Orthogonal Sets and Polar Methods in Linear Algebra: Applications to Matrix Calculations, Systems of Equations, Inequalities, and Linear Programming ... Wiley Series of Texts, Monographs and Tracts)

430 Pages·1999·6.69 MB·English

by  Enrique Castillo

Orthogonal Sets and
Polar Methods
in Linear Algebra
PURE AND APPLIED MATHEMATICS
A Wiley-Interscience Series of Texts, Monographs, and Tracts
Founded by RICHARD COURANT
Editor Emeritus: PETER HILTON and HARRY HOCHSTADT
Editors: MYRON B. ALLEN III, DAVID A. COX, PETER LAX,
JOHN TOLAND
A complete list of the titles in this series appears at the end of this volume.
Orthogonal Sets and
Polar Methods
in Linear Algebra
Applications to Matrix Calculations,
Systems of Equations, Inequalities, and
Linear Programming
ENRIQUE CASTILLO, ANGEL COBO,
FRANCISCO JUBETE, and ROSA EVA PRUNEDA
A Wiley-Interscience Publication
JOHN WILEY & SONS, INC.
New York / Chichester / Weinheim / Brisbane / Singapore / Toronto
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Copyright © 1999 by John Wiley & Sons, Inc.
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Library of Congress Cataloging in Publication Data:
Orthogonal sets and polar methods in linear algebra : applications to
matrix calculations, systems of equations, inequalities, and linear
programming / Enrique Castillo... [et al.].
p. cm. — (Pure and applied mathematics)
" AWiley-Interscience publication."
Includes index.
ISBN 0-471-32889-8 (acid-free paper)
1. Algebras, Linear. 2. Orthogonalization methods. I.Castillo,
Enrique, 1946- II. Series: Pure and applied mathematics (John
Wiley & Sons : Unnumbered)
QA 184.077 1999
512\5—dc21 98-38391
CIP
Printed in the United States of America
10 9 8 7 6 5 4 3 21
Preface
Inverting matrices, calculating determinants, solving linear systems of equa-
tions and/or inequalities, and linear programming problems are mathematical
problems one has to deal with very frequently. One is prepared to solve them
using many available tools, procedures, or computer packages. In fact, these
simple problems are studied in detail in standard courses. However, one may
not be prepared to solve other frequently appearing problems such as:
1. Calculating the inverse of a matrix after modifying, removing, and/or
adding rows or columns, based on the inverse of the initial matrix.
2. Calculating the inverse of a symbolic matrix.
3. Solving systems of linear equations in some selected variables.
4. Obtaining the solution of a system of linear equations when some equa-
tions and/or variables are changed, removed, or added, without starting
from scratch, that is, using the solution of the initial system.
5. Analyzing the compatibility of systems of linear equations and inequal-
ities with restricted variables.
In this book we give methods that allow solving a much wider range of
problems, which, apart from the above, include systems of linear equations
and inequalities with unrestricted, restricted, or strictly restricted variables.
This allows us to solve very interesting practical problems.
In this book we do not deal with the standard methods of analysis, which
are very well covered in many other books. Instead, we present new methods
v
vi PREFACE
and algorithms for solving standard and new problems based on the concept
of orthogonality.
When analyzing and discussing several problems in linear algebra involving
linear spaces, one possibility is to analyze them using the appropriate defi-
nitions and concepts directly, and stating the conditions to be satisfied by
the problem under consideration. This is referred to as the primal approach.
Alternatively, one can deal with orthogonal sets and state the corresponding
conditions using the concept of orthogonality. This, the dual approach, which
initially seems to be much more involved but is computationally simpler and
more elegant and appealing, will be used in this book.
Thus, this book does not deal with standard methods or solutions. The
reader looking for standard methods or references to published works with
this orientation should consult one of the very many books on the topic (see
References). On the contrary, in this book the above problems are discussed
from a different point of view and the corresponding methods are given.
In addition to obtaining solutions, mathematicians and engineers are inter-
ested in analyzing conditions leading to well-defined problems. In this context,
the problems of compatibility and uniqueness of solutions play a central role.
In fact, they lead to very interesting physical or engineering conclusions, which
relate the physical conditions, obtained from reality, and the corresponding
conditions behind the mathematical models.
This book is organized in four parts. In Part I we deal with linear spaces,
matrices, and systems of equations (including the problems of compatibility
and uniqueness) and obtaining their solutions.
Part II deals with cones and systems of linear equations and inequalities
in unrestricted, restricted, and strictly restricted variables, where again the
problems of compatibility, uniqueness of solution, and obtaining solutions are
treated.
We also show that a linear space of finite dimension is simply a particular
case of a cone. Thus, the cone concept is broader than the linear space
concept, and, what is more important, it offers the possibility of replacing the
usual methods of classical linear algebra with the specific methods of cones
and polarity, which are more efficient.
Part III is devoted to linear programming problems. However, instead of
describing well-known methods, such as the simplex, the primal-dual, and
the interior point methods, we present the new exterior point method, which
progresses from outside the feasible region.
In addition to this innovative method, we also discuss how the solution
of a linear programming problem can be obtained when removing or adding
constraints and/or variables, without starting from the very beginning and
thus eliminating a huge amount of computation.
Part IV includes a wide collection of examples, which are treated by a
mixture of mathematics, physics, and engineering points of view. Examples
such as the matrix analysis of structures, the transportation and production-
PREFACE vii
scheduling problems, the input-output tables, the diet problem and the net-
work flow problems are used to illustrate concepts and motivate applications.
All of the above models have arisen from real life. In other words, they
are mathematical models of physical, economic, or engineering problems. Se-
lection of the adequate model reproducing the reality is a crucial step for a
satisfactory solution to a real problem. The mathematical structures are not
arbitrary, but a consequence of reality itself. In this book we make a great
effort to connect physical and mathematical realities. We show the reader
the reasoning that leads to the analysis of the different structures and con-
cepts. This becomes apparent in the illustrative examples, which show the
connection between model and reality.
This book can be used as a reference or consulting book and as a textbook
in upper-division undergraduate courses or in graduate-level courses. Included
in the book are numerous illustrative examples and end-of-chapter exercises.
We have also developed some computer programs in Java and Mathematica to
implement the various algorithms and methodologies presented in this book.
The current version of these programs, together with a brief User's Guide,
can be obtained from the World Wide Web site
http ://ccaix3.unican.es/~AIGroup/Orthogonal.html.
This book is addressed to a wide audience, including mathematicians, en-
gineers, and applied scientists. There are few prerequisites for the reader,
though a previous knowledge of linear algebra and some familiarity with ma-
trices are essential.
Several colleagues read earlier versions of the manuscript for this book
and have provided us with valuable comments and suggestions. Their con-
tributions have given rise to the current substantially improved version. In
particular, we acknowledge the help of Ali S. Hadi and Juan Angel Diaz Her-
nando, who made a very careful reading of the manuscript and gave many
suggestions.
Special thanks are given to the University of Cantabria, Iberdrola and José
Antonio Garrido for their finantial support.
ENRIQUE CASTILLO, ANGEL COBO,
FRANCISCO JUBETE, AND ROSA EVA PRUNEDA
Santander, Spain
January 1999
Contents
Part I Linear Spaces and Systems of Equations
Basic Concepts 5
1.1 Introduction 5
1.2 Linear space 6
1.3 The Euclidean Space En 11
l.A Orthogonal Sets and Decompositions U
1.5 Matrices 16
1.6 Systems of Linear Equations 20
Exercises 21
Orthogonal Sets 23
2.1 Introduction and Motivation 23
2.2 Orthogonal Decompositions 24
2.3 The Orthogonalization Module 31
2.1 Mathematica Program 36
Exercises 39
Matrix Calculations Using Orthogonal Sets 41
ix
x CONTENTS
3.1 Introduction 41
3.2 Inverting a Matrix 4%
3.3 The Rank of a Matrix 43
3.4 Calculating the Determinant of a Matrix 44
3.5 Algorithm for Matrix Calculations 46
3.6 Complexity 50
3.7 Inverses and Determinants of Row-Modified
Matrices 51
3.8 Inverses of Symbolic Matrices 55
3.9 Extensions to Partitioned Matrices 56
3.10 Inverses of Modified Matrices 60
3.11 Mathematica Programs 67
Exercises 69
More Applications of Orthogonal Sets 73
4-1 Intersection of Two Linear Subspaces 73
4-2 Reciprocals Images in Linear Transformations 77
4-3 Other Applications 79
4-4 Mathematica Programs 81
Exercises 83
Orthogonal Sets and Systems of Linear Equations 85
5.1 Introduction 85
5.2 Compatibility of a System of Linear Equations 86
5.3 Solving a System of Linear Equations 90
5.4 Complexity 95
5.5 Checking Systems Equivalence 96
5.6 Solving a System in Some Selected Variables 99
5.7 Modifying Systems of Equations 101
5.8 Applications 124
5.9 Mathematica Programs 135
Exercises 137
Appendix: Proof of Lemma 5.2 140
Part II Cones and Systems of Inequalities
6 Polyhedral Convex Cones 147
6.1 Introduction 147