Generalized Inverses: Theory and Applications

436 Pages·2003·1.923 MB·English

by  Adi Ben-Israel, Thomas N.E. Greville

Editors-in-Chief
Re´dacteurs-en-chef
Jonathan Borwein
Peter Borwein
This page intentionally left blank
Adi Ben-Israel Thomas N.E. Greville
Generalized Inverses
Theory and Applications
Second Edition
AdiBen-Israel ThomasN.E.Greville(deceased)
RUTCOR—RutgersCenterfor
OperationsResearch
RutgersUniversity
Piscataway,NJ08854-8003
USA
[email protected]
Editors-in-Chief
Re´dacteurs-en-chef
JonathanBorwein
PeterBorwein
CentreforExperimentalandConstructiveMathematics
DepartmentofMathematicsandStatistics
SimonFraserUniversity
Burnaby,BritishColumbiaV5A1S6
Canada
[email protected]
With1figure.
MathematicsSubjectClassification(2000):15A09,65Fxx,47A05
LibraryofCongressCataloging-in-PublicationData
Ben-Israel,Adi.
Generalizedinverses:theoryandapplications/AdiBen-Israel,ThomasN.E.Greville.—
2nded.
p.cm.—(CMSbooksinmathematics;15)
Includesbibliographicalreferencesandindex.
ISBN0-387-00293-6(alk.paper)
1.Matrixinversion. I.Greville,T.N.E.(ThomasNallEden),1910–1998 II.Title.
III.Series.
QA188.B462003
512.9′434—dc21 2002044506
ISBN0-387-00293-6 Printedonacid-freepaper.
FirsteditionpublishedbyWiley-Interscience,1974.
2003Springer-VerlagNewYork,Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the
writtenpermissionofthepublisher(Springer-VerlagNewYork,Inc.,175FifthAvenue,NewYork,
NY10010,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Use
inconnection withany formof informationstorageand retrieval,electronic adaptation,computer
software,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if
theyarenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornot
theyaresubjecttoproprietaryrights.
PrintedintheUnitedStatesofAmerica.
9 8 7 6 5 4 3 2 1 SPIN10905616
Typesetting:Pagescreatedbytheauthorsusing 2e.
www.springer-ny.com
Springer-Verlag NewYork Berlin Heidelberg
AmemberofBertelsmannSpringerScience+BusinessMediaGmbH
Preface to the Second Edition
The field of generalized inverses has grown much since the appearance of
the first edition in 1974 and is still growing. I tried to account for these
developments while maintaining the informal and leisurely style of the first
edition. New material was added, including a preliminary chapter (Chap-
ter 0), a chapter on applications (Chapter 8), an Appendix on the work of
E.H. Moore, and new exercises and applications.
While preparing this volume I compiled a bibliography on generalized
inverses,postedinthewebpageoftheInternational Linear Algebra Society
http://www.math.technion.ac.il/iic/research.html
Thison-linebibliography,containingover2000items,willbeupdatedfrom
time to time. For reasons of space, many important works that appear in
the on-line bibliography are not included in the bibliography of this book.
I apologize to the authors of these works.
Many colleagues helped this effort. Special thanks go to R. Bapat, S.
Campbell, J. Miao, S.K. Mitra, Y. Nievergelt, R. Puystjens, A. Sidi, G.-R.
Wang, and Y. Wei.
Tom Greville, my friend and coauthor, passed away before this project
started. His scholarship and style marked the first edition and are sadly
missed.
I dedicate this book with love to my wife Yoki.
Piscataway, New Jersey Adi Ben-Israel
January 2002
v
This page intentionally left blank
From the Preface to the First Edition
This book is intended to provide a survey of generalized inverses from a
unified point of view, illustrating the theory with applications in many ar-
eas. It contains more than 450 exercises at different levels of difficulty,
many of which are solved in detail. This feature makes it suitable either
for reference and self–study or for use as a classroom text. It can be used
profitably by graduate students or advanced undergraduates, only an ele-
mentary knowledge of linear algebra being assumed.
Thebookconsistsofanintroductionandeightchapters,sevenofwhich
treatgeneralizedinversesoffinitematrices,whiletheeighthintroducesgen-
eralized inverses of operators between Hilbert spaces. Numerical methods
are considered in Chapter 7 and in Section 9.7.
Whileworkingintheareaofgeneralizedinverses,theauthorshavehad
the benefit of conversations and consultations with many colleagues. We
would like to thank especially A. Charnes, R.E. Cline, P.J. Erdelsky, I.
Erd´elyi, J.B. Hawkins, A.S. Householder, A. Lent, C.C. MacDuffee, M.Z.
Nashed, P.L. Odell, D.W. Showalter, and S. Zlobec. However, any errors
that may have occurred are the sole responsibility of the authors.
This book is dedicated to Abraham Charnes and J. Barkley Rosser.
Haifa, Israel Adi Ben-Israel
Madison, Wisconsin Thomas N.E. Greville
September 1973
vii
This page intentionally left blank
Contents
Preface to the Second Edition v
From the Preface to the First Edition vii
Glossary of Notation xiii
Introduction 1
1. The Inverse of a Nonsingular Matrix 1
2. Generalized Inverses of Matrices 1
3. Illustration: Solvability of Linear Systems 2
4. Diversity of Generalized Inverses 3
5. Preparation Expected of the Reader 4
6. Historical Note 4
7. Remarks on Notation 5
Suggested Further Reading 5
Chapter 0. Preliminaries 6
1. Scalars and Vectors 6
2. Linear Transformations and Matrices 10
3. Elementary Operations and Permutations 22
4. The Hermite Normal Form and Related Items 23
5. Determinants and Volume 28
6. Some Multilinear Algebra 32
7. The Jordan Normal Form 34
8. The Smith Normal Form 38
9. Nonnegative Matrices 39
Suggested Further Reading 39
Chapter 1. Existence and Construction of Generalized Inverses 40
1. The Penrose Equations 40
2. Existence and Construction of {1}-Inverses 41
3. Properties of {1}-Inverses 42
4. Existence and Construction of {1,2}-Inverses 45
5. Existence and Construction of {1,2,3}-, {1,2,4}-, and
{1,2,3,4}-Inverses 46
6. Explicit Formula for A† 48
7. Construction of {2}-Inverses of Prescribed Rank 49
Notes on Terminology 51
Suggested Further Reading 51
ix