Elements of the Theory of Generalized Inverses of Matrices
The purpose of this monograph is to provide a concise introduction to the theory of generalized inverses of matrices that is accessible to undergraduate mathematics majors. Although results from this active area of research have appeared in a number of ex
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ELEMENTS OF THE THEORY OF GENERALIZED INVERSES FOR MATRICES Randall E. Cline Mathematics Department University of Tennessee Knoxville, Tennessee 37916
The Project acknowledges Robert M. Thrall, Chairman of the UMAP Monograph Editorial Board, for his help in the development and review of this monograph.
Modules and Monographs in Undergraduate Mathematics and its Applications Project The goal of UMAP is to develop, through a community of users and developers, a system of instructional modules and monographs in undergraduate mathematics which may be used to supplement existing courses and from which complete courses may eventua1ly be bnilt. The Project is gnided by a National Steering Committee of mathematicians, scientists, and educators. UMAP is funded by a grant from the National Science Foundation to Education Development Center, Inc., a publicly supported, nonprofit corporation engaged in educational research in the U.S. and abroad. The Project acknowledges the help of the Monograph Editorial Board in the development and review of this monograph. Members of the Monograph Editorial Board include: Robert M. Thrall, Chairman, of Rice University; Clayton Aucoin, Clemson University; James C. Frauenthal, SUNY at Stony Brook; Helen Marcus-Roberts, Montclair State College; Ben Noble, University of Wisconsin; Paul C. Rosenbloom, Columbia University. Ex-officio members: Michael Anbar, SUNY at Buffalo; G. Robert Boynton, University of Iowa; Kenneth R. Rebman, California State University; Carroll O. Wilde, Naval Postgraduate School; Douglas A. Zahn, Florida State University. The Project wishes to thank Thomas N.E. Greville of the University of Wisconsin at Madison for his review of this manuscript and Edwina R. Michener, UMAP Editorial Consultant, for her assistance. Project administrative staff: Ross L. Finney, Director; Solomon Garfunkel, Associate Director/Consortium Coordinator; Felicia DeMay, Associate Director for Administration; Barbara Kelczewski, Coordinator for Materials Production.
ISBN-13: 978-0-8176-3013-3 DOl: 10.1007/978-1-4684-6717-8
e-ISBN-13: 978-1-4684-6717-8
Copyright©1979 by Education Development Center, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.
TABLE OF CONTENTS
1.
2.
Preface .
v
INTRODUCTION
1
1.1
Preliminary Remarks
1
1.2 1.3
Matrix Notation and Terminology A Rationale for Generalized Inverses
2 4
SYSTEMS OF EQUATIONS AND THE MOORE-PENROSE INVERSE OF A MATRIX . . . . . . . 2.1 2.2 2.3 2.4
3.
4.
5.
Zero, One or Many Solutions of Ax Full Rank Factorizations and the Moore-Penrose Inverse Some Geometric Illustrations Miscellaneous Exercises
=
b
9 9
12 21 27
MORE ON MOORE-PENROSE INVERSES
29
3.1 3.2 3.3
29 40 56
Basic Properties of A+ . . Applications with Matrices of Special Structure Miscellaneous Exercises
DRAZIN INVERSES 4.1 The Drazin Inverse of a
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