The Theory of Matrices in Numerical Analysis

Alston S. Householder

• Publisher: Dover Publications
• Year: 2013
• eBook ISBN: 9780486145631
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

This text explores aspects of matrix theory that are most useful in developing and appraising computational methods for solving systems of linear equations and for finding characteristic roots. Suitable for advanced undergraduates and graduate students, it assumes an understanding of the general principles of matrix algebra, including the Cayley-Hamilton theorem, characteristic roots and vectors, and linear dependence.
An introductory chapter covers the Lanczos algorithm, orthogonal polynomials, and determinantal identities. Succeeding chapters examine norms, bounds, and convergence; localization theorems and other inequalities; and methods of solving systems of linear equations. The final chapters illustrate the mathematical principles underlying linear equations and their interrelationships. Topics include methods of successive approximation, direct methods of inversion, normalization and reduction of the matrix, and proper values and vectors. Each chapter concludes with a helpful set of references and problems.

CHAPTER 1

Some Basic Identities and Inequalities

1.0. Objectives; Notation. In this chapter and the next will be developed some of the basic tools and relations that will be utilized repeatedly in subsequent chapters. At the outset a matrix of quite simple form will be introduced. It is the result of subtracting a matrix of rank 1 at most from the identity matrix, and any matrix of this form will be called here an “elementary matrix.” Such matrices will play a fundamental role in inversion, and also in the reduction of a matrix by similarity transformations to facilitate the evaluation of its characteristic roots. These will be applied immediately to obtain certain factorization theorems, which are of basic importance both for inversion and for equation solving.

The method of least squares can be interpreted geometrically as the projection of an arbitrary given vector upon a certain subspace; and each step in certain iterative methods can likewise be interpreted as a projection. Hence a general form of a projection operator will be obtained, and the formation of this operator makes some use of the factorization theorems just, obtained. A further application of the factorization theorems will be made in the following section, where certain classical determinantal identities and inequalities will be derived. These will be required in the analysis and derivation of certain iterative methods, and also for more general theoretical considerations.

In the last two sections of this chapter will be introduced certain polynomials associated with an arbitrary matrix and vector. One class is defined by an algorithm due to Lanczos who introduced it originally as a first step toward finding the characteristic values and vectors of a matrix; indeed, the algorithm is much used for this purpose. But the same algorithm occurs in the method of conjugate ...

Table of contents

• Title Page
• Copyright Page
• Dedication
• Preface
• Table of Contents
• CHAPTER 1 - Some Basic Identities and Inequalities
• CHAPTER 2 - Norms, Bounds, and Convergence
• CHAPTER 3 - Localization Theorems and Other Inequalities
• CHAPTER 4 - The Solution of Linear Systems: Methods of Successive Approximation
• CHAPTER 5 - Direct Methods of Inversion
• CHAPTER 6 - Proper Values and Vectors: Normalization and Reduction of the Matrix
• CHAPTER 7 - Proper Values and Vectors: Successive Approximation
• Bibliography
• Index