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Preface
The basic objectives that motivated the first edition of the book serve as the primary
motivations for the second edition as well. While the original content of this book has been
left essentially unchanged, this revision has been aimed at making the terminology used in
the description of some basic mathematical concepts and operations more "up-to-date."
This also makes this technical terminology more consistent with the usage now current in
other fields and disciplines of study (e.g., mathematics, computer science, statistics, social
and behavioral sciences, as well as marketing and business related sciences). A large
number of these revisions occur in Chapter 5, "Decompositions of Matrix Transformations:
Eigenstructures and Quadratic Forms," but can be found throughout the book.
The student willing to learn something about multivariate analysis will find no dearth of
textbooks and monographs on the subject. From introductory to advanced, theoretical to
applied, general to specific, the field has been well covered.
However, most of these books assume certain mathematical prerequisites—typically
matrix algebra and introductory calculus. Single-chapter reviews of the topics are usually
provided but, in turn, presuppose a fair amount of advance preparation. What appears to be
needed for the student who has received less exposure is a somewhat more elementary and
leisurely approach to developing the necessary mathematical foundations of applied
multivariate analysis.
The present book has been prepared to help students with those aspects of
transformational geometry, matrix algebra, and the calculus that seem most relevant for
the study of multivariate analysis. Since the author's interest is in applications, both the
material selected for inclusion and the point of view from which it is presented reflect
that orientation.
The book has been prepared for students who have either taken no matrix algebra at
all or, if they have, need a refresher program that is between a full-fledged matrix
algebra course and the highly condensed review chapter that is often found in
multivariate textbooks. The book can serve as a textbook for courses long enough to
permit coverage of precursory mathematical material or as a supplement to general
textbooks on multivariate analysis.
xi
Xll
PREFACE
The title was chosen rather carefully and helps demarcate what the book is not as much
as what it is. First, those aspects of linear algebra, geometry, and the calculus that are
covered here are treated from a pragmatic viewpoint-as tools for helping the applications
researcher in the behavioral and business disciplines. In particular, there are virtually no
formal proofs. In some cases outlines of proofs have been sketched, but usually small
numerical examples of the various concepts are presented. This decision has been
deliberate and it is the author's hope that the instructor will complement the material with
more formal presentations that reflect his interests and perceptions of the technical
backgrounds of his students.
The book consists of six chapters and two appendices. Chapter 1 introduces the topic of
multivariate analysis and presents three small problems in multiple regression, principal
components analysis, and multiple discriminant analysis to motivate the mathematics that
subsequent chapters are designed to supply. Chapter 2 presents a fairly standard treatment of
the mechanics of matrix algebra including definitions and operations on vectors, matrices,
and determinants. Chapter 3 goes through much of this same material but from a
geometrically oriented viewpoint. Each of the main ideas in matrix algebra is illustrated
geometrically and numerically (as well as algebraically).
Chapter 4 and 5 deal with the central topics of linear transformations and eigenstructures
that are essential to the understanding of multivariate techniques. In Chapter 4, the theme of
Chapter 3 receives additional attention as various matrix transformations are illustrated
geometrically. This same (geometric) orientation is continued in Chapter 5 as eigenstructures
and quadratic forms are described conceptually and illustrated numerically. A large number
of terminological changes made in this edition of the book occur in Chapter 5.
Chapter 6 completes the cycle by returning to the three applied problems presented in
Chapter 1. These problems are solved by means of the techniques developed in Chapters
2-5, and the book concludes with a further discussion of the geometric aspects of linear
transformations.
Appendix A presents supporting material from the calculus for deriving various matrix
equations used in the book. Appendix B provides a basic discussion on solving sets of linear
equations and includes an introduction to generalized inverses. Numerical exercises appear
at the end of each chapter and represent an integral part of the text. With the student's
interest in mind, solutions to all numerical problems are provided. (After all, it was those
even-numhcTQd
exercises that used to give us all the trouble!) The student is urged to work
through these exercises for purposes of conceptual as well as numerical reinforcement.
Completion of the book should provide both a technical base for tackling most
applications-oriented multivariate texts and, more importantly, a geometric perspective for
aiding one's intuitive grasp of multivariate methods. In short, this book has been written
for the student in the behavioral and administrative sciences—not the statistician or
mathematician. If it can help illuminate some of the material in current multivariate
textbooks that are designed for this type of reader, the author's objective will have been
well satisfied.
Acknowledgments
Many people helped bring this book to fruition. Literally dozens of masters and
doctoral students provided critical reviews of one or more chapters from the most
relevant perspective of all—their's. Those deserving special thanks are Ishmael Akaah,
Alain Blanch rude, Frank Deleo, J. A. English, Pascal Lang, and Gunter Seidel. Professor
David K. Hildebrand, University of Pennsylvania, prepared a thorough and useful critique
of the full manuscript. Helpful comments were also received from Professor Joel Huber,
Purdue University.
Production of the book was aided immeasurably by the competent and cheerful help
of Mrs. Joan Leary, who not only typed drafts and redrafts of a difficult manuscript, but
managed to do the extensive art work as well. The editorial staff of Academic Press also
deserves thanks for their production efforts and general cooperative spirit.
J. D. C.
P. E. G.
A. D. C.
xui
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