# Fundamentals of Matrix Analysis with Applications

# Fundamentals of Matrix Analysis with Applications

ISBN: 978-1-118-95369-3

Oct 2015

408 pages

$100.99

## Description

**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.

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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

"This is a straightforward modern introduction to matrices..... a very well done text, probably most suitable for engineering students." (Mathematical Association of America 2016)

"This is a straightforward modern introduction to matrices. As the title indicates, the emphasis is on the tool of matrices rather than the theory of linear spaces. There is a moderate amount on linear spaces, but this is oriented toward supporting some of the more advanced matrix operations rather than as a subject in itself.

The book starts out with a very detailed, almost loving, exposition of Gaussian elimination, and parlays that into the formalism of matrix algebra. Most of the rest of the book deals with useful matrix operations, and in particular with particular forms and decompositions of matrices, such as diagonalization through eigenvectors, LU and QR factorizations, Schur and Jordan forms, and SVD (singular value decomposition). At the end of each of the three sections of the book are several longer projects with realistic applications, mostly from electrical engineering with some mechanics and control theory. These a billed as group projects, but could just as well be individual projects. The last third of the book deals with differential equations, using these as an opportunity to introduce even more matrix techniques. There’s no companion web site for this book.

The book has copious exercises; most are numeric drill, with a few generalizations and simple proofs. Many of these are flagged to be done with a calculator or a computer and to examine the round-off errors. There are also a few “technical writing exercises” in each of the three sections; these ask the student to investigate something and write an explanation in words. There’s not any guidance on these in the text, so I think the instructor would have to give a lot of coaching to get anything useful. These are short exercises and would probably generate 1- to 2-page papers.

There are answers to the odd-numbered problems in the back. There is also a solutions manual for this book from the same authors and the same publisher. This has concise, complete solutions for all problems in the text. This manual is sold openly to anyone (on Amazon, for example) and is not one of those that is available only to instructors.

Very Good Feature: lots of examples using SVD.

Very Good Feature: the computational aspects are well-integrated into the narrative. (There is one silly statement about computers, though. On p. 95, in talking about the advantages of doing Gaussian elimination on a determinant rather than using Cramer’s rule, the book says, “So using elementary row operations the Tianhe-2 could calculate a 25-by-25 determinant in a fraction of a picosecond.” In truth, any current computer takes around a nanosecond (10−9

second) per operation, so doing anything in a picosecond (10^-12}

second) is impossible. The misapprehension comes because the Tianhe-2 is a massively parallel computer with over 3 million cores and an advertised top speed of 33.86 petaflops. Because of the parallelism, if all the cores are busy the *average* time per operation across the whole computer is under a picosecond, but no individual operation is anywhere near that fast.)

Bottom line: a very well done text, probably most suitable for engineering students. Math students would be better served by a book that combines matrices with a more thorough coverage of linear spaces; I like Strang’s *Introduction to Linear Algebra*. (Allen Stenger)

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"This book provides comprehensive coverage of matrix theory from a geometric and physical perspective, and the authors address the functionality of matrices and their ability to illustrate and aid in many practical applications. Readers are introduced to inverses and eigenvalues through physical examples such as rotations, reflections, and projections, and only then are computational details described and explored." (http://cds.cern.ch/record/2050353)