Theory of generalized inverses over commutative rings

The Theory of Generalized Inverses Over Commutative Rings

Algebra, Logic and Applications A series edited by R. Göbel Universität Gesamthochschule, Essen, Germany A. Macintyre University of Edinburgh, UK

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The Theory of Generalized Inverses Over Commutative Rings

K.P.S. Bhaskara Rao Southwestern College, Winfield, Kansas, USA

London and New York

First published 2002 by Taylor & Francis 11 New Fetter Lane, London EC4P 4EE Simultaneously published in the USA and Canada by Taylor & Francis Inc., 29 West 35th Street, New York, NY 10001 Taylor & Francis is an imprint of the Taylor & Francis Group This edition published in the Taylor & Francis e-Library, 2005. “To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.”

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ISBN 0-203-21887-6 Master e-book ISBN

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for two great kids my son, Swastik and my daughter, Swara

Contents

Foreword Preface

ix x

1 Elementary results on rings

1

2 Matrix algebra over rings 2.1 Elementary notions 2.2 Determinants 2.3 The ∂/∂aij notation 2.4 Rank of a matrix 2.5 Compound matrices

5 5 8 9 10 11

3 Regular elements in a ring 3.1 The Moore-Penrose equations 3.2 Regular elements and regular matrices 3.3 A Theorem of Von Neumann 3.4 Inverses of matrices 3.5 M-P inverses

15 15 16 19 20 25

4 Regularity – principal ideal rings 4.1 Some results on principal ideal rings 4.2 Smith Normal Form Theorem 4.3 Regular matrices over Principal Ideal Rings 4.4 An algorithm for Euclidean domains 4.5 Reflexive g-inverses of matrices 4.6 Some special integral domains 4.7 Examples

29 29 32 38 41 45 46 50

5 Regularity – basics 5.1 Regularity of rank one matrices 5.2 A basic result on regularity 5.3 A result of Prasad and Robinson

61 61 62 66

6 Regularity – integral domains 6.1 Regularity of matrices 6.2 All reflexive g-inverses

73 73 74 vii

viii

CONTENTS

6.3 6.4 6.5 6.6 6.7

M-P inverses over integral domains Generalized inverses of the form PCQ {1,2,3}- and {1,2,4}-inverses Group inverses over integral domains Drazin inverses over integral domains

77 82 85 89 93

7 Regularity – commutative rings 7.1 Commutative rings with zero divisors 7.2 Rank one matrices 7.3 Rao-regular matrices 7.4 Regular matrices over commutative rings 7.5 All generalized inverses 7.6 M-P inverses over commutative rings 7.7 Group inverses over commutative rings 7.8 Drazin inverses over commutative rings

101 101 102 106 114 119 122 123 124

8 Special topics 8.1 Generalized Cramer Rule 8.2 A rank condition for consistency 8.3 Minors of reflexive g-inverses 8.4 Bordering of regular matrices 8.5 Regularity over Banach algebras 8.6 Group inverses in a ring 8.7 M-P inverses in a ring 8.8 Group inverse of the companion matrix

127 127 129 130 135 141 144 146 148

Bibliography Index

153 167

Foreword

A possible definition of a “generalized inverse” of a linear operator A is an operator that has some useful inverse properties, and reduces to the inverse of A if A is invertible. The “useful properties” include solving linear equations Ax = a, selecting a particular (e.g. least norm) solution if the equation has more than one, or producing an approximate solution (e.g. a least squares solution) if the equation does not have any. Generalized inverses first arose in analysis in the study of integral equations (the “pseudo-inverse” of I. Fredholm 1903, and the “pseudo-resolvent” of W.A. Hurwitz, 1912). Although determinants, and limits of determinants, were used in some of these early studies (with linear operators being approximated by infinite matrices), the algebraic nature of generalized inverses was established later, in the works of E.H. Moore (1912, 1920, 1935), R. Penrose (1955), Drazin (1958) and others. Thousands of articles on generalized inverses appeared since Penrose’s seminal article, and most of them deal with matrices over the real and complex fields. This sufficed for many applications, and required only tools from linear algebra. Several researchers (notably D.R. Batigne, F.J. Hall, I.J. Katz, D.W. Robinson, R. Puystjens, R.B. Bapat, K.M. Prasad and K.P.S. Bhaskara Rao) studied generalized inverses in more general algebraic settings, fields and rings. The ever growing importance of discrete mathematics in modern applications have made their results timely. Professor K.P.S. Bhaskara Rao has collected the above results, until now scattered in the research literature, and added new ones, in this concise and wellwritten research monograph. I expect it to advance our knowledge of generalized inverses over fields and rings, by promoting and guiding future research. Adi Ben-Israel RUTCOR, Rutgers University USA

ix

Preface

My interest in the theory of generalized inverses was formed in my student days by my teacher Professor C.R. Rao. Since then I have learnt about many of the interesting features of g-inverses from the monographs of Ben-Israel and Greville and of Rao and Mitra. I was always more interested in the g-inverses of matrices over various algebraic structures than over classical real or complex fields. The present monograph is the result of my endeavor to present to the mathematical community various aspects of the theory of g-inverses of matrices over commutative rings. Though this subject is relatively young, it has many beautiful results and its development has reached a final and complete stage. The theory of generalized inverses of real or complex matrices is a well developed subject and the results of this theory have been chronicled in several monographs. See, for example, Generalized inverses: theory and applications by Adi Ben-Israel and Thomas N.E. Greville, Wiley (1974), Generalized inverse of matrices and its applications by C.R. Rao and S.K. Mitra, Wiley (1971) and the compendium Generalized inverses and applications edited by M.Z. Nashed, Academic Press (1976). In algebra, though the concept of regularity (that an element a in a ring R is regular if there is a g in R such that aga = a) was not new, it was studied very little with respect to matrices until about twenty years ago. In the mid thirties, Von Neumann proved that if every element of an associative ring R is regular then every matrix over R is regular. The problem of characterizing regular matrices over commutative rings was raised by the author in [20]. This became all the more important because of the interest of control theorists and systems theorists in polynomial matrices (see [8], [52], [93] and [94] and the references therein) and that of mathematicians working in operator algebras (see [43], [44], [46], [47] and [54] and the references therein). It was in the early eighties that problems were raised as to how much of the theory of g-inverses can be developed over the ring of integers; and the pursuit continued to principal ideal domains, integral domains and general commutative rings. I have presented the development as it happened giving the reader an insight into the intricacies of the subject. Mathematicians working in g-inverses of matrices, algebraists, system theorists and control theorists would be interested in the results presented here. Economists also deal with polynomial matrices and this monograph should be useful for them too. The results given here for matrices over Banach algebras would be of interest to mathematicians working in operator theory. x

PREFACE

xi

This monograph can be used to present a one or two semester course on g-inverses for final year undergraduate students who are interested in algebra. It can also form the basis for a sequel to algebra and linear algebra courses. There are several monographs on matrices over rings: Integral matrices by Morris Newman, Academic Press (1972). Linear algebra over commutative rings by B.R. McDonald, Marcel Dekker (1984), and Matrices over commutative rings by W.C. Brown, Marcel Dekker (1993). Besides being of independent interest the present monograph acts as a sequel to these monographs. Graduate students can utilize this to explore the yet unsolved problems (similar) for matrices over general associative rings. Several exercises, some of which are taken from the literature, are devised to enhance the understanding of the subject. A novel feature of this monograph is an annotated bibliography which also serves as “notes and comments” to the text. Dr. K.M. Prasad, many of whose results are also presented here, helped me in the initial stages of the preparation of this monograph. Professor Adi Ben-Israel and Professor D.W. Robinson kindly looked at the manuscript and I thank them for their constructive criticism and advice. Most of the work on this monograph was done while the author was at the Indian Statistical Institute, Bangalore and visiting North Dakota State University, Fargo. The author acknowledges the efforts of Mr. Dharmappa of ISIBC for wordprocessing this monograph. Dr. Surekha Rao, my wife, in spite of her own busy academic work, has been a constant source of encouragement to me throughout the preparation of this monograph. I am full of appreciation for her. K.P.S. Bhaskara Rao

Index

{1,3}-inverse 16 invertible matrix 20

adjoint 9 Annihilator 11 Associative Ring 1

Jacobi identity 133 bordering 135 Laplace expansion Theorem 8 left regular 18

Canonical Decomposition Theorem of Prasad 114 Cauchy-Binet Theorem 9 characteristic polynomial 92 commutative ring 1 commuting g-inverse 16 compound matrix 11 Cramer Rule 127

Moore-Penrose equations 15 Moore-Penrose inverse 16 M-P inverse 16 principal ideal domain 2 principal ideal ring 2 Projective free 48

Drazin inverse 16 rank factorization 24 rank function 130 Rao condition 51 Rao-idempotent 107 Rao-index 117 Rao-list of idempotents 117 Rao-list of ranks 117 Rao-regular 106 Rao-regular matrix 107 real closed field 59 regular 15, 17 regular inverse 15, 17 regular ring 19 right regular 18 Ring 1 Ring with an involution 1 Robinson’s Decomposition Theorem 116

Euclidean domain 2 Euclidean norm 2 formally real 58 Generalized Cramer Rule 129 generalized inverse 15, 17, 18 generalized Moore-Penrose inverse 98 g-inverse 15 greatest common divisor 2 group inverse 16 Hall matrix 157 ideal 2 idempotent 1 identity 1 index 93 inner inverse 15 integral domain 2 invariant factors 38 1-inverse 15 2-inverse 15

subring 1 symmetric involution 142 unimodular matrix 20 unit 1 zero divisors 1

167

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