• Preface
  • Syllabi and Prerequisites
  • Survey of Topics
  • Course Outlines
  • Comments on Individual Chapters
  • Changes from the First Edition
  • Exercises and Software
  • Conventions and Notations
  • History and Biography
  • Some Final Remarks
  • Acknowledgments
• Table of Contents
• Chapter 1: Linear Algebraic Systems
  • 1.1 Solution of Linear Systems
  • 1.2 Matrices and Vectors
    • Matrix Arithmetic
  • 1.3 Gaussian Elimination—Regular Case
    • Elementary Matrices
    • The LU Factorization
    • Forward and Back Substitution
  • 1.4 Pivoting and Permutations
    • Permutations and Permutation Matrices
    • The Permuted LU Factorization
  • 1.5 Matrix Inverses
    • Gauss–Jordan Elimination
    • Solving Linear Systems with the Inverse
    • The LDV Factorization
  • 1.6 Transposes and Symmetric Matrices
    • Factorization of Symmetric Matrices
  • 1.7 Practical Linear Algebra
    • Tridiagonal Matrices
    • Pivoting Strategies
  • 1.8 General Linear Systems
    • Homogeneous Systems
  • 1.9 Determinants
• Chapter 2: Vector Spaces and Bases
  • 2.1 Real Vector Spaces
  • 2.2 Subspaces
  • 2.3 Span and Linear Independence
    • Linear Independence and Dependence
  • 2.4 Basis and Dimension
  • 2.5 The Fundamental Matrix Subspaces
    • Kernel and Image
    • The Superposition Principle
    • Adjoint Systems, Cokernel, and Coimage
    • The Fundamental Theorem of Linear Algebra
  • 2.6 Graphs and Digraphs
• Chapter 3: Inner Products and Norms
  • 3.1 Inner Products
    • Inner Products on Function Spaces
  • 3.2 Inequalities
    • The Cauchy–Schwarz Inequality
    • Orthogonal Vectors
    • The Triangle Inequality
  • 3.3 Norms
    • Unit Vectors
    • Equivalence of Norms
    • Matrix Norms
  • 3.4 Positive Definite Matrices
    • Gram Matrices
  • 3.5 Completing the Square
    • The Cholesky Factorization
  • 3.6 Complex Vector Spaces
    • Complex Numbers
    • Complex Vector Spaces and Inner Products
• Chapter 4: Orthogonality
  • 4.1 Orthogonal and Orthonormal Bases
    • Computations in Orthogonal Bases
  • 4.2 The Gram–Schmidt Process
    • Modifications of the Gram–Schmidt Process
  • 4.3 Orthogonal Matrices
    • The QR Factorization
    • Ill-Conditioned Systems and Householder’s Method
  • 4.4 Orthogonal Projections and Orthogonal Subspaces
    • Orthogonal Projection
    • Orthogonal Subspaces
    • Orthogonality of the Fundamental Matrix Subspaces and the Fredholm Alternative
  • 4.5 Orthogonal Polynomials
    • The Legendre Polynomials
    • Other Systems of Orthogonal Polynomials
• Chapter 5: Minimization and Least Squares
  • 5.1 Minimization Problems
    • Equilibrium Mechanics
    • Solution of Equations
    • The Closest Point
  • 5.2 Minimization of Quadratic Functions
  • 5.3 The Closest Point
  • 5.4 Least Squares
  • 5.5 Data Fitting and Interpolation
    • Polynomial Approximation and Interpolation
    • Approximation and Interpolation by General Functions
    • Least Squares Approximation in Function Spaces
    • Orthogonal Polynomials and Least Squares
    • Splines
  • 5.6 Discrete Fourier Analysis and the Fast Fourier Transform
    • Compression and Denoising
    • The Fast Fourier Transform
• Chapter 6: Equilibrium
  • 6.1 Springs and Masses
    • Positive Definiteness and the Minimization Principle
  • 6.2 Electrical Networks
    • Batteries, Power, and the Electrical–Mechanical Correspondence
  • 6.3 Structures
• Chapter 7: Linearity
  • 7.1 Linear Functions
    • Linear Operators
    • The Space of Linear Functions
    • Dual Spaces
    • Composition
    • Inverses
  • 7.2 Linear Transformations
    • Change of Basis
  • 7.3 Affine Transformations and Isometries
    • Isometry
  • 7.4 Linear Systems
    • The Superposition Principle
    • Inhomogeneous Systems
    • Superposition Principles for Inhomogeneous Systems
    • Complex Solutions to Real Systems
  • 7.5 Adjoints, Positive Definite Operators, and Minimization Principles
    • Self-Adjoint and Positive Definite Linear Functions
    • Minimization
• Chapter 8: Eigenvalues and Singular Values
  • 8.1 Linear Dynamical Systems
    • Scalar Ordinary Differential Equations
    • First Order Dynamical Systems
  • 8.2 Eigenvalues and Eigenvectors
    • Basic Properties of Eigenvalues
    • The Gershgorin Circle Theorem
  • 8.3 Eigenvector Bases
    • Diagonalization
  • 8.4 Invariant Subspaces
  • 8.5 Eigenvalues of Symmetric Matrices
    • The Spectral Theorem
    • Optimization Principles for Eigenvalues of Symmetric Matrices
  • 8.6 Incomplete Matrices
    • The Schur Decomposition
    • The Jordan Canonical Form
  • 8.7 Singular Values
    • The Pseudoinverse
    • The Euclidean Matrix Norm
    • Condition Number and Rank
    • Spectral Graph Theory
  • 8.8 Principal Component Analysis
    • Variance and Covariance
    • The Principal Components
• Chapter 9: Iteration
  • 9.1 Linear Iterative Systems
    • Scalar Systems
    • Powers of Matrices
    • Diagonalization and Iteration
  • 9.2 Stability
    • Spectral Radius
    • Fixed Points
    • Matrix Norms and Convergence
  • 9.3 Markov Processes
  • 9.4 Iterative Solution of Linear Algebraic Systems
    • The Jacobi Method
    • The Gauss–Seidel Method
    • Successive Over-Relaxation
  • 9.5 Numerical Computation of Eigenvalues
    • The Power Method
    • The QR Algorithm
    • Tridiagonalization
  • 9.6 Krylov Subspace Methods
    • Krylov Subspaces
    • Arnoldi Iteration
    • The Full Orthogonalization Method
    • The Conjugate Gradient Method
    • The Generalized Minimal Residual Method
  • 9.7 Wavelets
    • The Haar Wavelets
    • Modern Wavelets
    • Solving the Dilation Equation
• Chapter 10: Dynamics
  • 10.1 Basic Solution Techniques
    • The Phase Plane
    • Existence and Uniqueness
    • Complete Systems
    • The General Case
  • 10.2 Stability of Linear Systems
  • 10.3 Two-Dimensional Systems
    • Distinct Real Eigenvalues
    • Complex Conjugate Eigenvalues
    • Incomplete Double Real Eigenvalue
    • Complete Double Real Eigenvalue
  • 10.4 Matrix Exponentials
    • Applications in Geometry
    • Invariant Subspaces and Linear Dynamical Systems
    • Inhomogeneous Linear Systems
  • 10.5 Dynamics of Structures
    • Stable Structures
    • Unstable Structures
    • Systems with Differing Masses
    • Friction and Damping
  • 10.6 Forcing and Resonance
    • Electrical Circuits
    • Forcing and Resonance in Systems
• References
• Symbol Index
• Subject Index

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