Table Of ContentMATRICES
AND
LINEAR
TRANSFORMATIONS
Second Edition
CHARLES G. CULLEN
University of Pittsburgh
DOVER PUBLICATIONS, INC., NEW YORK
Copyright © 1972 by Charles G. Cullen.
All rights reserved.
This Dover edition, first published in 1990, is an unabridged, corrected republication of the second
edition (1972) of the work originally published in 1966 by Addison-Wesley Publishing Company,
Reading, Massachusetts.
Library of Congress Cataloging-in-Publication Data
Cullen, Charles G.
Matrices and linear transformations / Charles G. Cullen.—2nd ed.
p. cm.—(Addison-Wesley series in mathematics) ISBN-13: 978-0-486-66328-9
ISBN-10: 0-486-66328-0
1. Matrices. 2. Transformations (Mathematics) I. Title. II. Series.
QA188.C85 1990
512.9'434—dc20
89-25677
CIP
Manufactured in the United States by Courier Corporation
66328008
www.doverpublications.com
Preface
to the
First
Edition
This textbook was originally developed for a one-term course in linear algebra
and matrix theory, which is offered each term at the University of Pittsburgh.
The students are mostly sophomores and juniors majoring in mathematics,
engineering, and the physical sciences. In addition, the course regularly attracts
freshman honors students as well as graduate students from the departments of
economics, public health, philosophy, education, as well as engineering and the
sciences.
My major purpose in writing this book was to provide a text, requiring a
minimum number of prerequisites, which would enable me, in a one-term
course, to cover these topics which are most frequently encountered in physical
applications. In addition to the standard material on systems of linear equations,
I wanted to include a fairly complete treatment of the characteristic value
problem, including a discussion of normal matrices, and culminating with a
discussion of the Jordan canonical form. Chapters 1 through 5 provide such a
text. For the first seven chapters only a first course in calculus and analytic
geometry is required and it is possible to study them simultaneously.
The major objects of study are matrices over an arbitrary field. Vector
spaces are introduced not only to help with this study but also because they are
of great interest in their own right. Even at the beginning level I have decided to
give the abstract definition of a vector space and to establish the isomorphism of
such systems to n-tuple spaces. Linear transformations on finite dimensional
vector spaces are introduced to provide a fruitful geometric interpretation for the
study of similarity of matrices and a geometric motivation for the characteristic
value problem. The connection between matrices and linear operators is treated
extensively, but when a choice exists I have normally given the matrix result
first and then interpreted it as a result about linear operators. A notable exception
is Chapter 5, where the Jordan canonical form is developed using invariant
subspaces and direct sum decompositions. By the time the student reaches
Chapter 5, he should have developed sufficient insight to be able to handle the
more abstract development given there.
Chapters 6 and 7 provide, among other things, an alternate matrix
development of the Jordan canonical form, which could be used in place of
Chapter 5. The most complete picture of the canonical forms for similarity is
provided by taking both paths to the Jordan canonical form. Unfortunately there
is seldom time for both treatments in a one-term course. If a two-term sequence
is given, it is certainly worthwhile to give both developments.
Chapters 8 and 9 provide introductions to matrix analysis and numerical
linear algebra respectively. These chapters are by no means complete and are
intended only to sample two interesting and important areas of further study.
Parts of these last two chapters require the student to have completed the normal
calculus and differential equations sequence.
Several topics are included which do not normally appear in elementary
texts. Notable among these are the detailed discussion of commutativity in
Section 5.6 and the complete treatment of the polar decomposition theorem in
Section 7.5. Both of these sections can be omitted during a first reading. Those
sections which are not in the main line of the development are marked with an
asterisk.
Although this book was primarily intended as a text for a one-term course, I
have included enough material for a two-term sequence. In this case all of the
text material can be covered and the instructor may have enough time left to
cover one or two of his favorite topics, which I have omitted.
In addition to Chapters 1 through 5 (Section 5.6 optional), there are several
rearrangements of the text material which would be suitable for one-term
courses. Among these are:
I. Chapters 1, 2, 3, 4, 6, 7 (Sections 7.4, 7.5 optional). (For those who prefer to
concentrate on matrix theory, keeping linear transformations and vector
spaces in the background.)
II. Chapters 1, 2, 3, 4, 5, Sections 6.1, 6.2, 6.3, 7.4, and 7.5, Chapters 8, 9. (For
well-prepared students for whom parts of Chapters 1 – 3 will be a review.)
III. Chapters 1, 2, 3, 4, Sections 7.4, 7.5, Chapter 5 (Sections 5.6 and 7.5
optional).
IV. Chapter 1, Sections 9.1, 9.2, Chapters 2, 3, 4, Sections 7.4 and/or 7.5.
Though mathematics is much easier to watch than to do, it is a most
unrewarding spectator sport. Thus there are over 375 exercises for the students,
which constitute an integral part of the text. Many of these are numerical
exercises designed to build skills and to illustrate the theory while others extend
or complete the theory. In many cases the exercises are designed to introduce the
student to the topics to follow. Answers to the numerical exercises are included.
The introduction of a certain amount of mathematical notation, mostly
standard, is unavoidable and in fact desirable. To help the student overcome any
initial difficulty with this symbolism I have included a glossary of mathematical
symbols immediately preceding the main index.
Many people have contributed to the preparation of this book and I am
indeed grateful. The Department of Mathematics at the University of Pittsburgh,
under Professor Mario Benedicty, has generously supplied the necessary
secretarial assistance and underwritten the cost of the first preliminary edition.
Miss Nancy Brown and Miss Grace Merante carefully typed all the preliminary
versions and their patience and cooperation is gratefully acknowledged. Several
of my colleagues have offered valuable and constructive criticism, which has
been most helpful. I particularly want to thank Professors Herbert Gindler,
George Byrne, and Louis Sacks. Several graduate students at the university have
been most helpful and I would particularly like to thank Ralph Carlson,
Catherine Falk, David Hott, and Allen Schweinsberg. My biggest debt is to the
several hundred students at the University of Pittsburgh who have acted as
guinea pigs for the preliminary versions of this book.
Pittsburgh, Pennsylvania C.G.C.
May 1966
Preface
to the
Second
Edition
The basic treatment of the subject in this edition is unchanged from that in the
first edition, although many developments have been clarified and expanded to
accommodate the reader for whom this is a first exposure to “abstract
mathematics.” Many more illustrative examples have been included, and the
number of exercises has been substantially increased. I have also tried to include
more “road signs”; to tell the reader where he should be and where the
development will take him if he follows along.
There has been a major change in notation from the first edition. At first
glance the expert may not like the new notation, but it has been my experience
that the students learn it easily and accurately and can transfer to other notation
later without difficulty. The original motivation for the changes came from the
success students have in learning the compiler languages (e.g. Fortran) used to
communicate with modem digital computers.
The treatment of metric concepts in Euclidean and unitary spaces has been
increased. Proofs of the Cauchy-Schwartz and triangle inequalities have been
included and the treatment of orthogonality has been substantially improved.
The number of inductive proofs has been increased and there is a greater use of
mapping diagrams and lattice diagrams to help the reader visualize the various
complicated situations which arise.
I am indebted to the many users of the first edition who have communicated
to me their suggestions for improving the text.
Pittsburgh, Pennsylvania C.G.C.
January 1972
CONTENTS
Chapter 1 Matrices and Linear Systems
1.1 Introduction
1.2 Fields and number systems
1.3 Matrices
1.4 Matrix addition and scalar multiplication
1.5 Transposition
1.6 Partitioned matrices
1.7 Special kinds of matrices
1.8 Row equivalence
1.9 Elementary matrices and matrix inverses
1.10 Column equivalence
1.11 Equivalence
Chapter 2 Vector Spaces
2.1 Introduction
2.2 Subspaces
2.3 Linear independence and bases
2.4 The rank of a matrix
2.5 Coordinates and isomorphisms
2.6 Uniqueness theorem for row equivalence
Chapter 3 Determinants
3.1 Definition of the determinant
3.2 The Laplace expansion
3.3 Adjoints and inverses
3.4 Determinants and rank
Chapter 4 Linear Transformations
4.1 Definition and examples
4.2 Matrix representation
4.3 Products and inverses
4.4 Change of basis and similarity
4.5 Characteristic vectors and characteristic values
4.6 Orthogonality and length
4.7 Gram-Schmidt process
4.8 Schur’s theorem and normal matrices
Chapter 5 Similarity: Part I
5.1 The Cayley-Hamilton theorem
5.2 Direct sums and invariant subspaces
5.3 Nilpotent linear operators
5.4 The Jordan canonical form
5.5 Jordan form—continued
5.6 Commutativity (the equation AX = XB)
Chapter 6 Polynomials and Polynomial Matrices
6.1 Introduction and review
6.2 Divisibility and irreducibility
6.3 Lagrange interpolation
6.4 Matrices with polynomial elements
6.5 Equivalence over [x]
6.6 Equivalence and similarity
Chapter 7 Similarity: Part II
7.1 Nonderogatory matrices
7.2 Elementary divisors
7.3 The classical canonical form
7.4 Spectral decomposition
7.5 Polar decomposition
Chapter 8 Matrix Analysis
8.1 Sequences and series
8.2 Primary functions
8.3 Matrices of functions
8.4 Systems of linear differential equations
Chapter 9 Numerical Methods
9.1 Introduction
9.2 Exact methods for solving AX = K
9.3 Iterative methods for solving AX = K
9.4 Characteristic values and vectors
Answers to Selected Exercises
Appendix
Glossary of Mathematical Symbols
Index
