Matrix partial orders, shorted operators and applications

MATRIX PARTIAL ORDERS, SHORTED OPERATORS AND APPLICATIONS

SERIES IN ALGEBRA Series Editors: Derek J S Robinson (University of Illinois at Urbana-Champaign, USA) John M Howie (University of St. Andrews, UK) W Douglas Munn (University of Glasgow, UK)

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Sylow Theory, Formations and Fitting Classes in Locally Finite Groups by Martyn R Dixon

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Groups with Prescribed Quotient Groups and Associated Module Theory by L Kurdachenko, J Otal and I Subbotin

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Ring Constructions and Applications by Andrei V Kelarev

Vol. 10 Matrix Partial Orders, Shorted Operators and Applications by Sujit Kumar Mitra*, P Bhimasankaram and Saroj B Malik

*

Deceased

SERIES

MATRIX PARTIAL ORDERS, SHORTED OPERATORS AND APPLICATIONS

IN

ALGEBRA VOLUME 10

Sujit Kumar Mitra Indian Statistical Institute, India

P Bhimasankaram University of Hyderabad, India

Saroj B Malik Hindu College, University of Delhi, India

World Scientific NEW JERSEY

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LONDON

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SINGAPORE

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BEIJING

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SHANGHAI

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HONG KONG

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TA I P E I

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CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office 57 Shelton Street, Covent Garden, London WC2H 9HE

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Series in Algebra — Vol. 10 MATRIX PARTIAL ORDERS, SHORTED OPERATORS AND APPLICATIONS Copyright © 2010 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN-13 978-981-283-844-5 ISBN-10 981-283-844-9

Printed in Singapore.

To Asha Latha Mitra, Sunil Kumar Mitra and Sheila Mitra - Parents and Wife of Sujit Kumar Mitra

Preface Matrix orders have fascinated mathematicians and applied scientists alike for many years. Several matrix orders have been developed during the past four decades by researchers working in Linear Algebra. Some of them are pre-orders such as the space pre-order and the Drazin pre-order, while others are partial orders like the minus order, the sharp order and the star order. These developments have grown symbiotically with advances in other areas such as Statistics and Matrix Generalized Inverses. Two closely connected concepts - parallel sums and shorted operators for non-negative definite matrices - play a major role in the study of electrical networks. Both of these share nice relationships with matrix partial orders as also between themselves. Several extensions of these concepts have been developed in the recent past by researchers working in these and related areas. There are many research articles in the areas of matrix orders and shorted operators that are scattered in various journals. This is the first full length monograph on these topics. The aim of this monograph is to present the developments in the fields of matrix orders and shorted operators for finite matrices in a unified way and illustrate them with suitable applications in Generalized Inverses, Statistics and Electrical Networks. In the process of this compilation, many new results have evolved. Virtually every chapter in the monograph contains results unpublished hitherto. In fact, Chapter 13 on partial orders of modified matrices comprises entirely of new material. We believe that dissection of matrices through matrix decompositions helps in clear understanding of the anatomy of various matrix orders in a transparent manner. We employ simultaneous decompositions such as the simultaneous normal form, the simultaneous singular value decomposition and the generalized singular value decomposition extensively in developing the properties of matrix orders. Accordingly, the reader will find new and

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Matrix Partial Orders, Shorted Operators and Applications

intuitive proofs to several known results in this monograph. We also pose some open problems which should be of interest to researchers in these topics. There are a number of exercises at the end of virtually every chapter. These are expected to serve the dual purpose of helping in the understanding of the topics covered in the text and of introducing other related results that have not been included in the main text. This monograph is aimed at (i) graduate students and researchers in Matrix Theory and (ii) researchers in Statistics and Electrical Engineering who may use these concepts and results as tools in their work. The monograph can be used as a text for a one-semester graduate course in advanced topics in Matrix Theory and it can also serve as self-study text for those who have knowledge in basic Linear Algebra, perhaps at the level of [Rao and Bhimasankaram (2000)]. We deliberately avoided the study of majorization as there is an excellent book [Marshall and Olkin (1979)] on this topic. In this monograph, we consider matrix orders and shorted operators for finite matrices over a field. We do not consider matrices over more general algebraic structures or operators over more general spaces. There has been a good deal of work in these directions - see [Drazin (1958)], [Hartwig (1979)], [Hartwig (1980)], [Jain, Srivastava, Blackwood and Prasad (2009)], [Morley and William (1990a,b)] and [Mitch (1986)], to name a few. We have not included them here in order to keep the monograph focused and less bulky. We have also not gone into order preserving and order reversing transformations (for the minus, star, sharp and one-sided orders) and application of the sharp order in the analysis of Markov chains. These are exciting topics of research that may develop over the next few years. This monograph was conceived by the first author, S. K. Mitra who was a pioneer in the development of theory of matrix partial orders and shorted operators. He introduced the sharp order, one-sided orders and a unified theory of matrix partial orders. He developed several approaches to the shorted operators along with applications in Generalized Inverses and Statistics. The first author who is no more, has been the mentor and the main source of inspiration for the other two authors of the monograph. However, we, the second and third authors are responsible for any errors in the monograph. July, 2009

P. Bhimasankaram and Saroj B. Malik

Acknowledgements

We are highly thankful to the SQC & OR Unit, Indian Statistical Institute, Hyderabad, the Department of Mathematics and Statistics, University of Hyderabad, Hyderabad, the Center for Analytical Finance, Indian School of Business, Hyderabad and Hindu College, Delhi University, Delhi for providing excellent facilities during different stages of the preparation of this monograph. We are deeply indebted to Professor Debasis Sengupta for his constructive comments on parts of the manuscript in its different stages of preparation which led to significant improvement in the presentation. We are also thankful to Professors Probal Chaudhuri, Thomas Mathew and PSSNV Prasada Rao for useful discussions. We thank Professors T. Amarnath, R. Tandon, V. Suresh and Shri ALN Murthy for their encouragement during the preparation of the manuscript. P. Bhimasankaram records his sincere thanks and appreciation for the encouragement and inspiration received from Professor Sankar De during the preparation of the manuscript. His family members, Amrita, Chandana, Shilpa, Chandu and Vijaya had to miss his company even during the weekends for years together during the preparation of the manuscript. But for the understanding, patience and cooperation received from them, he could never have completed the task satisfactorily. He expresses his deep sense of loving appreciation to them. Saroj Malik expresses her sincere thanks to the Department of Mathematics and Statistics, University of Hyderabad for providing local hospitality during her visits to University of Hyderabad during the preparation of the manuscript (under the SAP-UGC project). She also wishes to thank the National Academy of Sciences, Delhi for partial support during one such visit. While working on the monograph, she had several useful discussions

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Matrix Partial Orders, Shorted Operators and Applications

with Professor Ajeet I. Singh, Professor Emeritus, Stat.-Math. Division, Indian Statistical Institute, Delhi. She is deeply indebted to Professor Singh for all the help and encouragement. She wishes to thank her ex-Principal Dr. Kavita Sharma for her encouragement during all these years to bring this effort to a fruitful ending. We thank Manpreet Singh and Vishal Mangla for their help in preparing the S-diagram and Naveen Reddy for his help in editing the manuscript at various stages. We thank L. F. Kwong for her co-operation at every stage during the preparation of the manuscript. We also thank D. Rajesh Babu for the technical help. Finally, we wish to thank World Scientific Publishing Co., Singapore for providing us the necessary freedom and flexibility in completing the monograph.

Glossary of Symbols and Abbreviations

aij (aij ) At A? C(A) C(At ) C(A⊥ ) N (A) tr(A) ρ(A) d(S) det(A) A−1 L A−1 R A− {A− } A− r {A− r } A− com {A− com } A† A] AD A− ` {A− ` } A− m {A− m} A− ρ

- the (i, j)th element of the matrix A - the matrix whose (i, j)th element is aij - transpose of the matrix A - conjugate transpose of the matrix A - the column space of the matrix A - the row space of the matrix A - the orthogonal complement of C(A) - the null space of the matrix A - the trace of the matrix A - the rank of the matrix A - the dimension of the subspace S - the determinant of the matrix A - a left inverse of the matrix A - a right inverse of the matrix A - a g-inverse of the matrix A - the set of g-inverses of the matrix A - a reflexive g-inverse of the matrix A - the set of reflexive g-inverses of the matrix A - a commuting g-inverse of A - the set of commuting g-inverses of A - the Moore-Penrose inverse of the matrix A - the group inverse of the matrix A - the Drazin inverse of the matrix A - a least squares g-inverse of the matrix A - the set of least squares g-inverses of the matrix A - a minimum norm g-inverse of the matrix A - the set of minimum norm g-inverses of the matrix A - a commuting g-inverse of the matrix A with the propxi

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Matrix Partial Orders, Shorted Operators and Applications t

{A− ρ} A− χ {A− χ} λmax (A) λmin (A) σ(A) C R F Cn m×n C Fn d(V ) I1 I1,n F m×n H Hn I Ik diag(x1 , x2 , . . . , xn ) P(A|B) S(A|B) S(A|S, T )

G(A) G(A, B) Gr (A) ] G(A) ˜ Gr (A) P(F m×n ) m-column vector n-row vector

t erty that C(A− ρ ) ⊆ C(A ) - the set of all ρ-inverses of the matrix A - a commuting g-inverse of the matrix A with the property that C(A− χ ) ⊆ C(A) - the set of all χ-inverses of the matrix A - the maximum eigen-value of A - the minimum eigen-value of A - the set of all singular values A - the field of complex numbers - the field of real numbers - arbitrary field - the vector space of complex n-tuples - the set of all m × n matrices over C - the vector space of n-tuples over F - the dimension of vector space V - the set of all matrices of index ≤ 1 - the set of all n × n matrices of index ≤ 1 - the set of all m × n matrices over F - the set of all hermitian matrices - the set of all n × n hermitian matrices - the identity matrix - the k × k identity matrix - the n × n diagonal matrix with diagonal elements xi , i = 1, 2, . . . , n - the parallel sum of matrices A, B - the shorted matrix of A with respect to the matrix B - the shorted matrix of an m×n matrix A with respect to the subspace S of F m and the subspace T of F n , also read as the shorted matrix of an m × n matrix A indexed by subspaces S and T - a subset of {A− } - the set of {B− AB− : B− ∈ G(B)} - the set G(A) ∩ {A− r } - completion of G(A) ] ∩ {A− } - the set G(A) r - set of all subset of F m×n - an m × 1 matrix - a 1 × n matrix

Contents

Preface

vii

Acknowledgements

ix

Glossary of Symbols and Abbreviations

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1.

Introduction

1

1.1 Matrix orders . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Parallel sum and shorted operator . . . . . . . . . . . . . 1.3 A tour through the rest of the monograph . . . . . . . . . 2.

Matrix Decompositions and Generalized Inverses 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

3.

Introduction . . . . . . . . . . . . . . . . . Matrix decompositions . . . . . . . . . . . Generalized inverse of a matrix . . . . . . The group inverse . . . . . . . . . . . . . Moore-Penrose inverse . . . . . . . . . . . Generalized inverses of modified matrices Simultaneous diagonalization . . . . . . . Exercises . . . . . . . . . . . . . . . . . .

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The Minus Order 3.1 3.2 3.3 3.4

1 3 4

9 10 17 26 36 46 55 64 67

Introduction . . . . . . . . . . . . . . . . . . . . Space pre-order . . . . . . . . . . . . . . . . . . Minus order - Some characterizations . . . . . . Matrices above/below a given matrix under the minus order . . . . . . . . . . . . . . . . . . . . xiii

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67 68 72

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Matrix Partial Orders, Shorted Operators and Applications

3.5 Subclass of g-inverses A− of A such and AA− = BA− when A