Chapter 1: Matrices
Abstract
1.1 Matrix Arithmetic
1.2 Linear Transformations
1.3 Powers of Matrices
1.4 Nonsingular Matrices
1.5 The Matrix Transpose and Symmetric Matrices
1.6 Chapter Summary
1.7 Problems
Chapter 2: Linear Equations
Abstract
2.1 Introduction to Linear Equations
2.2 Solving Square Linear Systems
2.3 Gaussian Elimination
2.4 Systematic Solution of Linear Systems
2.5 Computing the Inverse
2.6 Homogeneous Systems
2.7 Application: A Truss
2.8 Application: Electrical Circuit
2.9 Chapter Summary
2.10 Problems
Chapter 3: Subspaces
Abstract
3.1 Introduction
3.2 Subspaces of n
3.3 Linear Independence
3.4 Basis of a Subspace
3.5 The Rank of a Matrix
3.6 Chapter summary
3.7 Problems
Chapter 4: Determinants
Abstract
4.1 Developing the Determinant of A 2 × 2 and A 3 × 3 matrix
4.2 Expansion by Minors
4.3 Computing a Determinant Using Row Operations
4.4 Application: Encryption
4.5 Chapter Summary
4.6 Problems
Chapter 5: Eigenvalues and Eigenvectors
Abstract
5.1 Definitions and Examples
5.2 Selected Properties of Eigenvalues and Eigenvectors
5.3 Diagonalization
5.4 Applications
5.5 Computing Eigenvalues and Eigenvectors Using Matlab
5.6 Chapter Summary
5.7 Problems
Chapter 6: Orthogonal Vectors and Matrices
Abstract
6.1 Introduction
6.2 The Inner Product
6.3 Orthogonal Matrices
6.4 Symmetric Matrices and Orthogonality
6.5 The L2 inner product
6.6 The Cauchy-Schwarz Inequality
6.7 Signal Comparison
6.8 Chapter Summary
6.9 Problems
Chapter 7: Vector and Matrix Norms
Abstract
7.1 Vector Norms
7.3 Submultiplicative Matrix Norms
7.4 Computing the Matrix 2-Norm
7.5 Properties of the Matrix 2-Norm
7.6 Chapter Summary
7.7 Problems
Chapter 8: Floating Point Arithmetic
Abstract
8.1 Integer Representation
8.2 Floating-Point Representation
8.3 Floating-Point Arithmetic
8.4 Minimizing Errors
8.5 Chapter summary
8.6 Problems
Chapter 9: Algorithms
Abstract
9.1 Pseudocode Examples
9.2 Algorithm Efficiency
9.3 The Solution to Upper and Lower Triangular Systems
9.4 The Thomas Algorithm
9.5 Chapter Summary
9.6 Problems
Chapter 10: Conditioning of Problems and Stability of Algorithms
Abstract
10.1 Why do we need numerical linear algebra?
10.2 Computation error
10.3 Algorithm stability
10.4 Conditioning of a problem
10.5 Perturbation analysis for solving a linear system
10.6 Properties of the matrix condition number
10.7 Matlab computation of a matrix condition number
10.8 Estimating the condition number
10.9 Introduction to perturbation analysis of eigenvalue problems
10.10 Chapter summary
10.11 Problems
Chapter 11: Gaussian Elimination and the LU Decomposition
Abstract
11.1 LU Decomposition
11.2 Using LU to Solve Equations
11.3 Elementary Row Matrices
11.4 Derivation of the LU Decomposition
11.5 Gaussian Elimination with Partial Pivoting
11.6 Using the LU Decomposition to Solve Axi=bi,1≤i≤k
11.7 Finding A–1
11.8 Stability and Efficiency of Gaussian Elimination
11.9 Iterative Refinement
11.10 Chapter Summary
11.11 Problems
Chapter 12: Linear System Applications
Abstract
12.1 Fourier Series
12.2 Finite Difference Approximations
12.3 Least-Squares Polynomial Fitting
12.4 Cubic Spline Interpolation
12.5 Chapter Summary
12.6 Problems
Chapter 13: Important Special Systems
Abstract
13.1 Tridiagonal Systems
13.2 Symmetric Positive Definite Matrices
13.3 The Cholesky Decomposition
13.4 Chapter Summary
13.5 Problems
Chapter 14: Gram-Schmidt Orthonormalization
Abstract
14.1 The Gram-Schmidt Process
14.2 Numerical Stability of the Gram-Schmidt Process
14.3 The QR Decomposition
14.3.1 Efficiency
14.3.2 Stability
14.4 Applications of The QR Decomposition
14.5 Chapter Summary
14.6 Problems
Chapter 15: The Singular Value Decomposition
Abstract
15.1 The SVD Theorem
15.2 Using the SVD to Determine Properties of a Matrix
15.3 SVD and Matrix Norms
15.4 Geometric Interpretation of the SVD
15.5 Computing the SVD Using MATLAB
15.6 Computing A–1
15.7 Image Compression Using the SVD
15.8 Final Comments
15.9 Chapter Summary
15.10 Problems
Chapter 16: Least-Squares Problems
Abstract
16.1 Existence and Uniqueness of Least-Squares Solutions
16.2 Solving Overdetermined Least-Squares Problems
16.3 Conditioning of Least-Squares Problems
16.4 Rank-Deficient Least-Squares Problems
16.5 Underdetermined Linear Systems
16.6 Chapter Summary
16.7 Problems
Chapter 17: Implementing the QR Decomposition
Abstract
17.1 Review of the QR Decomposition Using Gram-Schmidt
17.2 Givens Rotations
17.3 Creating a Sequence of Zeros in a Vector Using Givens Rotations
17.4 Product of a Givens Matrix with a General Matrix
17.5 Zeroing-Out Column Entries in a Matrix Using Givens Rotations
17.6 Accurate Computation of the Givens Parameters
17.7 THe Givens Algorithm for the QR Decomposition
17.8 Householder Reflections
17.9 Computing the QR Decomposition Using Householder Reflections
17.10 Chapter Summary
17.11 Problems
Chapter 18: The Algebraic Eigenvalue Problem
Abstract
18.1 Applications of The Eigenvalue Problem
18.2 Computation of Selected Eigenvalues and Eigenvectors
18.3 The Basic QR Iteration
18.4 Transformation to Upper Hessenberg Form
18.5 The Unshifted Hessenberg QR Iteration
18.6 The Shifted Hessenberg QR Iteration
18.7 Schur's Triangularization
18.8 The Francis Algorithm
18.9 Computing Eigenvectors
18.10 Computing Both Eigenvalues and Their Corresponding Eigenvectors
18.11 Sensitivity of Eigenvalues to Perturbations
18.12 Chapter Summary
18.13 Problems
Chapter 19: The Symmetric Eigenvalue Problem
Abstract
19.1 The Spectral Theorem and Properties of A Symmetric Matrix
19.2 The Jacobi Method
19.3 The Symmetric QR Iteration Method
19.4 The Symmetric Francis Algorithm
19.5 The Bisection Method
19.6 The Divide-And-Conquer Method
19.7 Chapter Summary
19.8 Problems
Chapter 20: Basic Iterative Methods
Abstract
20.1 Jacobi Method
20.2 The Gauss-Seidel Iterative Method
20.3 The Sor Iteration
20.4 Convergence of the Basic Iterative Methods
20.5 Application: Poisson's Equation
20.6 Chapter Summary
20.7 Problems
Chapter 21: Krylov Subspace Methods
Abstract
21.1 Large, Sparse Matrices
21.2 The CG Method
21.3 Preconditioning
21.4 Preconditioning For CG
21.5 Krylov Subspaces
21.6 The Arnoldi Method
21.16.1 An Alternative Formulation of the Arnoldi Decomposition
21.7 GMRES
21.8 The Symmetric Lanczos Method
21.9 The Minres Method
21.10 Comparison of Iterative Methods
21.11 Poisson's Equation Revisited
21.12 The Biharmonic Equation
21.13 Chapter Summary
21.14 Problems
Chapter 22: Large Sparse Eigenvalue Problems
Abstract
22.1 The Power Method
22.2 Eigenvalue Computation Using the Arnoldi Process
22.3 The Implicitly Restarted Arnoldi Method
22.4 Eigenvalue Computation Using the Lanczos Process
22.5 Chapter Summary
22.6 Problems
Chapter 23: Computing the Singular Value Decomposition
Abstract
23.1 Development of the One-Sided Jacobi Method For Computing the Reduced Svd
23.2 The One-Sided Jacobi Algorithm
23.3 Transforming a Matrix to Upper-Bidiagonal Form
23.4 Demmel and Kahan Zero-Shift QR Downward Sweep Algorithm
23.5 Chapter Summary
23.6 Problems

Numerical Linear Algebra with Applications is designed for those who want to gain a practical knowledge of modern computational techniques for the numerical solution of linear algebra problems, using MATLAB as the vehicle for computation. The book contains all the material necessary for a first year graduate or advanced undergraduate course on numerical linear algebra with numerous applications to engineering and science. With a unified presentation of computation, basic algorithm analysis, and numerical methods to compute solutions, this book is ideal for solving real-world problems.