2.1 Eigenvalues and eigenvectors
4 min read•august 16, 2024
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 of linear operators.
Eigenvalues and Eigenvectors
Fundamental Concepts
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- Eigenvalues represent scalar values satisfying the equation , where A denotes a linear operator and v a non-zero vector
- Eigenvectors constitute non-zero vectors v fulfilling the equation , with λ representing the corresponding
- Rewrite the eigenvalue equation as , where I signifies the identity operator
- Determine eigenvalues for finite-dimensional vector spaces by finding roots of the
- Spectrum of an operator in infinite-dimensional vector spaces encompasses eigenvalues and may include continuous spectrum and residual spectrum
- 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
- Solve the resulting characteristic equation to identify eigenvalues
- For each eigenvalue λ, solve the homogeneous system to determine corresponding eigenvectors
- Confirm obtained vectors satisfy the eigenvalue equation
- 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 for approximating eigenvalues and eigenvectors of slightly modified operators
Numerical Techniques
- Implement power iteration method for finding dominant eigenvalue and corresponding
- Repeatedly multiply a vector by the operator and normalize
- Converges to the eigenvector with the largest absolute eigenvalue
- Apply to find smallest eigenvalue and its eigenvector
- Use 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 for large sparse matrices
- Builds an orthonormal basis for the Krylov subspace
- Useful for finding a subset of eigenvalues
- Utilize 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
- 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
- Normal operators () have orthogonal eigenvectors but may exhibit complex eigenvalues
- 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
- 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
Key Terms to Review (26)
A v = λ v: The equation $$A v = \lambda v$$ represents an eigenvalue problem where $$A$$ is a linear operator or matrix, $$v$$ is a non-zero vector known as the eigenvector, and $$\lambda$$ is a scalar called the eigenvalue. This relationship indicates that when the linear operator is applied to the eigenvector, the output is simply a scaled version of the eigenvector, meaning that it does not change direction in the vector space. Understanding this equation is crucial because it lays the groundwork for exploring properties of linear transformations and their effects on vector spaces.
Algebraic Multiplicity: Algebraic multiplicity refers to the number of times an eigenvalue appears as a root of the characteristic polynomial of a matrix. It indicates the multiplicity with which an eigenvalue is counted when determining the eigenvalues of a linear transformation. This concept helps in understanding the structure of the corresponding eigenspaces and the behavior of the matrix under various transformations.
Arnoldi Iteration: Arnoldi iteration is an algorithm used to compute an orthonormal basis for the Krylov subspace generated by a matrix and a starting vector. This method is particularly useful for approximating eigenvalues and eigenvectors of large sparse matrices, as it transforms the problem into a smaller one that can be more easily handled. By building a sequence of vectors, Arnoldi iteration allows for efficient extraction of the dominant eigenvalues, which are crucial for various applications in numerical linear algebra.
Characteristic Polynomial: The characteristic polynomial is a polynomial that is derived from a square matrix and encapsulates important information about the eigenvalues of that matrix. Specifically, it is defined as the determinant of the matrix subtracted by a scalar multiple of the identity matrix, expressed as $$p(\lambda) = \text{det}(A - \lambda I)$$. The roots of this polynomial correspond to the eigenvalues, linking it directly to the concept of eigenvalues and eigenvectors, while also providing insight into the spectrum of an operator.
Compact Operator: A compact operator is a linear operator that maps bounded sets to relatively compact sets, meaning the closure of the image is compact. This property has profound implications in functional analysis, particularly concerning convergence, spectral theory, and various types of operators, including self-adjoint and Fredholm operators.
Determinant: The determinant is a scalar value that is calculated from a square matrix and provides important information about the matrix's properties. It can determine whether a matrix is invertible, as well as the volume scaling factor when the matrix is viewed as a linear transformation. In the context of eigenvalues and eigenvectors, the determinant plays a crucial role in finding these values by helping to form the characteristic polynomial.
Eigenspace: An eigenspace is a subspace associated with a specific eigenvalue of a linear transformation or matrix, consisting of all eigenvectors that correspond to that eigenvalue, along with the zero vector. This concept connects eigenvalues and eigenvectors, as the eigenspace provides a geometric interpretation of how the transformation acts on vectors, revealing crucial information about the behavior of the system under consideration.
Eigenvalue: An eigenvalue is a special scalar associated with a linear transformation or operator, representing the factor by which a corresponding eigenvector is stretched or compressed during that transformation. Eigenvalues play a crucial role in understanding the properties of operators and can be used to analyze stability, dynamics, and even solutions to differential equations.
Eigenvector: An eigenvector is a non-zero vector that changes only by a scalar factor when a linear transformation is applied to it. This property connects eigenvectors to eigenvalues, as the scalar factor represents the corresponding eigenvalue associated with that eigenvector. The significance of eigenvectors extends to understanding the spectrum of operators, particularly in the analysis of compact operators and self-adjoint operators, where they reveal important structural characteristics.
Generalized eigenvector: A generalized eigenvector is a vector that satisfies the equation $(A - \lambda I)^k v = 0$ for some positive integer $k$, where $A$ is a linear operator, $\lambda$ is an eigenvalue, and $I$ is the identity operator. This concept extends the idea of eigenvectors, allowing for vectors that correspond to an eigenvalue even when they are not solutions to the typical eigenvalue equation $(A - \lambda I)v = 0$. Generalized eigenvectors are particularly important in cases where an eigenvalue has algebraic multiplicity greater than its geometric multiplicity.
Geometric Multiplicity: Geometric multiplicity refers to the number of linearly independent eigenvectors associated with a given eigenvalue of a matrix. It provides insight into the eigenspace's dimension, which is essential for understanding the behavior of linear transformations. The geometric multiplicity can reveal whether a system of equations has enough solutions or if there is a possibility of unique solutions.
Inverse Iteration: Inverse iteration is a numerical method used to find eigenvalues and eigenvectors of a matrix, particularly effective for locating the eigenvalue that is closest to a given shift. This method refines an initial guess by repeatedly applying the inverse of the matrix adjusted by the shift, converging towards the desired eigenvector. It plays a critical role in exploring properties of matrices and helps to improve the accuracy of eigenvalue estimates.
Lanczos Algorithm: The Lanczos algorithm is an iterative method used to approximate the eigenvalues and eigenvectors of large symmetric matrices. It is particularly effective for finding a few of the smallest or largest eigenvalues, leveraging the fact that it transforms the original matrix into a much smaller tridiagonal matrix, which can be more easily analyzed. This algorithm plays a critical role in numerical linear algebra and has applications in various fields including quantum mechanics and structural engineering.
Linear Transformation: A linear transformation is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. This means that if you take two vectors and add them, applying the transformation gives you the same result as transforming each vector individually and then adding them. This concept is crucial in understanding various types of operators, including bounded operators, as well as key ideas such as eigenvalues and eigenvectors, and more specialized classes like Fredholm operators.
Normal Operator: A normal operator is a bounded linear operator on a Hilbert space that commutes with its adjoint, meaning that for an operator \(T\), it holds that \(T^*T = TT^*\). This property leads to several important characteristics, including the existence of an orthonormal basis of eigenvectors and the applicability of the spectral theorem. Normal operators encompass self-adjoint operators, unitary operators, and other types of operators that play a vital role in functional analysis.
Orthogonal Eigenvectors: Orthogonal eigenvectors are eigenvectors that correspond to distinct eigenvalues of a linear operator or matrix, and they are perpendicular to each other in the Euclidean space. This property allows for the simplification of various mathematical problems, especially in the context of diagonalization and spectral decomposition. When working with symmetric matrices, orthogonal eigenvectors can be particularly useful since they ensure that the resulting transformation maintains orthogonality.
Perturbation Theory: Perturbation theory is a mathematical approach used to analyze how a small change in a system's parameters affects its properties, particularly eigenvalues and eigenvectors. It plays a crucial role in understanding stability and the behavior of operators under slight modifications, making it essential for various applications in spectral theory and operator analysis.
Positive Definite Operator: A positive definite operator is a linear operator on a Hilbert space such that for any non-zero vector, the inner product with the operator applied to that vector is strictly greater than zero. This property indicates that the operator behaves nicely in terms of its eigenvalues, which are all positive, and allows for the definition of a unique positive square root of the operator, connecting it to various mathematical concepts.
Power Method: The power method is an iterative algorithm used to find the dominant eigenvalue and its corresponding eigenvector of a matrix. It works by repeatedly multiplying a vector by the matrix and normalizing it, which helps to amplify the influence of the largest eigenvalue while diminishing others. This method is particularly useful for large matrices where direct computation of eigenvalues is challenging.
QR Algorithm: The QR algorithm is a numerical method used for finding the eigenvalues and eigenvectors of a matrix. It works by decomposing a matrix into a product of an orthogonal matrix (Q) and an upper triangular matrix (R), which allows for iterative refinement of the eigenvalue estimates. This algorithm is crucial in computational linear algebra, especially when dealing with large matrices, and has applications in spectral theory for analyzing properties of operators.
Self-adjoint operator: A self-adjoint operator is a linear operator on a Hilbert space that is equal to its own adjoint. This property ensures that the operator has real eigenvalues and allows for various important results in functional analysis and quantum mechanics. Self-adjoint operators have deep connections with spectral theory, stability, and physical observables.
Spectral Theorem: The spectral theorem is a fundamental result in linear algebra and functional analysis that characterizes self-adjoint operators on Hilbert spaces, providing a way to diagonalize these operators in terms of their eigenvalues and eigenvectors. It connects various concepts such as eigenvalues, adjoint operators, and the spectral properties of bounded and unbounded operators, making it essential for understanding many areas in mathematics and physics.
Spectrum: In operator theory, the spectrum of an operator refers to the set of values (complex numbers) for which the operator does not have a bounded inverse. It provides important insights into the behavior of the operator, revealing characteristics such as eigenvalues, stability, and compactness. Understanding the spectrum helps connect various concepts in functional analysis, particularly in relation to eigenvalues and the behavior of compact and self-adjoint operators.
Trace of an operator: The trace of an operator is defined as the sum of its eigenvalues, each counted with their algebraic multiplicity. This concept connects deeply with the properties of linear transformations and matrices, providing insight into the behavior of operators in finite-dimensional vector spaces. Understanding the trace can also reveal important characteristics of an operator, such as its invariance under similarity transformations.
Unitary Operator: A unitary operator is a bounded linear operator on a Hilbert space that preserves inner products, meaning it keeps the length of vectors and the angles between them unchanged. This property makes unitary operators crucial in quantum mechanics and functional analysis, as they maintain the structure of the space and allow for transformations without loss of information. Additionally, they are linked to eigenvalues and eigenvectors, adjoint operators, polar decomposition, and various applications in spectral theory.
λ: In the context of linear algebra and operator theory, λ (lambda) typically represents an eigenvalue of a linear operator or matrix. An eigenvalue is a scalar that indicates how a linear transformation scales an eigenvector, which remains in the same direction after the transformation. The relationship between λ, eigenvectors, and matrices is fundamental in understanding the behavior of linear operators and their applications in various fields.