The Mathematics of Matrices,Second Edition | Date: 28 April 2011, 04:47
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Mathematics of Matrices
By Philip J. Davis
* Publisher: John Wiley & Sons Inc
* Number Of Pages: 348
* Publication Date: 1973-12
* ISBN-10 / ASIN: 0471009288
* ISBN-13 / EAN: 9780471009283
Contents
Chapter 1 What is a Matrix
1.1 In the Beginning
1.2 What is a Matrix
1.3 The Order of a Matrix
1.4 Various Ways of Writing Matrices
1.5 Tabulations of Information as Matrices
1.6 When Two Matrices are Equal
1.7 Switching Rows and Columns: The Transpose
of a Matrix
I.B Who's Who in the World of Matrices
Chapter 2 The Arithmetic of Matrices, Part I
2.1 Adding Matrices
2.2 Subtracting Matrices
2.3 Scalar Multiplication
2.4 Matrix Multiplication
2.5 Why and Wherefore
2.6 Multiplication has its Rules
2.7 Matrix Powers and Polynomials
2.B The Transpose of a Product
2.9 Matrix Multiplication at Work
2.10 Multiplication of a Rectangular Matrix by a Column
Matrix: Some Practical Problems
Chapter 3 The Arithmetic of Matrices, Part II
3.1 Remembrance of Things Past: The Solution of
Linear Equations
3.2 Elementary Row Transformations
3.3 Matrix Inversion
3.4 The Algebra of Inverses
3.5 Matrix Inversion by the Gauss-Jordan Scheme:
Reduction of Matrices
3.6 Tricks of the Matrix Trade
3.7 The Sense of Identity
3.8 Matrix Inversion and the Solution of Systems of
Linear Equations 114
3.9 Some Further Applications 119
Chapter 4 Linear Transformations of the Plane 125
4.1 Functions, Correspondences, Transformations,
and Mappings 125
4.2 Transformations of the Plane 132
4.3 The Simple Transformations 135
4.4 Matrices as Transformations of the Plane 149
4.5 Linear Homogeneous Transformations of the Plane 158
4.6 What Does It Do 162
4.7 Degenerate Linear Transformations 165
4.8 Projections 166
4.9 Nonhomogeneous Transformations 167
Chapter 5 Determinants 171
5.1 What is a Determinant 171
5.2 Determinants, Cofactors, and Inverses 179
5.3 Pushing the Postulates: An Alternate Approach to
Determinants 183
5.4 Determinants, Matrices, and Multiplication 186
5.5 Additional Algebra of Determinants 192
5.6 The Geometry behind the Multiplication Rule 194
5.7 Determinants and Linear Systems of Equations 204
Chapter 6 Vectors and Inner Products 207
6.1 Column Matrices as Vectors 207
6.2 Components, Magnitude, and Direction 213
6.3 The Inner Product and Angle Geometry 219
6.4 Inner Products, Lines, and Half-Planes 226
6.5 Convex Sets of Points 231
6.6 Vectors and Transformations 235
6.7 A Glimpse at n-Dimensional Space and its Geometry 239
Chapter 7 Matrices as Operators 245
7.1 The Concept of the Black Box 245
7.2 Linear Boxes: Matrices as Linear Boxes 248
7.3 Linear Analysis in Applied Mathematics 251
7.4 Some Simple Operations and their Matrices 253
7.5 Time Independent Boxes 258
Chapter 8 Characteristic Values and their Application 265
8.1 Characteristic Values of Matrices 265
8.2 High Powers of a Matrix 272
8.3 Progressions and Difference Equations 280
8.4 Difference Equations and Matrices 284
8.5 Some Applications of Difference Equations 291
Chapter 9 Matrices and Abstract Algebra 296
9.1 Complex Numbers and How to Live Without Them 296
9.2 The Mathematics of Symmetry: Groups of Matrices 305
Chapter 10 Pippins and Cheese 313
10.1 Additional Problems 313
10.2 Topics for Investigation 316
10.3 Historical and Biographical Notices 335
10.4 Bibliography 337
INDEX 341
Preface to the Second Edition
THE USE OF MATRICES has now extended beyond mathematics and the
physical sciences to business and economics, psychology, and the
social and political sciences. The wide use of the first edition of this book
confirms the opinion that matrix algebra can and should be introduced to
the student as early as possible.
In the years since preparing the first edition, time-sharing computer
facilities and interactive languages have become widely available. Many
of these languages (such as APL) have extremely convenient matrix
implementation which is close to classical matrix notation. This makes
available many simplifications in setting up computation and hence is a
further reason for learning matrix algebra.
I have recently been using this text for the matrix part of a freshman
course called "The Introduction to Applied Mathematics." This course
uses computer facilities routinely and-while by no means necessary-
believe the computer serves at once to drive home certain theoretical
points and to offer the opportunity of handling problems of realistic proportions.
In the Second Edition of this book, a number of minor errors present
in the first edition have been eliminated.
PHILIP J. D AVIS
March 17, 1972
Preface
I T IS INEVITABLE in an age of rapid change to tell children about the
horse drawn wagons of one's youth, and it is equally inevitable in writing
a text that makes matrix theory available to young students for one's
thought to turn to the position of the subject a few years back. A generation
ago, the subject was taught as an intermediate level or first-year
graduate course in college, and was taken by majors in mathematics and
theoretical physics. Before long, I suspect, the educated man on the street
may be making bad jokes about matrix inversion and characteristic values,
the way he now does about Pythagoras' Theorem or the Binomial Formula.
This is not a case of how the mighty have fallen; on the contrary, it is a
wonderful example of the gradual dissemination of knowledge that has been
going on since antiquity when the ancient scientific elite yielded their
secrets of arithmetic. It should make anyone who loves mathematics both
proud and humble, simultaneously.
A number of reasons can be advanced for the study of matrices and linear
algebra as a one-semester course. I n the first place, this theory is basic and
of wide applicability to advanced work in pure and applied mathematics.
In the second place, since the rules of matrix algebra are similar-but not
altogether similar-to the rules of ordinary algebra, its study can serve as a
review or a consolidation of ordinary algebra, and can lend new insights
to old processes.
In composing this text I have been guided by the principle that "sufficient
unto the day is the rigor, generality, and notation thereof." This means,
in practice, that I have tried not to be tediously doctrinaire about rigor,
generality, and notation, but have tried to allow the subject matter to
carry as much of them as seemed possible without their gaining the upper
hand. I have avoided the use of the sigma sign for summation, thinking
that the notation of matrix algebra is itself sufficiently burdensome to the
beginner. I have not proved any theorem involving "the general case" by
means of mathematical induction. The arguments used in these cases will
be convincing without induction; and the teacher who wants to drill on
the technique of induction will therefore find ample opportunity to do so
at these spots.
The learning of matrices at the elementary level consists of four things;
(1) the notation and terminology; (2) the formal algebra; (3) the interpretations
of matrices; and (4) the applications of matrices. I have tried to
arrange the material in such a way that the four aspects are intermingled
in an easy and natural association. When one goes to a foreign land, say to
Japan, it would be foolish to avoid communicating with the Japanese until
one had learned the subtleties of Haiku poetry.
I approach the subject of matrices with the eye of the analyst and the
experience of the applied mathematician. For this reason, I tend to regard
matrix theory as mathematical grammar, and not as fiction or belles-lettres.
On the other hand, I have been associated over the years with a number of
very distinguished algebraists who regard matrix theory, like virtue, as its
own reward. This association has led me to appreciate the role of matrices
in algebra, and I hope that I h
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