Courses on linear algebra and numerical analysis need each other. Often NA courses have some linear algebra topics, and LA courses mention some topics from numerical analysis/scientific computing. This text merges these two areas into one introductory undergraduate course. It assumes students have had multivariable calculus. A second goal of this text is to demonstrate the intimate relationship of linear algebra to applications/computations.
A rigorous presentation has been maintained. A third reason for writing this text is to present, in the first half of the course, the very important topic on singular value decomposition, SVD. This is done by first restricting consideration to real matrices and vector spaces. The general inner product vector spaces are considered starting in the middle of the text.
The text has a number of applications. These are to motivate the student to study the linear algebra topics. Also, the text has a number of computations. MATLAB® is used, but one could modify these codes to other programming languages. These are either to simplify some linear algebra computation, or to model a particular application.
Author(s): Robert E. White
Series: Textbooks in Mathematics
Publisher: CRC Press
Year: 2023
Language: English
Pages: 330
Cover
Half Title
Series Page
Title Page
Copyright Page
Contents
List of Figures
Preface
Introduction
Author Biography
1. Solution of Ax = d
1.1. Matrix Models
1.1.1. Column vectors and Rn
1.1.2. Matrices
1.1.3. Application to visualization of minimum cost
1.1.4. Application to two-bar truss
1.1.5. Application to two-loop circuit
1.1.6. Exercises
1.2. Matrix Products
1.2.1. Matrix-vector products
1.2.2. Matrix-matrix products
1.2.3. Application to heat conduction
1.2.4. Matrix computations using MATLAB®
1.2.5. Exercises
1.3. Special Cases of Ax = d
1.3.1. Five possible classes of “solutions”
1.3.2. Triangular matrices
1.3.3. Application to heat in wire with current
1.3.4. Matrix computations using MATLAB®
1.3.5. Exercises
1.4. Row Operations and Gauss Elimination
1.4.1. Introductory illustration
1.4.2. Three types of row operations
1.4.3. Gauss elimination for solving Ax = d
1.4.4. Application to six-bar truss
1.4.5. Gauss elimination using MATLAB®
1.4.6. Exercises
1.5. Inverse Matrices
1.5.1. Examples of inverse matrices
1.5.2. Gauss–Jordan method to find inverse matrices
1.5.3. Properties of inverse matrices
1.5.4. Inverse matrices and MATLAB®
1.5.5. Exercises
1.6. Determinants and Cramer's Rule
1.6.1. Determinants for 2 x 2 and 3 x 3 matrices
1.6.2. Determinant of an n x n matrix
1.6.3. Cramer's rule and inverses
1.6.4. Determinants using MATLAB®
1.6.5. Exercises
2. Matrix Factorizations
2.1. The Schur Complement
2.1.1. Heat diffusion in fin with two directions
2.1.2. Exercises
2.2. PA = LU and A Nonsingular
2.2.1. Exercises
2.3. A = LU, A-1 ≥ 0 and M-Matrix
2.3.1. Exercises
2.4. A = GGT and A SPD
2.4.1. SPD and minimization
2.4.2. Exercises
3. Least Squares and Normal Equations
3.1. Normal Equations
3.1.1. Exercises
3.2. MATLAB® Code price_expdata.m
3.2.1. Exercises
3.3. Basis of Subspace
3.3.1. Exercises
3.4. Projection to Subspace
3.4.1. Exercises
4. Ax = d with m < n
4.1. Examples in R3
4.2. Row Echelon Form
4.2.1. Solutions in R4
4.2.2. General solution of Ax = d
4.2.3. Exercises
4.3. Relationship of R(A), N(AT) and R(AT), N(A)
4.3.1. Construction of bases
4.3.2. Exercises
4.4. Null Space Method for Equilibrium Equations
4.4.1. Block Gauss elimination method
4.4.2. Null space method for equilibrium equations
4.4.3. Application to three-loop circuit
4.4.4. Application to six-bar truss
4.4.5. Application to fluid flow
4.4.6. Exercises
5. Orthogonal Subspaces and Bases
5.1. Orthogonal Subspace
5.1.1. Exercises
5.2. Fundamental Theorem: Rn = N(A) ⊕ R(AT)
5.2.1. Exercises
5.3. A = QR Factorization
5.3.1. MATLAB® code qr_col.m
5.3.2. Exercises
5.4. Orthonormal Basis
5.4.1. Exercises
5.5. Four Methods for QR Factors
5.5.1. Classical Gram–Schmidt
5.5.2. Givens transform
5.5.3. Householder transform
5.5.4. Exercises
6. Eigenvectors and Orthonormal Basis
6.1. Eigenvectors of Symmetric Matrix
6.1.1. Exercises
6.2. Approximation of Eigenvalues
6.2.1. Gerschgorin circles
6.2.2. Power iterations
6.2.3. QR iteration
6.2.4. Exercises
6.3. Spectral Theorem Factors AQ = QD
6.3.1. Exercises
6.4. Applications
6.4.1. Nonsingular Ax = d
6.4.2. Singular value decomposition
6.4.3. Exercises
7. Singular Value Decomposition
7.1. “Small” SVD
7.1.1. Exercises
7.2. “Full” SVD
7.2.1. MATLAB® code svd_ex.m
7.2.2. Exercises
7.3. “Truncated” SVD
7.3.1. Exercises
8. Three Applications of SVD
8.1. Image Compression
8.1.1. MATLAB® code svdimage.m
8.2. Search Engines
8.2.1. MATLAB® codes sengine.m, senginesparse.m
8.3. Noise Filter
8.3.1. MATLAB® code Image1dsvd.m
9. Pseudoinverse of A
9.1. Σ† and A† = VΣ†UT
9.1.1. Exercises
9.2. A† and Least Squares
9.2.1. Exercises
9.3. Ill-Conditioned Least Squares
9.3.1. Exercises
9.4. Application to Hazard Identification
9.4.1. MATLAB® code hazidsvd1.m
10. General Inner Product Vector Spaces
10.1. Vector Spaces
10.1.1. Exercises
10.2. Inner Products and Orthogonal Vectors
10.2.1. General inner products
10.2.2. Orthonormal vectors
10.2.3. Norms on vector spaces
10.2.4. Exercises
10.3. Schur Decomposition
10.3.1. Norms and spectral radius
10.3.2. Normal matrices
10.3.3. Cayley–Hamilton theorem
10.3.4. Exercises
10.4. Self-Adjoint Differential Operators
10.4.1. Linear operators
10.4.2. Sturm-Liouville problem
10.4.3. Exercises
10.5. Self-Adjoint Positive Definite BVP
10.5.1. Exercises
11. Iterative Methods
11.1. Inverse Matrix Approximations
11.1.1. Exercises
11.2. Regular Splittings for M-Matrices
11.2.1. Exercises
11.3. P-Regular Splittings for SPD Matrices
11.3.1. SOR for diffusion in 3D
11.3.2. MATLAB® implementation of SOR
11.3.3. Exercises
11.4. Conjugate Gradient for SPD Matrices
11.4.1. MATLAB® implementations of CG
11.4.2. Exercises
11.5. Generalized Minimum Residual
11.5.1. MATLAB® implementations of GMRES
11.5.2. Exercises
12. Nonlinear Problems and Least Squares
12.1. Picard Approximation
12.1.1. MATLAB® code piccool.m
12.2. Newton Method
12.2.1. MATLAB® code newtcool.m
12.3. Levenberg-Marquardt Method
12.3.1. MATLAB® code levmarqprice.m
12.4. SIRD Epidemic Models
12.4.1. MATLAB® code sird_parid.m
12.5. The Cumulated Infection Version of SIRD
12.5.1. US COVID-19: An aggregated model
Bibliography
Index
