2.1 Eigenvalues and eigenvectors

Eigenvalues and eigenvectors are key concepts in linear operator theory. They help us understand how operators transform vectors and provide insights into their behavior.

In finite dimensions, we can find eigenvalues using characteristic polynomials. Infinite-dimensional spaces require more advanced techniques. This topic lays the groundwork for understanding the spectrum of linear operators.

Eigenvalues and Eigenvectors

Fundamental Concepts

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• Eigenvalues represent scalar values λ satisfying the equation $Av = λv$, where A denotes a linear operator and v a non-zero vector
• Eigenvectors constitute non-zero vectors v fulfilling the equation $Av = λv$, with λ representing the corresponding eigenvalue
• Rewrite the eigenvalue equation as $(A - λI)v = 0$, where I signifies the identity operator
• Determine eigenvalues for finite-dimensional vector spaces by finding roots of the characteristic polynomial $det(A - λI) = 0$
• Spectrum of an operator in infinite-dimensional vector spaces encompasses eigenvalues and may include continuous spectrum and residual spectrum
• Eigenspace of a particular eigenvalue comprises all corresponding eigenvectors and the zero vector

Differences in Finite and Infinite Dimensions

• Finite-dimensional spaces allow for straightforward calculation of eigenvalues through characteristic polynomials
• Infinite-dimensional spaces require more advanced spectral theory techniques (functional analysis, spectral decomposition)
• Finite-dimensional operators always have at least one eigenvalue in the complex field
• Infinite-dimensional operators may have empty point spectrum (no eigenvalues)
• Compact operators on infinite-dimensional spaces exhibit properties similar to finite-dimensional operators (discrete spectrum)

Calculating Eigenvalues and Eigenvectors

Analytical Methods

• Compute the characteristic polynomial by evaluating $det(A - λI) = 0$
• Solve the resulting characteristic equation to identify eigenvalues
• For each eigenvalue λ, solve the homogeneous system $(A - λI)v = 0$ to determine corresponding eigenvectors
• Confirm obtained vectors satisfy the eigenvalue equation $Av = λv$
• Investigate existence of generalized eigenvectors for repeated eigenvalues
• Apply spectral theory techniques for infinite-dimensional operators to determine spectrum and eigenfunctions
  • Use resolvent operators
  • Analyze spectral measures
• Utilize perturbation theory for approximating eigenvalues and eigenvectors of slightly modified operators

Numerical Techniques

• Implement power iteration method for finding dominant eigenvalue and corresponding eigenvector
  • Repeatedly multiply a vector by the operator and normalize
  • Converges to the eigenvector with the largest absolute eigenvalue
• Apply inverse iteration to find smallest eigenvalue and its eigenvector
• Use QR algorithm for computing all eigenvalues and eigenvectors of a matrix
  • Perform QR decomposition repeatedly
  • Converges to an upper triangular matrix with eigenvalues on the diagonal
• Employ Arnoldi iteration for large sparse matrices
  • Builds an orthonormal basis for the Krylov subspace
  • Useful for finding a subset of eigenvalues
• Utilize Lanczos algorithm for symmetric matrices
  • Special case of Arnoldi iteration
  • More efficient for self-adjoint operators

Geometric Interpretation of Eigenvalues

Transformation Properties

• Eigenvectors represent directions where the linear operator acts as scalar multiplication
• Corresponding eigenvalue indicates the scaling factor applied by the operator to the eigenvector
• Eigenvectors in 2D and 3D transformations signify invariant lines or planes under the transformation
• Positive real eigenvalues correspond to stretching (λ > 1) or compression (0 < λ < 1) along eigenvector direction
• Negative real eigenvalues represent reflection followed by stretching (λ < -1) or compression (-1 < λ < 0)
• Complex eigenvalues indicate rotation combined with scaling in the plane spanned by real and imaginary parts of the corresponding eigenvector
• Determinant of the operator equals the product of its eigenvalues, representing volume scaling factor

Visualization and Applications

• Eigenvectors form a natural coordinate system for describing the action of the operator
• In image processing, eigenfaces (eigenvectors of covariance matrix) represent principal components of facial features
• Stress tensors in mechanics use eigenvectors to identify principal stress directions
• Quantum mechanics employs eigenvectors of Hamiltonian operators to represent stationary states
• Principal component analysis utilizes eigenvectors of covariance matrix to identify directions of maximum variance in data
• Markov chains use eigenvectors corresponding to eigenvalue 1 to find steady-state distributions

Properties of Eigenvalues for Special Operators

Self-Adjoint and Normal Operators

• Self-adjoint (Hermitian) operators possess real eigenvalues and orthogonal eigenvectors
• Normal operators ($AA* = A*A$) have orthogonal eigenvectors but may exhibit complex eigenvalues
• Spectral theorem guarantees diagonalizability of self-adjoint operators on finite-dimensional spaces
• Compact self-adjoint operators on infinite-dimensional spaces ensure a complete orthonormal set of eigenvectors (spectral theorem)
• Positive definite operators have strictly positive eigenvalues
• Unitary operators possess eigenvalues with absolute value 1, lying on the complex unit circle
• Trace of an operator equals the sum of its eigenvalues, counting multiplicities

Other Special Cases

• Nilpotent operators have only 0 as an eigenvalue
• Projection operators have eigenvalues 0 and 1
• Companion matrices have characteristic polynomial equal to the polynomial they represent
• Toeplitz operators exhibit symmetry in their eigenvalue distribution
• Fredholm operators have discrete spectrum with possible accumulation point at 0
• Compact operators on infinite-dimensional spaces have countable spectrum with 0 as the only possible accumulation point