ebook img

Elementary Matrix Algebra PDF

797 Pages·2012·26.89 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Elementary Matrix Algebra

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

Description:
Fully rigorous treatment starts with basics and progresses to sweepout process for obtaining complete solution of any given system of linear equations and role of matrix algebra in presentation of useful geometric ideas, techniques, and terminology. Also, commonly used properties of determinants, li
See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.