Mathematical Tools for Applied Multivariate

This revised edition presents the relevant aspects of transformational geometry, matrix algebra, and calculus to those who may be lacking the necessary mathematical foundations of applied multivariate analysis. It brings up-to-date many definitions of mathematical concepts and their operations. It also clearly defines the relevance of the exercises to concerns within the business community and the social and behavioral sciences. Readers gain a technical background for tackling applications-oriented multivariate texts and receive a geometric perspective for understanding multivariate methods. Mathematical Tools for Applied Multivariate Analysis, Revised Edition illustrates major concepts in matrix algebra, linear structures, and eigenstructures geometrically, numerically, and algebraically. The authors emphasize the applications of these techniques by discussing potential solutions to problems outlined early in the book. They also present small numerical examples of the various concepts.

Preface

xi

(2)

Acknowledgments

xiii

Chapter 1 The Nature of Multivariate Data Analysis

1

(25)

1.1 Introduction

1

(1)

1.2 Multivariate Methods in Research

2

(2)

1.3 A Classification of Techniques for Analyzing Associative Data

4

(4)

1.4 Organizing the Techniques

8

(3)

1.5 Illustrative Applications

11

(3)

1.6 Some Numerical Examples

14

(9)

1.7 Format of Succeeding Chapters

23

(1)

1.8 Summary

24

(1)

Review Questions

25

(1)

Chapter 2 Vector and Matrix Operations for Multivariate Analysis

26

(51)

2.1 Introduction

26

(1)

2.2 Vector Representation

27

(1)

2.3 Basic Definitions and Operations on Vectors

27

(13)

2.4 Matrix Representation

40

(1)

2.5 Basic Definitions and Operations on Matrices

40

(12)

2.6 Some Special Matrices

52

(6)

2.7 Determinants of Matrices

58

(11)

2.8 Applying Matrix Operations to Statistical Data

69

(4)

2.9 Summary

73

(1)

Review Questions

74

(3)

Chapter 3 Vector and Matrix Concepts from a Geometric Viewpoint

77

(50)

3.1 Introduction

77

(1)

3.2 Euclidean Space and Rectangular Cartesian Coordinates

78

(7)

3.3 Geometric Representation of Vectors

85

(16)

3.4 Linear Dependence of Vectors

101

(10)

3.5 Orthogonal Transformations

111

(7)

3.6 Geometric Aspects of Cross-Product Matrices and Determinants

118

(6)

3.7 Summary

124

(1)

Review Questions

125

(2)

Chapter 4 Linear Transformations from a Geometric Viewpoint

127

(67)

4.1 Introduction

127

(1)

4.2 Simultaneous Equations and Matrix Transformations

128

(8)

4.3 Matrix Inversion

136

(11)

4.4 Geometric Relationships Involving Matrix Transformations

147

(9)

4.5 Composite Transformations

156

(7)

4.6 Invertible Transformations and Matrix Rank

163

(13)

4.7 Methods for Rank Determination and Matrix Inversion

176

(13)

4.8 Summary

189

(1)

Review Questions

190

(4)

Chapter 5 Decomposition of Matrix Transformations: Eigenstructures and Quadratic Forms

194

(65)

5.1 Introduction

194

(1)

5.2 An Overview of Matrix Eigenstructures

195

(12)

5.3 Transformations of Covariance Matrices

207

(3)

5.4 Eigenstructure of a Symmetric Matrix

210

(9)

5.5 Properties of Matrix Eigenstructures

219

(6)

5.6 Eigenstructures and Matrix Rank

225

(5)

5.7 The Basic Structure of a Matrix

230

(10)

5.8 Quadratic Forms

240

(7)

5.9 Eigenstructures of Nonsymmetric Matrices in Multivariate Analysis

247

(8)

5.10 Summary

255

(1)

Review Questions

256

(3)

Chapter 6 Applying the Tools to Multivariate Data

259

(36)

6.1 Introduction

259

(1)

6.2 The Multiple Regression Problem

260

(10)

6.3 Other Forms of the General Linear Model

270

(2)

6.4 The Factor Analysis Problem

272

(6)

6.5 The Multiple Discriminant Analysis Problem

278

(8)

6.6 A Parting Look at Multivariate Technique Classification

286

(7)

6.7 Summary

293

(1)

Review Questions

294

(1)

Appendix A Symbolic Differentiation and Optimization of Multivariable Functions

295

(28)

A.1 Introduction

295

(1)

A.2 Differentiation of Functions of One Argument

296

(8)

A.3 Differentiation of Functions of Two Arguments

304

(8)

A.4 Symbolic Differentiation

312

(6)

A.5 Application of the Calculus to Multivariate Analysis

318

(2)

A.6 Summary

320

(1)

Review Questions

321

(2)

Appendix B Linear Equations and Generalized Inverses

323

(29)

B.1 Introduction

323

(1)

B.2 Simultaneous Linear Equations

324

(10)

B.3 Introductory Aspects of Generalized Inverses

334

(9)

B.4 The g Inverse

343

(7)

B.5 Summary

350

(1)

Review Questions

350

(2)

Answers to Numerical Problems

352

(12)

References

364

(5)

Index

369

What is included with this book?

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.

FREE SHIPPING

Amazon no longer offers textbook rentals. We do!

Mathematical Tools for Applied Multivariate Analysis

Summary

Table of Contents

Supplemental Materials