Fundamentals of Matrix Analysis with Applications

Providing comprehensive coverage of matrix theory from a geometric and physical perspective, Fundamentals of Matrix Analysis with Applications describes the functionality of matrices and their ability to quantify and analyze many practical applications. Written by a highly–qualified author team, the book presents tools for matrix analysis and is illustrated with extensive examples and software implementations.

Beginning with a detailed exposition and review of the Gauss elimination method, the authors maintain readers interest with refreshing discussions regarding the issues of operation counts, computer speed and prevision, complex arithmetic formulations, parametrization of solutions, and the logical traps that dictate strict adherence to Gauss s instructions. The book heralds matrix formulation both as notational shorthand and a quantifier of physical operations such as rotations, projections, reflections, and the Gauss reductions. Inverses and eigenvectors are visualized first in an operator context before being addressed computationally. In addition to coverage on lease squares theory s manifestations such as optimization, orthogonality, computational accuracy, and even function theory, Fundamentals of Matrix Analysis with Applications features:

Preface

Part I

Introduction: Three Examples

Chapter 1. SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS

1.1 Linear Algebraic Equations

1.2 Matrix Representation of Linear Systems and the Gauss Jordan Algorithm

1.3 The Complete Gauss Elimination Algorithm

1.4 Echelon Form and Rank

1.5 Computational Considerations

Chapter 2. MATRIX ALGEBRA

2.1 Matrix Multiplication

2.2 Some Applications of Matrix Operators

2.3 The Inverse and the Transpose

2.4 Determinants

2.5 Three Important Determinant Rules

Review Problems for Part I

Technical Writing Exercises for Part I

Group Projects for Part I

A. LU Factorization

B. Two Point Boundary Value Problems

C. Electrostatic Voltage

D. Kirchhoff′s Laws

E. Global Positioning Systems

Part II

Introduction: The Structure of General Solutions to Linear Algebraic Equations

Chapter 3. VECTOR SPACES

3.1 General Spaces, Subspaces, and Spans

3.2 Linear Dependence

3.3 Bases, Dimension, and Rank

Chapter 4. ORTHOGONALITY

4.1 Orthogonal Vectors and the Gram Schmidt Algorithm Norm

4.2 Orthogonal Matrices

4.3 Least Squares

4.4 Function Spaces

Review Problems for Part II

Magic square

Controllability

Technical Writing Exercises for Part II

Group Projects for Part II

A. Orthogonal Matrices, Rotations, and Reflections

B. Householder Reflectors and the QR Factorization

C. Infinite Dimensional Matrices

Part III

Introduction: Reflect on This

Chapter 5. Eigenvalues and Eigenvectors

5.1 Eigenvector Basics

5.2 Calculating Eigenvalues and Eigenvectors

5.3 Symmetric and Hermitian Matrices

Chapter 5. Summary

Chapter 6. Similarity

6.1 Similarity Transformations and Diagonalizability

6.2 Principal Axes Normal Modes

6.3 Schur Decomposition and Its Implications

6.4 The Power Method and the QR Algorithm

Chapter 7. Linear Systems of Differential Equations

7.1 First Order Linear Systems of Differential Equations

7.2 The Matrix Exponential Function

7.3 The Jordan Normal Form

Review Problems for Part III

Technical Writing Exercises for Part III

Group Projects for Part III

A. Positive Definite Matrices

B. Hessenberg Form

C. The Discrete Fourier Transform and Circulant Matrices

Answers to Odd Numbered Problems

Index

Edward Barry Saff, PhD, is Professor in the Department of Mathematics and Director of the Center for Constructive Approximation at Vanderbilt University.  Dr. Saff is an Inaugural Fellow of the American Mathematical Society, Foreign Member of the Bulgarian Academy of Science, and the recipient of both a Guggenheim and Fulbright Fellowship.  He is Editor–in–Chief of two research journals, Constructive Approximation and Computational Methods and Function Theory, and has authored or co–authored over 250 journal articles and eight books. Dr. Saff also serves as an organizer for a sequence of international research conferences that help to foster the careers of mathematicians from developing countries.  

Arthur David Snider, PhD, is Professor Emeritus at the University of South Florida (USF), where he served on the faculties of the Departments of Mathematics, Physics, and Electrical Engineering. Previously an analyst at the Massachusetts Institute of Technology s Draper Lab and recipient of the USF Krivanek Distinguished Teacher Award, he consults in industry and has authored or co–authored over 100 journal articles and eight books. With the support of National Science Foundation, Dr. Snider also pioneered a course in fine art appreciation for engineers.