Generalized Inverses: Theory and Computations

This book begins with the fundamentals of the generalized inverses, then moves to more advanced topics.It presents a theoretical study of the generalization of Cramer's rule, determinant representations of the generalized inverses, reverse order law of th

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Guorong Wang · Yimin Wei  Sanzheng Qiao

Generalized Inverses: Theory and Computations

Developments in Mathematics Volume 53

Series editors Krishnaswami Alladi, Gainesville, USA Hershel M. Farkas, Jerusalem, Israel

More information about this series at http://www.springer.com/series/5834

Guorong Wang Yimin Wei Sanzheng Qiao •

Generalized Inverses: Theory and Computations

123

Guorong Wang Department of Mathematics Shanghai Normal University Shanghai China

Sanzheng Qiao Department of Computing and Software McMaster University Hamilton, ON Canada

Yimin Wei Department of Mathematics Fudan University Shanghai China

ISSN 1389-2177 ISSN 2197-795X (electronic) Developments in Mathematics ISBN 978-981-13-0145-2 ISBN 978-981-13-0146-9 (eBook) https://doi.org/10.1007/978-981-13-0146-9 Jointly Published with Science Press, Beijing, China The print edition is not for sale in China Mainland. Customers from China Mainland please order the print book from: Science Press, Beijing, China. Library of Congress Control Number: 2018938386 Mathematics Subject Classification (2010): 15A09, 65Fxx, 47A05 © Springer Nature Singapore Pte Ltd. and Science Press 2018 This work is subject to copyright. All rights are reserved by the Publishers, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. part of Springer Nature The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

The concept of the generalized inverses was first introduced by I. Fredholm [81] in 1903. He proposed a generalized inverse of an integral operator, called pseudoinverse. The generalized inverses of differential operators were brought up in D. Hilbert’s [107] discussion of the generalized Green’s functions in 1904. For a history of the generalized inverses of differential operators, the reader is referred to W.

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