Elementary Matrix Algebra

797 Pages·2012·26.89 MB·English

by  Franz E. Hohn

ELEMENTARY MATRIX
ALGEBRA
ELEMENTARY MATRIX
ALGEBRA
Third Edition
FRANZ E. HOHN
DOVER PUBLICATIONS, INC.
Mineola, New York
PREFACE TO THE THIRD EDITION
The third edition of Elementary Matrix Algebra is designed for the same
audience and has the same scope as earlier editions. As before, full and clear
exposition, complemented by many illustrative examples, is a basic
characteristic of this book. The purpose is to provide a text that can teach by
itself. The detailed exposition and the many examples, which the student can
study on his own, make it possible to progress very rapidly through the
following pages.
There are, however, two major changes. The first of these is that substantial
geometrical material has been introduced early in the text. This helps to give a
good intuitive basis for the following treatment of vector spaces. The second and
more important change is that the chapter on determinants has been deferred
until it is essential for following work. This serves to place maximum emphasis
on the basic concepts and methods of linear algebra.
Other changes include simplification of the notation wherever possible, a
more coherent arrangement of the material, and expansion of the lists of
exercises, many of which refer to applications.
Chapter 1, “Introduction to Matrix Algebra,” presents the most basic laws of
matrix algebra. Chapter 2, “Linear Equations,” introduces the sweepout process
for obtaining the complete solution of any given system of linear equations,
homogeneous or nonhomogeneous. This permits extensive use of concrete
examples and exercises to facilitate the learning of following abstract concepts.
In Chapter 3, “Vector Geometry in ,” and Chapter 4, “Vector Geometry in n-
Dimensional Space,” matrix algebra is used to present useful geometric ideas,
techniques, and terminology as well as to lay a strong intuitive foundation for
Chapter 5, “Vector Spaces,” in which the approach is fully general. Since the
reader can solve linear equations and invert matrices, computational exercises
can be used to facilitate the learning of the abstract ideas. Chapter 6, “The Rank
of a Matrix,” reviews and unifies much that precedes and provides a complete
treatment of the structure of the solution space of a system of linear equations.
Chapter 7, “Determinants,” presents the most commonly used properties of
determinants. It also uses this tool to provide alternative solutions to algebraic
and geometric problems treated earlier, thus providing valuable review. Chapter
8, “Linear Transformations,” treats both linear operators and linear
transformations of coordinates, using geometrical ideas extensively to give the
material intuitive content. Unitary transformations and projections receive
special emphasis. Chapter 9, “The Characteristic Value Problem,” presents this
topic in extensive detail and makes use of virtually all the preceding material. In
Chapter 10, “Quadratic, Bilinear, and Hermitian Forms,” the main emphasis is
placed on real quadratic forms. Definiteness receives special attention because of
its importance in applications.
There is considerably more material here than can be offered in a one-
semester course unless the students have already been introduced to linear
algebra in their calculus courses. The purpose is to provide a degree of choice so
differing interests can be met, to make the book useful for students with varying
backgrounds, and to provide a fairly comprehensive reference volume. The
approach is frankly computational since the book is intended for those who use
matrix algebra as a tool. However, the treatment is fully rigorous and the
sequencing of topics is mathematically sound. The intent is to guarantee that the
study of this book will make a following, mature study of linear algebra a more
rewarding experience than it might otherwise have been.
A book like this cannot be prepared without substantial assistance from
others. Thanks for good help go to Mrs. Carolyn Bloemker, who did the typing,
to Professor Donald R. Sherbert, who read the proof critically, to the editors of
The Macmillan Company, in particular Mr. Leo Malek, who have produced a
most attractive volume, to the very capable printers and compositors, and to my
wife, Mrs. Marian Hohn, who has been patient and helpful throughout the long
processes of revision and production. To all of them, I am deeply grateful.
FRANZ E. HOHN
Urbana, Illinois
PREFACE TO THE SECOND EDITION
This second edition of Elementary Matrix Algebra retains the point of view and
the scope of the first edition. Some of the topics have been reordered to make the
exposition simpler. For example, the partitioning of matrices is now treated in
Chapter One so that it can be used to greater advantage. The theorem on the
determinant of the product of two matrices is proved without use of the Laplace
expansion in Chapter Two. Thus the material on the Laplace expansion may be
omitted entirely if that is desired since it is not essential to later developments.
Similar changes occur in following chapters. A number of topics which are
useful in applications (projections, for example) have been introduced in various
places. This has been done in such a way that these topics need not be made part
of the classroom work. An exceptionally detailed index, referencing also
symbols and those exercises which contain important contributions to the theory,
has been provided. Finally, a number of errors have been corrected and a great
many new exercises have been included. It is hoped that the net effect of all
these changes is to make the book easier to read and more useful as a text and as
a reference. Exactly the same type of course can be taught from it as before.
The author wishes to thank his colleagues, Professors Richard L. Bishop and
Hiram Paley, for their careful reading of the revised manuscript and of the proof,
respectively. Many corrections and improvements are the direct result of their
efforts. Professor Gene Golub provided a number of interesting and important
exercises. Thanks are also due the many users of the first edition who have sent
the author helpful suggestions, exercises, corrections, and letters of
encouragement. Finally, the author is most grateful to the staff of The Macmillan
Company and to the printers for their exceptionally cordial and able assistance.
FRANZ E. HOHN
Philo, Illinois
PREFACE TO THE FIRST EDITION
This text has been developed over a period of years for a course in Linear
Transformations and Matrices given at the University of Illinois. The students
have been juniors, seniors, and graduates whose interests have included such
diverse subjects as aeronautical engineering, agricultural economics, chemistry,
econometrics, education, electrical engineering, high speed computation,
mechanical engineering, metallurgy, physics, psychology, sociology, statistics,
and pure mathematics.
The book makes no pretense of being in any sense “complete.” On the other
hand, to meet as well as possible the needs of so varied a group, I have searched
the literature of the various applications to find what aspects of matrix algebra
and determinant theory are most commonly used. The book presents this most
essential material as simply as possible and in a logical order with the objective
of preparing the reader to study intelligently the applications of matrices in his
special field. The topics are separated so far as possible into distinct, self-
contained chapters in order to make the book more useful as a reference volume.
With the same purpose in view, the principal results are listed as numbered
theorems, identified as to chapter and section, and printed in italics. Formulas
are similarly numbered, but the numbers are always enclosed in parentheses so
as to distinguish formulas from theorems. Again for reference purposes, I have
added appendices on the Σ and Π notations and on the algebra of complex
numbers, for many readers will no doubt be in need of review of these matters.
The exercises often present formal aspects of certain applications, but no
knowledge of the latter is necessary for working any problem. To keep down the
size of the volume, detailed treatment of applications was omitted. The exercises
range from purely formal computation and extremely simple proofs to a few
fairly difficult problems designed to challenge the reader. I hope that every
reader will find it possible to work most of the simpler exercises and at least to
study the rest, for many useful results are contained in them, and, also, there is
no way to learn the techniques of computation and of proof except through
practice. The exercises marked with an asterisk (*) are of particular importance
for immediate or later use and should not be overlooked.
In order to make the learning and the teaching of matrix algebra as easy as
possible, I have tried always to proceed by means of ample explanations from
the familiar and the concrete to the abstract. Abstract algebraic concepts
themselves are not the prime concern of the volume. However, I have not
hesitated to make an important mathematical point where the need and the
motivation for it are clear, for it has been my purpose that in addition to learning
useful methods of manipulating matrices, the reader should progress
significantly in mathematical maturity as a result of careful study of this book. In
fact, since the definitions of fields, groups, and vector spaces as well as of other
abstract concepts appear and are used here, I believe that a course of this kind is
not only far more practical but is also better preparation for later work in abstract
algebra than is the traditional course in the theory of equations. I also believe
that some appreciation of these abstract ideas will help the student of
applications to read and work in his own field with greater insight and
understanding.
The chief claim to originality here is in the attempt to reduce this material to
the junior-senior level. Although a few proofs and many exercises are believed
to be new, my debt to the standard authors—M. Bôcher, L. E. Dickson, W. L.
Ferrar, C. C. McDuffee, F. J. Murnaghan, G. Birkhoff and S. MacLane, O.
Schreier and E. Sperner, among others—is a very great one, and I acknowledge
it with respect and gratitude.
I am particularly indebted to Professor A. B. Coble for permission to adapt to
my needs his unpublished notes on determinants. The credit is his for all merit in
the organization of Chapter Two. I am also indebted to Professor William G.
Madow who helped to encourage and guide my efforts in their early stages. The
value of the critical assistance of Professors Paul Bateman, Albert Wilansky, and
Wilson Zaring cannot be overemphasized. Without their severe but kindly
criticisms, this book would have been much less acceptable. However, I alone
am responsible for any errors of fact or judgment which still persist. For
additional critical aid, and for many of the problems, I owe thanks to a host of
students and colleagues who have had contact with this effort. Finally, I owe
thanks to Mrs. Betty Kaplan and to Mrs. Rachel Dyal for their faithful and
competent typing of the manuscript, to Wilson Zaring and Russell Welker for
assistance with reading proof, and to the staff of The Macmillan Company for
their patient and helpful efforts during the production of this book.
FRANZ E. HOHN
Urbana, Illinois
CONTENTS
CHAPTER
1 Introduction to Matrix Algebra
1.1 Matrices
1.2 Equality of Matrices
1.3 Addition of Matrices
1.4 Commutative and Associative Laws of Addition
1.5 Subtraction of Matrices
1.6 Scalar Multiples of Matrices
1.7 The Multiplication of Matrices
1.8 The Properties of Matrix Multiplication
1.9 Exercises
1.10 Linear Equations in Matrix Notation
1.11 The Transpose of a Matrix
1.12 Symmetric, Skew-Symmetric, and Hermitian Matrices
1.13 Scalar Matrices
1.14 The Identity Matrix
1.15 The Inverse of a Matrix
1.16 The Product of a Row Matrix into a Column Matrix
1.17 Polynomial Functions of Matrices
1.18 Exercises
1.19 Partitioned Matrices
1.20 Exercises
2 Linear Equations
2.1 Linear Equations
2.2 Three Examples
2.3 Exercises
2.4 Equivalent Systems of Equations
2.5 The Echelon Form for Systems of Equations
2.6 Synthetic Elimination
2.7 Systems of Homogeneous Linear Equations
2.8 Exercises
2.9 Computation of the Inverse of a Matrix
2.10 Matrix Inversion by Partitioning
2.11 Exercises
2.12 Number Fields
2.13 Exercises
2.14 The General Concept of a Field
2.15 Exercises
3 Vector Geometry in
3.1 Geometric Representation of Vectors in
3.2 Operations on Vectors
3.3 Isomorphism
3.4 Length, Direction, and Sense
3.5 Orthogonality of Two Vectors
3.6 Exercises
3.7 The Vector Equation of a Line
3.8 The Vector Equation of a Plane
3.9 Exercises
3.10 Linear Combinations of Vectors in
3.11 Linear Dependence of Vectors; Bases
3.12 Exercises
4 Vector Geometry in n-Dimensional
Space
4.1 The Real n-Space
4.2 Vectors in
4.3 Lines and Planes in