Basic Matrix Theory - Softcover

Fuller, Leonard E.

Softcover

ISBN 10: 0486818462 ISBN 13: 9780486818467

Verlag: Dover Publications Inc., 2017

Inhaltsangabe

Written as a guide to using matrices as a mathematical tool, this text is geared toward physical and social scientists, engineers, economists, and others who require a model for procedure rather than an exposition of theory. Knowledge of elementary algebra is the only mathematical prerequisite. Detailed numerical examples illustrate the treatment's focus on computational methods.
The first four chapters outline the basic concepts of matrix theory. Topics include the development of the concept of elementary operations and a systematic procedure for simplifying matrices as well as a method for evaluating the determinant of a given square matrix. Subsequent chapters explore important numerical procedures, including the process for approximating characteristic roots and vectors plus direct and iterative methods for inverting matrices and solving systems of equations. Solutions to the problems are included.

Über diesen Titel

Über die Autorin bzw. den Autor

Leonard E. Fuller was Professor of Mathematics at Kansas State University and the author of Linear Algebra with Applications.

Auszug. © Genehmigter Nachdruck. Alle Rechte vorbehalten.

Basic Matrix Theory

By Leonard E. Fuller

Dover Publications, Inc.

Copyright © 2017 Dover Publications, Inc.
All rights reserved.

ISBN: 978-0-486-81846-7

Contents

1. Basic Properties of Matrices, 
2. Elementary Matrix Operations, 
3. Vector Spaces and Linear Transformations, 
4. Determinants, 
5. Characteristic Roots and Vectors, 
6. Inversion of Matrices, 
7. Inversion of Matrices by Iteration, 
8. Homogeneous Forms, 
Bibliography, 
Answers to Problems, 
Index, 

CHAPTER 1

Basic Properties of Matrices

1.1 Introduction

One of the most widely used mathematical concepts is that of a system of linear equations in certain unknowns. Such a system arises in many diverse situations and in a variety of subjects. For such a system, a set of values for the unknowns that will "satisfy" all the equations is desired. In the language of matrices, a system of linear equations can be written in a very simple form. The use of properties of matrices then makes the solution of the system easier to find.

However, this is not the only reason for studying matrix algebra. The sociologist uses matrices whose elements are zeros or ones in talking about dominance within a group. Closely allied to this application are the matrices arising in the study of communication links between pairs of people. In genetics, the relationship between frequencies of mating types in one generation and those in another can be expressed using matrices. In electrical engineering, network analysis is greatly aided by the use of matrix representations.

Today the language of matrices is spreading to more and more fields as its usefulness is becoming recognized. The reader can probably already call to mind instances in his own field where matrices are used. It is hoped that many more applications will occur to him after this study of matrix algebra is completed.

1.2 The Form of a Matrix

A proper way to begin a discussion of matrices would be to give a definition. However, before doing this, it should be noted that a simple definition cannot begin to convey the concept that is involved. For this reason, a two part discussion will follow the definition. The first part will be concerned with trying to convey the nature of the form of a matrix. The second part of the discussion will be concerned with the properties of what are called matrix addition, matrix multiplication, and scalar multiplication. This will establish the basic algebra of matrices. Consider the following definition.

Definition 1.1. A matrix is a rectangular array of numbers of some algebraic system.

What does this mean? It simply means that a matrix is first of all a set of numbers arranged in a pattern that suggests the geometric form of a rectangle. Most of the time this will actually be a square. The algebraic system from which the numbers are chosen will be discussed in more detail later. Some simple examples of matrices are as follows:

[MATHEMATICAL EXPRESSION OMITTED]

The [] that are used to enclose the array bring out the rectangular form. Sometimes large () are used, whereas other authors prefer double vertical lines instead of the []. Regardless of the notation, the numbers of the array are set apart as an entity by the symbolism. These numbers are often referred to as the elements of the matrix. The numbers in a horizontal line constitute a row of the matrix, those in a vertical line a column. The rows are numbered from the top to the bottom, while the columns are numbered from left to right.

It is sometimes necessary in a discussion to refer to a matrix that has been given. To avoid having to write it out completely every time, it is customary to label matrices with capital letters A,B,C, etc. as was done in the example above. In case the matrix being referred to is a general one, then its elements are often denoted with the corresponding small letters with numerical subscripts. The next example will illustrate this symbolism. With this notation one knows at once that capital letters refer to matrices and small letters to the elements.

When it is necessary in a discussion to talk about a general matrix A, it will be assumed that A consists of a set of mn numbers arranged in n rows with m numbers in each row. The individual elements will be denoted as ars where the r denotes the row in which the element belongs, and the s denotes the column. In other words, a23 will be the element in the second row and third column. The double subscript is the address of the element; it tells in which row and in which column the element may be found. This is definite since there can be, in each row, only one element that is also in a given column. Consider then the following examples.

[MATHEMATICAL EXPRESSION OMITTED]

In these general matrices, notice that the first subscript does denote the row in which the element occurs, whereas the second indicates the column of the entry. When the number of rows and the number of columns are known, the shorter notation A = (ars) is often used. The real significance of this idea will appear several times in this chapter.

Another quite useful concept connected with the form of matrices is given by the following definition.

Definition 1.2. The dimension of a matrix A with n rows and m columns is n × m.

In the numerical examples given before, the dimensions are 2 × 3, 3 × 1, 3 × 2, and 3 × 3, respectively. In the examples of the general matrices, A is of dimension 4 × 3 and B is of dimension 3 × 4. The dimension of a matrix is often referred to as the "size" of the matrix.

A special kind of a matrix is one that has only one row or only one column. These are useful enough to have a special name given to them. This designation is indicated in the next definition.

Definition 1.3. A row vector is a 1 × m matrix. A column vector is an n × 1 matrix.

Using these concepts, a matrix can be thought of as being composed of a set of row vectors placed one under the other. These can be numbered in order from top to bottom so that, in the double subscript notation, the first number refers to the row vector to which the element belongs. Similarly, a matrix can be considered as a set of column vectors placed side by side. If these are numbered from left to right, the second subscript of the address of each element would then refer to a column vector in this set.

There are occasions when reference will be made to the row vectors of a matrix or to the column vectors. In this case, the matrix is to be considered as indicated above. Sometimes the term "vectors" of a matrix will be used. In this case, the reference is to either row vectors or column vectors or both.

1.3 The Transpose of a Matrix

Associated with every 1 × w row vector is an m × 1 column vector. This column vector has the same numbers appearing in the same order as in the row vector. The only difference is that they are written vertically for the column vector and horizontally for the row vector. The column vector is referred to as the transpose of the row vector. Also, the row vector is called the transpose of the column vector.

This concept is readily extensible to matrices. With a given matrix A, one can associate a matrix...

Details zur Ausgabe

• Verlag: Dover Publications Inc.
• Erscheinungsdatum: 2017
• Sprache: Englisch
• ISBN 10: 0486818462
• ISBN 13: 9780486818467
• Einband: Taschenbuch
• Anzahl der Seiten: 256
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: Soc. Agr. La Raia Ss; http://www.la-raia.it; ; Strada Monterotondo, 79; Novi Ligure; 15067; Italien