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Matrix Algebra From a Statistician’s Perspective

David A. Harville Matrix Algebra From a Statistician’s Perspective 1 3

David A. Harville IBM T. J. Watson Research Center Mathematical Sciences Department Yorktown Heights, NY 10598-0218 USA harville@us.ibm.com ISBN 978-0-387-78356-7 e-ISBN 978-0-387-22677-4 Library of Congress Control Number: 2008927514 c 2008 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identiﬁed as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper. 9 8 7 6 5 4 3 2 1 springer.com

Preface Matrix algebra plays a very important role in statistics and in many other disci- plines. In many areas of statistics, it has become routine to use matrix algebra in the presentation and the derivation or veriﬁcation of results. One such area is linear statistical models; another is multivariate analysis. In these areas, a knowledge of matrix algebra is needed in applying important concepts, as well as in studying the underlying theory, and is even needed to use various software packages (if they are to be used with conﬁdence and competence). On many occasions, I have taught graduate-level courses in linear statistical models. Typically, the prerequisites for such courses include an introductory (un- dergraduate) course in matrix (or linear) algebra. Also typically, the preparation provided by this prerequisite course is not fully adequate. There are several rea- sons for this. The level of abstraction or generality in the matrix (or linear) algebra course may have been so high that it did not lead to a “working knowledge” of the subject, or, at the other extreme, the course may have emphasized computations at the expense of fundamental concepts. Further, the content of introductory courses on matrix (or linear) algebra varies widely from institution to institution and from instructor to instructor. Topics such as quadratic forms, partitioned matrices, and generalized inverses that play an important role in the study of linear statistical models may be covered inadequately if at all. An additional difﬁculty is that sev- eral years may have elapsed between the completion of the prerequisite course on matrix (or linear) algebra and the beginning of the course on linear statistical models. This book is about matrix algebra. A distinguishing feature is that the content, the ordering of topics, and the level of generality are ones that I consider appro- priate for someone with an interest in linear statistical models and perhaps also The content of the paperback version is essentially the same as that of the earlier, hard- cover version. The paperback version differs from the earlier version in that a number of (mostly minor) corrections and alterations have been incorporated. In addition, the typography has been improved—as a side effect, the content and the numbering of the individual pages differ somewhat from those in the earlier version.

vi Preface for someone with an interest in another area of statistics or in a related discipline. I have tried to keep the presentation at a level that is suitable for anyone who has had an introductory course in matrix (or linear) algebra. In fact, the book is essentially self-contained, and it is hoped that much, if not all, of the material may be comprehensible to a determined reader with relatively little previous exposure to matrix algebra. To make the material readable for as broad an audience as pos- sible, I have avoided the use of abbreviations and acronyms and have sometimes adopted terminology and notation that may seem more meaningful and familiar to the non-mathematician than those favored by mathematicians. Proofs are provided for essentially all of the results in the book. The book includes a number of results and proofs that are not readily available from standard sources and many others that can be found only in relatively high-level books or in journal articles. The book can be used as a companion to the textbook in a course on linear statistical models or on a related topic—it can be used to supplement whatever results on matrices may be included in the textbook and as a source of proofs. And, it can be used as a primary or supplementary text in a second course on matrices, including a course designed to enhance the preparation of the students for a course or courses on linear statistical models and/or related topics. Above all, it can serve as a convenient reference book for statisticians and for various other professionals. While the motivation for the writing of the book came from the statistical ap- plications of matrix algebra, the book itself does not include any appreciable dis- cussion of statistical applications. It is assumed that the book is being read because the reader is aware of the applications (or at least of the potential for applications) or because the material is of intrinsic interest—this assumption is consistent with the uses discussed in the previous paragraph. (In any case, I have found that the discussions of applications that are sometimes interjected into treatises on matrix algebra tend to be meaningful only to those who are already knowledgeable about the applications and can be more of a distraction than a help.) The book has a number of features that combine to set it apart from the more tra- ditional books on matrix algebra—it also differs in signiﬁcant respects from those matrix-algebra books that share its (statistical) orientation, such as the books of Searle (1982), Graybill (1983), and Basilevsky (1983). The coverage is restricted to real matrices (i.e., matrices whose elements are real numbers)—complex matrices (i.e., matrices whose elements are complex numbers) are typically not encountered in statistical applications, and their exclusion leads to simpliﬁcations in terminol- ogy, notation, and results. The coverage includes linear spaces, but only linear spaces whose members are (real) matrices—the inclusion of linear spaces facili- tates a deeper understanding of various matrix concepts (e.g., rank) that are very relevant in applications to linear statistical models, while the restriction to linear spaces whose members are matrices makes the presentation more appropriate for the intended audience. The book features an extensive discussion of generalized inverses and makes heavy use of generalized inverses in the discussion of such standard topics as the solution of linear systems and the rank of a matrix. The discussion of eigenvalues

Preface vii and eigenvectors is deferred until the next-to-last chapter of the book—I have found it unnecessary to use results on eigenvalues and eigenvectors in teaching a ﬁrst course on linear statistical models and, in any case, ﬁnd it aesthetically displeasing to use results on eigenvalues and eigenvectors to prove more elementary matrix results. And the discussion of linear transformations is deferred until the very last chapter—in more advanced presentations, matrices are regarded as subservient to linear transformations. The book provides rather extensive coverage of some nonstandard topics that have important applications in statistics and in many other disciplines. These in- clude matrix differentiation (Chapter 15), the vec and vech operators (Chapter 16), the minimization of a second-degree polynomial (in n variables) subject to linear constraints (Chapter 19), and the ranks, determinants, and ordinary and general- ized inverses of partitioned matrices and of sums of matrices (Chapter 18 and parts of Chapters 8, 9, 13 16, 17, and 19). An attempt has been made to write the book in such a way that the presentation is coherent and non-redundant but, at the same time, is conducive to using the various parts of the book selectively. With the obvious exception of certain of their parts, Chapters 12 through 22 (which comprise approximately three-quarters of the book’s pages) can be read in arbitrary order. The ordering of Chapters 1 through 11 (both relative to each other and relative to Chapters 12 through 22) is much more critical. Nevertheless, even Chapters 1 through 11 include sections or subsections that are prerequisites for only a small part of the subsequent material. More often than not, the less essential sections or subsections are deferred until the end of the chapter or section. The book does not address the computational aspects of matrix algebra in any systematic way, however it does include descriptions and discussion of certain computational strategies and covers a number of results that can be useful in dealing with computational issues. Matrix norms are discussed, but only to a limited extent. In particular, the coverage of matrix norms is restricted to those norms that are deﬁned in terms of inner products. In writing the book, I was inﬂuenced to a considerable extent by Halmos’s (1958) book on ﬁnite-dimensional vector spaces, by Marsaglia and Styan’s (1974) paper on ranks, by Henderson and Searle’s (1979, 1981b) papers on the vec and vech operators, by Magnus and Neudecker’s (1988) book on matrix differential calculus, and by Rao and Mitra’s (1971) book on generalized inverses. And I ben- eﬁted from conversations with Oscar Kempthorne and from reading some notes (on linear systems, determinants, matrices, and quadratic forms) that he had pre- pared for a course (on linear statistical models) at Iowa State University. I also beneﬁted from reading the ﬁrst two chapters (pertaining to linear algebra) of notes prepared by Justus F. Seely for a course (on linear statistical models) at Oregon State University. The book contains many numbered exercises. The exercises are located at (or near) the ends of the chapters and are grouped by section—some exercises may require the use of results covered in previous sections, chapters, or exercises. Many of the exercises consist of verifying results supplementary to those included in the body of the chapter. By breaking some of the more difﬁcult exercises into parts

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