➗Abstract Linear Algebra II Unit 3 – Eigenvalues and Eigenvectors

Key Concepts and Definitions

• Eigenvalues are scalar values $\lambda$ that satisfy the equation $Av = \lambda v$ for a square matrix $A$ and a non-zero vector $v$
• Eigenvectors are non-zero vectors $v$ that, when multiplied by a square matrix $A$, result in a scalar multiple of themselves $Av = \lambda v$
  • The scalar multiple $\lambda$ is the corresponding eigenvalue
• The set of all eigenvalues of a matrix is called its spectrum
• Algebraic multiplicity of an eigenvalue is the number of times it appears as a root of the characteristic polynomial
• Geometric multiplicity of an eigenvalue is the dimension of its corresponding eigenspace (the space spanned by its eigenvectors)
• A matrix is diagonalizable if it is similar to a diagonal matrix, meaning it can be written as $A = PDP^{-1}$, where $D$ is a diagonal matrix and $P$ is an invertible matrix
• The eigenspace of an eigenvalue $\lambda$ is the nullspace of the matrix $A - \lambda I$, where $I$ is the identity matrix

Eigenvalue Equation and Characteristic Polynomial

• The eigenvalue equation is $Av = \lambda v$, where $A$ is a square matrix, $v$ is a non-zero vector, and $\lambda$ is a scalar
• This equation can be rewritten as $(A - \lambda I)v = 0$, where $I$ is the identity matrix
• For a non-zero solution to exist, the determinant of $(A - \lambda I)$ must be zero
• The characteristic polynomial of a matrix $A$ is defined as $p(\lambda) = \det(A - \lambda I)$
  • It is a polynomial in $\lambda$ whose roots are the eigenvalues of $A$
• The degree of the characteristic polynomial is equal to the size of the matrix $A$
• The coefficients of the characteristic polynomial are determined by the entries of the matrix $A$
• The characteristic equation is the equation obtained by setting the characteristic polynomial equal to zero: $p(\lambda) = 0$

Geometric Interpretation of Eigenvectors

• Eigenvectors represent the directions in which a linear transformation (represented by a matrix) acts as a scaling operation
• When a matrix $A$ is applied to one of its eigenvectors $v$, the result is a scalar multiple of $v$, with the scalar being the corresponding eigenvalue $\lambda$
  • Geometrically, this means that the eigenvector's direction remains unchanged, but its magnitude is scaled by $\lambda$
• Eigenvectors corresponding to distinct eigenvalues are linearly independent
• The eigenspace of an eigenvalue $\lambda$ is the set of all vectors $v$ (including the zero vector) that satisfy $Av = \lambda v$
  • Geometrically, the eigenspace is a subspace of the vector space on which the linear transformation acts as a scaling by $\lambda$ in all directions
• If a matrix has a full set of linearly independent eigenvectors, it can be diagonalized, meaning that the linear transformation can be represented as a scaling along mutually orthogonal axes (eigenvectors)

Properties of Eigenvalues and Eigenvectors

• The sum of the eigenvalues of a matrix $A$ is equal to the trace of $A$ (the sum of its diagonal entries)
• The product of the eigenvalues of a matrix $A$ is equal to the determinant of $A$
• If $\lambda$ is an eigenvalue of $A$, then $\lambda^k$ is an eigenvalue of $A^k$ for any positive integer $k$
• If $A$ and $B$ are similar matrices ($B = P^{-1}AP$ for some invertible matrix $P$), then they have the same eigenvalues
  • Their eigenvectors are related by the similarity transformation: if $v$ is an eigenvector of $A$, then $P^{-1}v$ is an eigenvector of $B$
• If $A$ is a real symmetric matrix, then all its eigenvalues are real, and its eigenvectors corresponding to distinct eigenvalues are orthogonal
• If $A$ is a Hermitian matrix (complex analogue of a real symmetric matrix), then all its eigenvalues are real, and its eigenvectors corresponding to distinct eigenvalues are orthogonal
• The algebraic and geometric multiplicities of an eigenvalue are always positive integers, with the geometric multiplicity never exceeding the algebraic multiplicity

Diagonalization and Its Applications

• A matrix $A$ is diagonalizable if it can be written as $A = PDP^{-1}$, where $D$ is a diagonal matrix and $P$ is an invertible matrix
  • The columns of $P$ are the eigenvectors of $A$, and the diagonal entries of $D$ are the corresponding eigenvalues
• Diagonalization allows for the simplification of matrix powers: if $A = PDP^{-1}$, then $A^k = PD^kP^{-1}$ for any positive integer $k$
  • This is because $D^k$ is easy to compute (just raise each diagonal entry to the power $k$)
• Diagonalization is useful in solving systems of linear differential equations
  • If the coefficient matrix of the system is diagonalizable, the system can be decoupled into independent equations that are easier to solve
• Diagonalization is also used in principal component analysis (PCA), a technique for dimensionality reduction and data compression
  • The eigenvectors of the covariance matrix of a dataset represent the principal components, and the corresponding eigenvalues indicate the amount of variance captured by each component
• Markov chains, which model systems that transition between states according to certain probabilities, can be analyzed using diagonalization of the transition matrix
  • The long-term behavior of the system is determined by the eigenvalues and eigenvectors of the transition matrix

Spectral Theorem and Symmetric Matrices

• The spectral theorem states that if $A$ is a real symmetric matrix, then $A$ is diagonalizable by an orthogonal matrix $P$
  • This means $A = PDP^{-1}$, where $D$ is a diagonal matrix and $P$ is an orthogonal matrix ($P^{-1} = P^T$)
  • The columns of $P$ are orthonormal eigenvectors of $A$, and the diagonal entries of $D$ are the corresponding eigenvalues
• The spectral theorem guarantees that a real symmetric matrix has a full set of orthonormal eigenvectors and real eigenvalues
• The eigenvalues of a real symmetric matrix are always real because the characteristic polynomial has real coefficients and the matrix is diagonalizable
• The eigenvectors corresponding to distinct eigenvalues of a real symmetric matrix are orthogonal
  • This is because the matrix is diagonalizable by an orthogonal matrix, and the columns of an orthogonal matrix are orthonormal
• The spectral theorem has important applications in physics and engineering, such as in the analysis of vibrations and the study of quantum mechanics
  • In quantum mechanics, observables are represented by Hermitian operators, and their eigenvalues correspond to the possible measurement outcomes, while the eigenvectors represent the corresponding quantum states

Computational Methods and Algorithms

• The power method is an iterative algorithm for finding the dominant eigenvalue (eigenvalue with the largest absolute value) and its corresponding eigenvector
  • It starts with an initial vector $v_0$ and iteratively computes $v_{k+1} = Av_k / ||Av_k||$ until convergence
  • The dominant eigenvalue is approximated by the Rayleigh quotient $\lambda \approx (v_k^T A v_k) / (v_k^T v_k)$
• The QR algorithm is a more sophisticated method for computing all eigenvalues and eigenvectors of a matrix
  • It involves iteratively decomposing the matrix into a product of an orthogonal matrix $Q$ and an upper triangular matrix $R$ (QR decomposition)
  • The algorithm converges to a matrix whose diagonal entries are the eigenvalues, and the columns of the accumulated $Q$ matrices are the eigenvectors
• The Jacobi method is an iterative algorithm for diagonalizing a real symmetric matrix
  • It works by performing a series of orthogonal similarity transformations to reduce the off-diagonal elements to zero
  • Each transformation is a rotation that eliminates one off-diagonal element and modifies the others
  • The algorithm converges to a diagonal matrix containing the eigenvalues, and the product of the rotation matrices converges to the matrix of eigenvectors
• Krylov subspace methods, such as the Arnoldi iteration and the Lanczos algorithm, are used for computing a subset of eigenvalues and eigenvectors of large sparse matrices
  • These methods work by iteratively building an orthonormal basis for a Krylov subspace (a subspace spanned by powers of the matrix applied to a starting vector) and extracting approximate eigenvalues and eigenvectors from a projected matrix of smaller size

Advanced Topics and Extensions

• Generalized eigenvalue problems involve finding scalar values $\lambda$ and non-zero vectors $v$ that satisfy $Av = \lambda Bv$, where $A$ and $B$ are matrices
  • This arises in problems such as vibration analysis of structures with mass and stiffness matrices
• Singular value decomposition (SVD) is a generalization of eigendecomposition to rectangular matrices
  • It decomposes a matrix $A$ into $A = U\Sigma V^T$, where $U$ and $V$ are orthogonal matrices and $\Sigma$ is a diagonal matrix containing the singular values
  • The columns of $U$ and $V$ are called left and right singular vectors, respectively
• Perturbation theory studies how eigenvalues and eigenvectors change when the matrix is subjected to small perturbations
  • It provides bounds on the sensitivity of eigenvalues and eigenvectors to errors or uncertainties in the matrix entries
• Pseudospectra are sets in the complex plane that provide insight into the behavior of non-normal matrices (matrices whose eigenvectors are not orthogonal)
  • They are defined as the sets of complex numbers $z$ for which the norm of the resolvent $(zI - A)^{-1}$ is greater than a specified threshold
  • Pseudospectra can reveal the sensitivity of eigenvalues to perturbations and the transient behavior of the dynamical system represented by the matrix
• Eigenvalue optimization problems seek to find matrices with eigenvalues that satisfy certain constraints or optimize certain objectives
  • Examples include finding the matrix with the largest or smallest eigenvalue subject to structural constraints (e.g., sparsity or symmetry) or finding the matrix that best approximates a given set of desired eigenvalues
• Eigenvalue problems arise in various applications beyond linear algebra, such as in graph theory (spectral graph theory), quantum mechanics (Schrödinger equation), and dynamical systems (stability analysis)
  • In these contexts, eigenvalues and eigenvectors often have specific physical or practical interpretations that guide the analysis and understanding of the underlying systems