➗Abstract Linear Algebra II Unit 6 – Canonical Forms

Key Concepts and Definitions

• Linear transformations map vectors from one vector space to another while preserving linear combinations and the zero vector
• Matrices represent linear transformations with respect to a chosen basis and enable computation and analysis of the transformation's properties
• Similarity of matrices indicates they represent the same linear transformation under different bases and share key properties (eigenvalues, rank, determinant)
  • Two matrices $A$ and $B$ are similar if there exists an invertible matrix $P$ such that $A = P^{-1}BP$
• Eigenvalues are scalar values $\lambda$ that satisfy the equation $Av = \lambda v$ for a square matrix $A$ and non-zero vector $v$
  • Eigenvectors are the corresponding non-zero vectors $v$ that, when multiplied by $A$, result in a scalar multiple of themselves
• Characteristic polynomial of a matrix $A$ is defined as $p(x) = \det(xI - A)$ and its roots are the eigenvalues of $A$
• Diagonalizable matrices can be expressed as $A = PDP^{-1}$, where $D$ is a diagonal matrix containing the eigenvalues and $P$ is formed by the corresponding eigenvectors
• Canonical forms aim to simplify the representation of a matrix or linear transformation using a standardized format that reveals key properties and structure

Matrix Transformations and Similarity

• Matrix transformations linearly map vectors from one space to another, enabling the study of linear systems and their properties
• Composition of matrix transformations corresponds to matrix multiplication, allowing for the analysis of complex transformations through simpler components
• Similar matrices $A$ and $B$ satisfy the equation $A = P^{-1}BP$ for some invertible matrix $P$, called the change of basis matrix
  • Similar matrices share the same eigenvalues, trace, determinant, and rank
• Diagonalization is a special case of similarity where a matrix $A$ is similar to a diagonal matrix $D$, i.e., $A = PDP^{-1}$
  • Diagonalizable matrices have a full set of linearly independent eigenvectors that form the columns of $P$
• Similarity transformations preserve the structure and properties of a matrix while expressing it in a more convenient or insightful basis
• Symmetric matrices are always diagonalizable and have real eigenvalues, making them important in applications (quadratic forms, optimization)
• Orthogonal matrices $Q$ satisfy $Q^T = Q^{-1}$ and represent rotations or reflections in a vector space, preserving angles and lengths

Eigenvalues and Eigenvectors Revisited

• Eigenvalues and eigenvectors capture the essential behavior of a linear transformation or matrix, revealing its stretching, shrinking, or rotation effects on specific vectors
• Eigenvalues are found by solving the characteristic equation $\det(A - \lambda I) = 0$, which can be derived from the definition $Av = \lambda v$
  • The algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic polynomial
• Eigenvectors corresponding to an eigenvalue $\lambda$ are non-zero vectors that satisfy $(A - \lambda I)v = 0$ and span the eigenspace associated with $\lambda$
  • The geometric multiplicity of an eigenvalue is the dimension of its eigenspace
• A matrix is diagonalizable if and only if the sum of the geometric multiplicities of its eigenvalues equals its size (i.e., it has a full set of eigenvectors)
• Spectral theorem states that a real symmetric matrix has an orthonormal basis of eigenvectors and is diagonalizable by an orthogonal matrix
• Eigendecomposition expresses a matrix as $A = Q\Lambda Q^{-1}$, where $\Lambda$ is a diagonal matrix of eigenvalues and $Q$ contains the corresponding eigenvectors
• Eigenvalues and eigenvectors have numerous applications, such as principal component analysis, differential equations, and stability analysis of dynamical systems
  • The dominant eigenvalue (largest in magnitude) often determines the long-term behavior or convergence properties of a system

Jordan Canonical Form

• Jordan canonical form (JCF) is a unique matrix representation that captures the structure of a linear transformation or matrix using Jordan blocks
  • JCF exists for any square matrix over an algebraically closed field (complex numbers)
• Jordan block is a square upper triangular matrix with an eigenvalue $\lambda$ on the diagonal, ones on the superdiagonal, and zeros elsewhere
  • Size of a Jordan block is determined by the geometric multiplicity of its eigenvalue
• JCF of a matrix $A$ is a block diagonal matrix $J = \text{diag}(J_1, \ldots, J_k)$, where each $J_i$ is a Jordan block corresponding to an eigenvalue of $A$
• Similarity transformation $A = PJP^{-1}$ relates the original matrix $A$ to its JCF $J$, where $P$ is an invertible matrix formed by generalized eigenvectors
• Generalized eigenvectors are vectors that satisfy $(A - \lambda I)^kv = 0$ for some positive integer $k$ and are used to construct the Jordan basis
• Algebraic multiplicity of an eigenvalue equals the sum of the sizes of its corresponding Jordan blocks in the JCF
• Geometric multiplicity of an eigenvalue is the number of Jordan blocks associated with it in the JCF
• A matrix is diagonalizable if and only if all its Jordan blocks have size one, i.e., the algebraic and geometric multiplicities are equal for each eigenvalue

Rational Canonical Form

• Rational canonical form (RCF) is a unique matrix representation that exists over any field, making it more general than the Jordan canonical form
• Companion matrix of a monic polynomial $p(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$ is a square matrix of the form: $C_p = \begin{bmatrix} 0 & 0 & \cdots & 0 & -a_0 \\ 1 & 0 & \cdots & 0 & -a_1 \\ 0 & 1 & \cdots & 0 & -a_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & -a_{n-1} \end{bmatrix}$
• RCF of a matrix $A$ is a block diagonal matrix $R = \text{diag}(C_{p_1}, \ldots, C_{p_k})$, where each $C_{p_i}$ is a companion matrix corresponding to an invariant factor of $A$
• Invariant factors of a matrix $A$ are the unique monic polynomials obtained by factoring the characteristic polynomial of $A$ over the given field
  • Invariant factors satisfy divisibility relations: each one divides the next, and the product of all invariant factors equals the characteristic polynomial
• Similarity transformation $A = PRP^{-1}$ relates the original matrix $A$ to its RCF $R$, where $P$ is an invertible matrix formed by basis vectors of cyclic subspaces
• Cyclic subspace generated by a vector $v$ under a matrix $A$ is the subspace spanned by $\{v, Av, A^2v, \ldots\}$ and has dimension equal to the degree of the minimal polynomial of $v$
• Minimal polynomial of a vector $v$ under a matrix $A$ is the monic polynomial $p(x)$ of least degree such that $p(A)v = 0$
• RCF provides a canonical way to study the structure and properties of a matrix over any field, making it useful in algebraic geometry and number theory

Applications in Linear Systems

• Linear systems of differential equations $\frac{d\mathbf{x}}{dt} = A\mathbf{x}$ can be solved using eigenvalues and eigenvectors of the coefficient matrix $A$
  • Solutions are of the form $\mathbf{x}(t) = c_1e^{\lambda_1t}\mathbf{v}_1 + \cdots + c_ne^{\lambda_nt}\mathbf{v}_n$, where $\lambda_i$ and $\mathbf{v}_i$ are eigenvalues and eigenvectors of $A$
• Stability of a linear system is determined by the eigenvalues of $A$: the system is stable if all eigenvalues have negative real parts, and unstable otherwise
• Markov chains model systems transitioning between states, with transition probabilities encoded in a stochastic matrix $P$
  • Long-term behavior of a Markov chain is determined by the eigenvalues and eigenvectors of $P$, particularly the dominant eigenvalue of 1
• Principal component analysis (PCA) uses eigenvectors of the covariance matrix to identify the most important directions of variation in a dataset
  • Eigenvectors corresponding to the largest eigenvalues capture the principal components, which can be used for dimensionality reduction and data visualization
• Singular value decomposition (SVD) factorizes a matrix $A$ into $A = U\Sigma V^T$, where $U$ and $V$ are orthogonal matrices and $\Sigma$ is diagonal with singular values
  • SVD has applications in data compression, noise reduction, and recommender systems, as it reveals the most significant factors or features in a matrix
• Quantum mechanics heavily relies on eigenvalues and eigenvectors to describe the states and observables of a quantum system
  • Eigenvalues correspond to possible measurement outcomes, while eigenvectors represent the associated quantum states
• Fourier analysis decomposes functions into a sum of simpler trigonometric functions, which can be viewed as eigenfunctions of the Laplace operator
  • This has applications in signal processing, image compression, and solving partial differential equations

Computational Methods and Algorithms

• Power iteration is an iterative algorithm for approximating the dominant eigenvalue and its corresponding eigenvector of a matrix
  • Involves repeated matrix-vector multiplication and normalization until convergence
• Inverse iteration is similar to power iteration but uses the inverse of a shifted matrix to approximate eigenvectors corresponding to a specific eigenvalue
• QR algorithm is an iterative method for computing the eigenvalues and eigenvectors of a matrix by repeatedly performing QR decomposition
  • Converges to a triangular matrix whose diagonal entries are the eigenvalues, with eigenvectors obtained from the accumulated orthogonal matrices
• Jacobi method for symmetric matrices iteratively applies rotations to diagonalize the matrix, with the resulting diagonal entries being the eigenvalues
• Krylov subspace methods (Arnoldi, Lanczos) efficiently compute a subset of eigenvalues and eigenvectors by constructing an orthonormal basis for the Krylov subspace
• Singular value decomposition (SVD) can be computed using the Golub-Kahan-Reinsch algorithm, which combines bidiagonalization with the QR algorithm
• Schur decomposition $A = QTQ^*$ factorizes a matrix into a unitary matrix $Q$ and an upper triangular matrix $T$, useful for computing eigenvalues and invariant subspaces
• Numerical stability and accuracy are important considerations when implementing eigenvalue algorithms, as rounding errors can accumulate and lead to inaccurate results
  • Techniques such as shifting, deflation, and balancing can help mitigate these issues and improve the stability of the algorithms

Advanced Topics and Extensions

• Generalized eigenvalue problem $Av = \lambda Bv$ arises when considering pencils of matrices $(A, B)$ and has applications in vibration analysis and control theory
• Pseudospectra of a matrix $A$ are sets in the complex plane defined by the norm of the resolvent $(zI - A)^{-1}$ and provide insight into the sensitivity of eigenvalues to perturbations
• Matrix functions $f(A)$ extend scalar functions to matrices and can be defined using the Jordan canonical form or eigendecomposition of $A$
  • Examples include the matrix exponential $e^A$, logarithm $\log(A)$, and power $A^p$, which have applications in differential equations and network analysis
• Spectral graph theory studies the eigenvalues and eigenvectors of matrices associated with graphs (adjacency, Laplacian) to infer properties of the graph structure
  • Connections to graph connectivity, partitioning, and random walks on graphs
• Infinite-dimensional operators on Hilbert spaces generalize the concept of matrices and have eigenvalues and eigenvectors defined by the spectral theorem
  • Applications in quantum mechanics, partial differential equations, and functional analysis
• Lie algebras are vector spaces equipped with a bilinear operation called the Lie bracket, generalizing the commutator of matrices
  • Eigenvalues and eigenvectors of elements in a Lie algebra provide insight into the structure and representations of the associated Lie group
• Random matrix theory studies the statistical properties of eigenvalues and eigenvectors of large random matrices, with applications in physics, statistics, and number theory
• Multilinear algebra extends linear algebra concepts to tensors, which can be viewed as higher-dimensional analogues of matrices
  • Eigenvalues and eigenvectors of tensors have applications in data analysis, signal processing, and machine learning