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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 elementary 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. Erdélyi, 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