eigenvectors, and linear operator
  • Cayley-Hamilton theorem states every linear operator satisfies its own characteristic equation
    • Replace scalar λ with operator T in characteristic equation
  • Eigenvalues relate to matrix properties
    • Product of eigenvalues equals determinant of matrix
    • Sum of eigenvalues equals trace of matrix

Applications and Examples

  • Diagonalization of matrices relies on eigenvalues and eigenvectors
    • Matrix A diagonalizable if it has n linearly independent eigenvectors (n = dimension of matrix)
  • Markov chains use eigenvalues to determine long-term behavior of system
    • Largest eigenvalue (in absolute value) determines convergence properties
  • Vibration analysis in engineering employs eigenvalues to find natural frequencies of systems
    • Eigenvectors represent mode shapes of vibration
  • Google's PageRank algorithm utilizes eigenvalue problem to rank web pages
    • Principal eigenvector of web graph provides ranking of pages
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© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.