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Generalized Inverses

Theory and Applications

  • Textbook
  • © 2003

Overview

Authors:
  1. Adi Ben-Israel

    1. RUTCOR-Rutgers Center for Operations Research, Rutgers University, Piscataway, USA

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  2. Thomas N. E. Greville
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Part of the book series: CMS Books in Mathematics (CMSBM)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xv
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  2. Introduction

    Pages 1-5

  3. Preliminaries

    Pages 6-39

  4. Existence and Construction of Generalized Inverses

    Pages 40-51

  5. Linear Systems and Characterization of Generalized Inverses

    Pages 52-103

  6. Minimal Properties of Generalized Inverses

    Pages 104-151

  7. Spectral Generalized Inverses

    Pages 152-174

  8. Generalized Inverses of Partitioned Matrices

    Pages 175-200

  9. A Spectral Theory for Rectangular Matrices

    Pages 201-256

  10. Computational Aspects of Generalized Inverses

    Pages 257-281

  11. Miscellaneous Applications

    Pages 282-329

  12. Generalized Inverses of Linear Operators between Hilbert Spaces

    Pages 330-369

  13. Back Matter

    Pages 370-420
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Keywords

About this book

1. The Inverse of a Nonsingular Matrix It is well known that every nonsingular matrix A has a unique inverse, ?1 denoted by A , such that ?1 ?1 AA = A A =I, (1) where I is the identity matrix. Of the numerous properties of the inverse matrix, we mention a few. Thus, ?1 ?1 (A ) = A, T ?1 ?1 T (A ) =(A ) , ? ?1 ?1 ? (A ) =(A ) , ?1 ?1 ?1 (AB) = B A , T ? where A and A , respectively, denote the transpose and conjugate tra- pose of A. It will be recalled that a real or complex number ? is called an eigenvalue of a square matrix A, and a nonzero vector x is called an eigenvector of A corresponding to ?,if Ax = ?x. ?1 Another property of the inverse A is that its eigenvalues are the recip- cals of those of A. 2. Generalized Inverses of Matrices A matrix has an inverse only if it is square, and even then only if it is nonsingular or, in other words, if its columns (or rows) are linearly in- pendent. In recent years needs have been felt in numerous areas of applied mathematics for some kind of partial inverse of a matrix that is singular or even rectangular.

Reviews

From the reviews of the second edition:

"The book under review which is the second edition of the 30 years ago published one provides a detailed survey of generalized inverses and their main properties … . An important feature of this book is the over 600 exercises … . Each chapter ends with the section ‘Suggested further reading’. These sections provide excellent additional references on topics treated … . it can be used profitably by graduate or advanced undergraduate students of mathematics and computer science, and by PhD students … ." (Róbert Rajkó, Acta Scientiarum Mathematicarum, Vol. 71, 2005)

"Each chapter is accompanied by suggestions for further reading, while the bibliography contains 901 references. … The book contains 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 undergraduate students … ." (Nicholas Karampetakis, Zentralblatt MATH, Vol. 1026, 2004)

Authors and Affiliations

  • RUTCOR-Rutgers Center for Operations Research, Rutgers University, Piscataway, USA

    Adi Ben-Israel

About the authors

 

Adi Ben-Israel is Professor of Operations Research, Business and Mathematics at Rutgers University, New Brunswick, NJ. Previously he was Professor of Applied Mathematics at the University of Delaware, Northwestern University, and the Technion-Israel Institute of Technology.

The late Thomas N.E. Greville was Professor of Mathematics, and a member of the US Army Mathematics Research Center at the University of Wisconsin, Madison, WI.

Bibliographic Information

  • Book Title: Generalized Inverses

  • Book Subtitle: Theory and Applications

  • Authors: Adi Ben-Israel, Thomas N. E. Greville

  • Series Title: CMS Books in Mathematics

  • DOI: https://doi.org/10.1007/b97366

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 2003

  • Hardcover ISBN: 978-0-387-00293-4Published: 16 June 2003

  • Softcover ISBN: 978-1-4419-1814-7Published: 29 November 2010

  • eBook ISBN: 978-0-387-21634-8Published: 18 April 2006

  • Series ISSN: 1613-5237

  • Series E-ISSN: 2197-4152

  • Edition Number: 2

  • Number of Pages: XVI, 420

  • Additional Information: Originally published by Wiley-Interscience, 1974

  • Topics: Linear and Multilinear Algebras, Matrix Theory, Operator Theory, Numerical Analysis

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