# Fundamentals of Matrix Analysis with Applications

## Books

An accessible and clear introduction to linear algebra with a focus on matrices and engineering 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 precision, complex arithmetic formulations, parameterization of solutions, and the logical traps that dictate strict adherence to Gauss’s instructions. The book heralds matrix formulation both as notational shorthand and as 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. Least squares theory is expounded in all its manifestations including optimization, orthogonality, computational accuracy, and even function theory. Fundamentals of Matrix Analysis with Applications also features:

• Novel approaches employed to explicate the QR, singular value, Schur, and Jordan decompositions and their applications
• Coverage of the role of the matrix exponential in the solution of linear systems of differential equations with constant coefficients
• Chapter-by-chapter summaries, review problems, technical writing exercises, select solutions, and group projects to aid comprehension of the presented concepts

Fundamentals of Matrix Analysis with Applications is an excellent textbook for undergraduate courses in linear algebra and matrix theory for students majoring in mathematics, engineering, and science. The book is also an accessible go-to reference for readers seeking clarification of the fine points of kinematics, circuit theory, control theory, computational statistics, and numerical algorithms.

Preface ix

Part I Introduction: Three Examples 1

1 Systems of Linear Algebraic Equations 5

1.1 Linear Algebraic Equations 5

1.2 Matrix Representation of Linear Systems and the Gauss-Jordan Algorithm 17

1.3 The Complete Gauss Elimination Algorithm 27

1.4 Echelon Form and Rank 38

1.5 Computational Considerations 46

1.6 Summary 55

2 Matrix Algebra 58

2.1 Matrix Multiplication 58

2.2 Some Physical Applications of Matrix Operators 69

2.3 The Inverse and the Transpose 76

2.4 Determinants 86

2.5 Three Important Determinant Rules 100

2.6 Summary 111

Group Projects for Part I

A. LU Factorization 116

B. Two-Point Boundary Value Problem 118

C. Electrostatic Voltage 119

D. Kirchhoff’s Laws 120

E. Global Positioning Systems 122

F. Fixed-Point Methods 123

Part II Introduction: The Structure of General Solutions to Linear Algebraic Equations 129

3 Vector Spaces 133

3.1 General Spaces Subspaces and Spans 133

3.2 Linear Dependence 142

3.3 Bases, Dimension, and Rank 151

3.4 Summary 164

4 Orthogonality 165

4.1 Orthogonal Vectors and the Gram–Schmidt Algorithm 165

4.2 Orthogonal Matrices 174

4.3 Least Squares 180

4.4 Function Spaces 190

4.5 Summary 197

Group Projects for Part II

A. Rotations and Reflections 201

B. Householder Reflectors 201

C. Infinite Dimensional Matrices 202

Part III Introduction: Reflect on This 205

5 Eigenvectors and Eigenvalues 209

5.1 Eigenvector Basics 209

5.2 Calculating Eigenvalues and Eigenvectors 217

5.3 Symmetric and Hermitian Matrices 225

5.4 Summary 232

6 Similarity 233

6.1 Similarity Transformations and Diagonalizability 233

6.2 Principle Axes and Normal Modes 244

6.3 Schur Decomposition and Its Implications 257

6.4 The Singular Value Decomposition 264

6.5 The Power Method and the QR Algorithm 282

6.6 Summary 290

7 Linear Systems of Differential Equations 293

7.1 First-Order Linear Systems 293

7.2 The Matrix Exponential Function 306

7.3 The Jordan Normal Form 316

7.4 Matrix Exponentiation via Generalized Eigenvectors 333

7.5 Summary 339

Group Projects for Part III

A. Positive Definite Matrices 342

B. Hessenberg Form 343

C. Discrete Fourier Transform 344

D. Construction of the SVD 346

E. Total Least Squares 348

F. Fibonacci Numbers 350

Answers to Odd Numbered Exercises 351

Index 393

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