Applied Dimensional Analysis and Modeling

By Thomas Szirtes

Applied Dimensional Analysis and Modeling provides the full mathematical background and step-by-step procedures for employing dimensional analyses, along with a wide range of applications to problems in engineering and applied science, such as fluid dynamics, heat flow, electromagnetics, astronomy and economics. This new edition offers additional worked-out examples in mechanics, physics, geometry, hydrodynamics, and biometry.

• Covers 4 essential aspects and applications: principal characteristics of dimensional systems, applications of dimensional techniques in engineering, mathematics and geometry, applications in biosciences, biometry and economics, applications in astronomy and physics
• Offers more than 250 worked-out examples and problems with solutions
• Provides detailed descriptions of techniques of both dimensional analysis and dimensional modeling

• Language: English
• Publisher: Elsevier Science
• Release date: Apr 27, 2007
• ISBN: 9780080555454

Book preview

APPLIED DIMENSIONAL ANALYSIS AND MODELING

Second Edition

Thomas Szirtes, Ph.D., P.Eng.

Thomas Szirtes and Associates Inc., Toronto, Ontario, Canada

Senior Staff Engineer (ret.) SPAR Aerospace Ltd., Toronto, Ontario, Canada

(Predecessor of McDonald Dettwiler and Associates, Brampton, Ontario, Canada)

with a chapter on mathematical preliminaries

by

Pál Rózsa, D.Sc.

Professor and former Chairman, Department of Mathematics and Computer Science, Technical University of Budapest, Hungary

Table of Contents

Cover image

Title page

Copyright

Dedication

About the Author

LIST OF TITLED EXAMPLES AND PROBLEMS

FOREWORD TO THE FIRST EDITION

FOREWORD TO THE SECOND EDITION

ACKNOWLEDGMENTS

PREFACE TO THE FIRST EDITION

PREFACE TO THE SECOND EDITION

ORGANIZATION, NOTATION, AND CONVENTIONS

Chapter 1: MATHEMATICAL PRELIMINARIES

1.1 MATRICES AND DETERMINANTS

1.2 OPERATIONS WITH MATRICES

1.3 THE RANK OF A MATRIX

1.4 SYSTEMS OF LINEAR EQUATIONS

1.5 LIST OF SELECTED PUBLICATIONS DEALING WITH LINEAR ALGEBRA AND MATRICES

Chapter 2: FORMATS AND CLASSIFICATION

2.1 FORMATS FOR PHYSICAL RELATIONS

2.2 CLASSIFICATION OF PHYSICAL QUANTITIES

Chapter 3: DIMENSIONAL SYSTEMS

3.1 GENERAL STATEMENTS

3.2 CLASSIFICATION

3.3 THE SI

3.4 OTHER THAN SI DIMENSIONAL SYSTEMS

3.5 A NOTE ON THE CLASSIFICATION OF DIMENSIONAL SYSTEMS

Chapter 4: TRANSFORMATION OF DIMENSIONS

4.1 NUMERICAL EQUIVALENCES

4.2 TECHNIQUE

4.3 EXAMPLES

4.4 PROBLEMS

Chapter 5: ARITHMETIC OF DIMENSIONS

Chapter 6: DIMENSIONAL HOMOGENEITY

6.1 EQUATIONS

6.2 GRAPHS

6.3 PROBLEMS (Solutions are in Appendix 6)

Chapter 7: STRUCTURE OF PHYSICAL RELATIONS

7.1 MONOMIAL POWER FORM

7.2 THE DIMENSIONAL MATRIX

7.3 GENERATING PRODUCTS OF VARIABLES OF DESIRED DIMENSIONS

7.4 NUMBER OF INDEPENDENT SETS OF PRODUCTS OF GIVEN DIMENSIONS (I)

7.5 COMPLETENESS OF THE SET OF PRODUCTS OF VARIABLES

7.6 SPECIAL CASE: MATRIX A IS SINGULAR

7.7 NUMBER OF INDEPENDENT SETS OF PRODUCTS OF A GIVEN DIMENSION (II); BUCKINGHAM’S THEOREM

7.8 SELECTABLE AND NONSELECTABLE DIMENSIONS IN A PRODUCT OF VARIABLES

7.9 MINIMUM NUMBER OF INDEPENDENT PRODUCTS OF VARIABLES OF GIVEN DIMENSION

7.10 CONSTANCY OF THE SOLE DIMENSIONLESS PRODUCT

7.11 NUMBER OF DIMENSIONS EQUALS OR EXCEEDS THE NUMBER OF VARIABLES

7.12 PROBLEMS

Chapter 8: SYSTEMATIC DETERMINATION OF COMPLETE SET OF PRODUCTS OF VARIABLES

8.1 DIMENSIONAL SET; DERIVATION OF PRODUCTS OF VARIABLES OF A GIVEN DIMENSION

8.2 CHECKING THE RESULTS

8.3 THE FUNDAMENTAL FORMULA

Chapter 9: TRANSFORMATIONS

9.1 THEOREMS RELATEDTO SOME SPECIFIC TRANSFORMATIONS

9.2 TRANSFORMATION BETWEEN SYSTEMS OF DIFFERENT D MATRICES

9.3 TRANSFORMATION BETWEEN DIMENSIONAL SETS

9.4 INDEPENDENCE OF DIMENSIONLESS PRODUCTS OF THE DIMENSIONAL SYSTEM USED

Chapter 10: NUMBER OF SETS OF DIMENSIONLESS PRODUCTS OF VARIABLES

10.1 DISTINCT AND EQUIVALENT SETS

10.2 CHANGES IN A DIMENSIONAL SET NOT AFFECTING THE DIMENSIONLESS VARIABLES

10.3 PROHIBITED CHANGES IN A DIMENSIONAL SET

10.4 NUMBER OF DISTINCT SETS

10.5 EXCEPTIONS

10.6 PROBLEMS

Chapter 11: RELEVANCY OF VARIABLES

11.1 DIMENSIONAL IRRELEVANCY

11.2 PHYSICAL IRRELEVANCY

11.3 PROBLEMS

Chapter 12: ECONOMY OF GRAPHICAL PRESENTATION

12.1 NUMBER OF CURVES AND CHARTS

12.2 PROBLEMS

Chapter 13: FORMS OF DIMENSIONLESS RELATIONS

13.1 GENERAL CLASSIFICATION

13.2 MONOMIAL IS MANDATORY

13.3 MONOMIAL IS IMPOSSIBLE – PROVEN

13.4 MONOMIAL IS IMPOSSIBLE—NOT PROVEN

13.5 RECONSTRUCTIONS

13.6 PROBLEMS

Chapter 14: SEQUENCE OF VARIABLES IN THE DIMENSIONAL SET

14.1 DIMENSIONLESS PHYSICAL VARIABLE IS PRESENT

14.2 PHYSICAL VARIABLES OF IDENTICAL DIMENSIONS ARE PRESENT

14.3 INDEPENDENT AND DEPENDENT VARIABLES

14.4 PROBLEMS

Chapter 15: ALTERNATE DIMENSIONS

Chapter 16: METHODS OF REDUCING THE NUMBER OF DIMENSIONLESS VARIABLES

16.1 REDUCTION OF THE NUMBER OF PHYSICAL VARIABLES

16.2 FUSION OF DIMENSIONLESS VARIABLES

16.3 INCREASING THE NUMBER OF DIMENSIONS

16.4 PROBLEMS

Chapter 17: DIMENSIONAL MODELING

17.1 INTRODUCTORY REMARKS

17.2 HOMOLOGY

17.3 SPECIFIC SIMILARITIES

17.4 DIMENSIONAL SIMILARITY

17.5 SCALE EFFECTS

17.6 PROBLEMS

Chapter 18: FIFTY-TWO ADDITIONAL APPLICATIONS

References

REFERENCES IN NUMERICAL ORDER

REFERENCES IN ALPHABETICAL ORDER OF AUTHORS’ SURNAMES

Appendices

APPENDIX 1: RECOMMENDED NAMES AND SYMBOLS FOR SOME PHYSICAL QUANTITIES

APPENDIX 2: SOME MORE-IMPORTANT PHYSICAL CONSTANTS

APPENDIX 3: SOME OF THE MORE-IMPORTANT NAMED DIMENSIONLESS VARIABLES

APPENDIX 4: NOTES ATTACHED TO FIGURES

APPENDIX 5: ACRONYMS

APPENDIX 6: SOLUTIONS TO PROBLEMS

APPENDIX 7: PROOFS FOR SELECTED THEOREMS AND EQUATIONS

APPENDIX 8: BLANK MODELING DATA TABLE

Indices

SUBJECT INDEX

SURNAME INDEX

Copyright

Butterworth-Heinemann is an imprint of Elsevier

30 Corporate Drive, Suite 400, Burlington, MA 01803, USA

Linacre House, Jordan Hill, Oxford OX2 8DP, UK

Copyright © 2007, Elsevier Inc. All rights reserved.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.

Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: [phone], fax: [phone], E-mail: [e-mail]. You may also complete your request on-line via the Elsevier homepage (http://elsevier.com), by selecting Support & Contact then Copyright and Permission and then Obtaining Permissions.

Recognizing the importance of preserving what has been written, Elsevier prints its books on acid-free paper whenever possible.

Library of Congress Cataloging-in-Publication Data

Szirtes, Thomas.

Applied dimensional analysis and modeling / Thomas Szirtes.

p. cm.

Includes bibliographical references and index.

ISBN 0-07-062811-4

1. Dimensional analysis. 2. Engineering models. I. Title

TA347.D5S95 1997

530.8—dc21 97-26056

CIP

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

ISBN 13: 978-0-12-370620-1

ISBN 10: 0-12-370620-3

For information on all Butterworth–Heinemann publications visit our Web site at www.books.elsevier.com

Printed in the United States of America

06 07 08 09 10 11 12 13     10 9 8 7 6 5 4 3 2 1

Dedication

I dedicate this book to the memory of my late teacher, Professor Ádám Muttnyánszky of the Technical University of Budapest, who, by his style, supreme educating skill, and human greatness, taught me to love my profession.

Thomas Szirtes

About the Author

The author with a small-scale model

Thomas Szirtes is a professional engineer who devotes his career to consulting, writing, and teaching. He is a former Senior Staff Engineer at SPAR Aerospace Ltd. in Toronto, and was one of the Project Engineers of the Shuttle Robotic Manipulator Arm (Canadarm), for which he received NASA’s Achievement Award.

Dr. Szirtes has published over 60 scientific and engineering papers, as well as a college text on mathematical logic. He has taught at the Technical University of Budapest, McGill University, and Loyola College (Montreal) and was the founding editor of the SPAR Journal of Engineering and Technology.

LIST OF TITLED EXAMPLES AND PROBLEMS

Solutions to problems are in Appendix 6. Some titles are listed in abbreviated form.

Ex. = example; Pr. = problem.

FOREWORD TO THE FIRST EDITION

The student being introduced to dimensional analysis for the first time is always amazed by the demonstration, without recourse to full physical analysis, that the period of oscillation of a simple pendulum must be proportional to the square root of the pendulum length and independent of its mass. The rationale for this relationship is, of course, based on the simple argument that each term of a properly constructed physical equation needs to be dimensionally homogeneous with the others. Likewise, the student is also impressed by the application of such results to predicting full-scale behavior from measurements using a scale model. From this simple example, dimensional arguments can be taken to increasing levels of complexity, and can be applied to a wide range of situations in science and engineering.

This book develops the ideas of dimensions, dimensional homogeneity, and model laws to considerable depth and with admirable rigor, enabling Dr. Szirtes to provide us with intriguing insights into an impressive range of physical examples—panning such topics as the impact velocity of a meteorite, the lengths of dogs’ tails in urban areas, the price of land, the cracking of window glass, and so on—and all without the tedium so often involved in a conventional step-by-step physical analysis.

Dr. Szirtes makes it clear that he regards dimensional analysis and modeling as an art. His intention here is to remind us that the practitioner of dimensional analysis is offered a refreshing scope for personal initiative in the approach to any given problem. It is, then, inevitable that the reader will be inspired by the creative spirit that illuminates the following pages, and will be struck by the elegance of the techniques that are developed.

This book gives an in-depth treatment of all aspects of dimensional analysis and modeling, and includes an extensive selection of examples from a variety of fields in science and engineering. I am sure that this text will be well received and will prove to be an invaluable reference to researchers and students with an interest in dimensional analysis and modeling and those who are engaged in design, testing, and performance evaluation of engineering and physical systems.

MICHAEL ISAACSON, Ph.D., P.Eng.,     Dean, Faculty of Applied Science, and, Professor of Civil Engineering, University of British Columbia, Vancouver, B.C., Canada

FOREWORD TO THE SECOND EDITION

I was honoured to have written the foreword to the first edition of the book Applied Dimensional Analysis and Modeling, and I am equally honoured to be invited to write the foreword to the second edition. The first edition was very well received by the engineering and scientific communities world-wide. As one indication of this, a Hungarian translation of the first edition is under development and will serve as a university text in Hungary.

Dimensional analysis is often indispensable in treating problems that would otherwise prove intractable. One of its most attractive features is the manner in which it may be applied across a broad range of disciplines—including engineering, physics, biometry, physiology, economics, astronomy, and even social sciences. As a welcome feature of the book, the first edition included more than 250 examples drawn from this broad range of fields.

It would be difficult to improve on the comprehensive, in-depth treatment of the subject, including the extensive selection of examples that appeared in the first edition. But remarkably Dr. Szirtes has managed to accomplish this. The second edition contains a number of changes and improvements to the original version, including a range of additional applications of dimensional analysis. These include examples from biometry, mechanics, and fluid dynamics. For instance, Dr. Szirtes elegantly demonstrates that the size of the human foot is not proportional to body height, but instead is proportional to the body height to the power of 1.5!

Overall, the second edition of the book once more provides an in-depth treatment of all aspects of dimensional analysis, and includes an extensive selection of examples drawn from a variety of fields. I am sure that the second edition will be well received, and once more will prove to be an invaluable reference to students, researchers, and professionals across a range of disciplines.

MICHAEL ISAACSON, Ph.D., P.Eng.,     Dean, Faculty of Applied Science, and, Professor of Civil Engineering, University of British Columbia, Vancouver, B.C., Canada

ACKNOWLEDGMENTS

The author wishes to thank his former colleague Bruce Sayer, of SPAR Aerospace Ltd., Toronto, Ontario, for the expert preparation of line illustrations, and John H. Connor, of the Centre for Research in Earth and Space Technology, Toronto, Ontario, for his reading of the first draft of this book and for his subsequent valuable comments and suggestions.

Bertrand Russell reputedly once said: An interesting problem has more value than an interesting solution; for while the former generates the thinking of many, the latter registers the thinking of only one. With this in mind, the author is appreciative of another former colleague, Robert Ferguson, P.Eng., now of Taiga Engineering Group, Inc., Bolton, Ontario, for proposing two interesting problems that appear as Problem 4/13 and Example 18-43. The latter was found to be especially stimulating, mainly for its seemingly intricate, but in fact relatively simple, outcome.

Sincere appreciation is expressed to Robert Houserman, former Senior Editor of McGraw-Hill, for his initiative in proposing this book, for his subsequent and sustained general helpfulness, and, in particular, for his two argument-free acquies-cences to extensions of the manuscript’s deadline.

For his help in securing reference material and subsequent detailed observations and suggestions, the author conveys his indebtedness to Eugene Brach, P.Eng., Senior Technical Adviser for the Government of Canada. His constructive scrutiny greatly reduced the numerous minor, and not so minor, numerical errors and inconsistencies in the draft text.

For his assistance in reading the manuscript from a professional mathematician’s perspective and for his numerous and detailed suggestions and comments, much indebtedness is expressed to Prof. Pál Rózsa of the Technical University of Budapest, Hungary, whose teaching during the author’s undergraduate years greatly enhanced his affinity for mathematical techniques.

The author acknowledges the enthusiastic, conscientious, and timely efforts of Dr. Simon Yiu of Seneca College, Toronto, in checking the text for errors in arithmetic, references, and typography.

The author offers his special thanks and great appreciation to his wife, Penny, who meticulously corrected the large number of misplaced or misused definite articles, inappropriate pronouns, faulty pluralities, flawed use of tenses, wrongly placed adjectives, erroneous adverbs, dangling modifiers, and other serious linguistic offenses.

Ideas for a number of examples and problems appearing in this book were obtained from a multitude of sources, and where appropriate, credit is given. However, the author may have missed mentioning some sources, and for this he sincerely apologizes to those who may feel slighted.

PREFACE TO THE FIRST EDITION

I have often been impressed by the scanty attention paid even by original workers in physics to the great principle of similitude. It happens not infrequently that results in the form of laws are put forward as novelties on the basis of elaborate experiments, which might have been predicted a priori after a few minutes of consideration.

Lord Rayleigh (Ref. 71)

In the late sixties at RCA Victor Ltd. in Montreal, where the author was working as an engineer, Peter Foldes—the antenna man in the organization—gave him a problem: he was to determine the deformation and force-related characteristics of a very large, log-periodic and as yet nonexistent antenna for a set of widely differing load conditions (a log-periodic antenna is essentially a three-dimensional cable structure of a complex V-shaped configuration).

An analytical approach was out of the question, due to the geometric intricacies and uneven concentrated and distributed mechanical load patterns. So, what to do? In quasidesperation, the author went to the McGill University Engineering Library to find some material—any material—that would help to extricate him from his predicament. What greatly hampered his effort in the library, of course, was the fact that he did not have any notion of what he wanted to find! It is always difficult to find something if the seeker has no idea what he is looking for. But, in this instance, by luck and good fortune, he found H. Langhaar’s great book Dimensional Analysis and Theory of Models (Ref. 19) from which he could, albeit with some effort, learn how to size and construct a small scale model of the antenna, determine the loads to be put on this model, and on it measure the corresponding mechanical parameters. Then, with these results and by formulating the appropriate scale factors, he was able to determine all the deformations and reactive forces for the full-scale product.

When the results were presented to the in-house customer, he asked how the impressive set of data was obtained. He was told that they were derived by modeling experimentation. His response was: Very good, and since it seems that you have such specialized knowledge, you should be a specialist here! And indeed, within a couple of weeks your chronicler was promoted to Engineering Specialist, at that time a rather exalted title at RCA. This episode proves that the dimensional technique can be beneficial to its user in more than one way!

Since then, the author has been hooked on the fine art of dimensional techniques and modeling. He has used the method a great many times and in widely varying circumstances. It has unfailingly provided him with reliable results, often achieved with astonishing speed and little effort. He has also found the method to be potent, since it could usually produce results even when all other means failed. In short, the dimensional method has proven to be an extremely elegant technique, and elegance is perhaps the most important characteristic in a technical or scientific discipline.

This book is the result of the author’s study of existing dimensional technology, his practical experience in the field, and his own independent work in further developing and refining the technique. These sources have enabled him to construct and include in the text over two hundred illustrative examples and problems, which he strongly advises the motivated reader to study carefully. For these examples and problems not only demonstrate in detail the great diversity of the practical applications of the technique, but they also serve as powerful learning routines that will develop and further illuminate the essential ideas.

Since the chapter titles in this book are rather descriptive and self-explanatory, the author feels he does not need to further explain the contents of each chapter here, as is frequently done in prefaces to technical books. It is pointed out, nevertheless, that the author’s main contribution to the dimensional method is in Chapter 8, where the compact and efficient technique, culminating in the Fundamental Formula, is presented. By this formula, the Dimensional Set is constructed, from which the generated dimensionless variables can be simply read off. These variables are the indispensable building blocks of all dimensional methods, including analysis and modeling. The introduced technique is then consistently employed throughout the balance of the book.

Apart from an inquisitive mind and a knowledge of basic mechanics and electricity, the only requisite to understanding and fully benefiting from the material presented is some familiarity with determinants and elementary matrix arithmetic, especially inversion and multiplication. Although these operations can be easily performed on most hand-held scientific calculators, for those readers who wish to go a little deeper into the subject and who feel that their relevant proficiency might require some brushing-up, an outline of the essentials of matrices and determinants is presented in Chapter 1. This chapter was written by Prof. Pál Rózsa, whose contributions to the field of linear algebra are known internationally, and whose roster of students includes the author.

Considerable effort was made to find and eliminate numerical errors, omissions, etc. from the text. However, in a work of this size it is virtually unavoidable that some mistakes will remain undetected. Therefore, the author would be grateful to anyone who, finding an error, communicates this to him through the publisher.

This book contains a great deal of information, and at times demands concentration. But this should not discourage the reader, for the effort spent will be amply rewarded by the acquired skills, proficiency, and even pleasure derived from possessing an extremely powerful tool in engineering and scientific disciplines.

The author wishes the reader an enriching, stimulating and rewarding journey through the ideas and thoughts presented in these pages.

Thomas Szirtes,     Toronto, Ontario, Canada, 1996 May

PREFACE TO THE SECOND EDITION

Apart from the correction of minor errors and misprints in the first edition, this second edition contains (in Chapter 18) nine more applications of dimensional analysis. These new examples are from biometry, applied and theoretical physics, electric circuitry, mechanics, hydrodynamics, and geometry.

In facilitating the publication of this book, the author greatly appreciates the expert and generous assistance of Joel Stein, Senior Editor of Elsevier Science, Highland Park, NJ, USA.

The author sincerely thanks Lev Gorelic, D.Sc., Chicago, IL, USA, and Eugene Brach, P.Eng., Ottawa, Canada, for their proficient scrutiny of the first edition, and as a result the identification of the numerous typographical and other similar errors.

Also, for expediting and preparing the graphics in the added text, the author acknowledges the valued contributions of Lynne Vanin and Richard McDonald at MacDonald Dettwiler and Associates, Brampton, Ontario, Canada.

Finally, the author sincerely thanks his wife, Penny, for careful scrutiny of the appended text, resulting in the elimination of grammatical and punctuation errors and other grave linguistic misdeeds.

Thomas Szirtes,     Toronto, Ontario, Canada, 2006 April

ORGANIZATION, NOTATION, AND CONVENTIONS

This book is composed of chapters and appendices. The chapters are consecutively numbered from 1 to 18 and are frequently composed of articles. Articles are denoted by single or, if necessary, multiple digits separated by periods. For example 3.4.2 designates Article 4.2 in Chapter 3. To identify or refer to this article, we write Art. 3.4.2. Appendices are consecutively numbered from 1 to 8. For example, the fifth appendix is referred to as Appendix 5.

Examples are consecutively numbered starting with 1 in each chapter. The designation consists of two (single or double-digit) numbers separated by a hyphen. These numbers identify, in order, the chapter and the example’s sequential position in the chapter. Thus, Example 18-12 is the twelfth example in Chapter 18. Note that the numbers are not in parentheses. The last line of every example is double underlined ==== followed by the symbol ⇑.

Problems are consecutively numbered starting from 1 in each chapter. The designation of a typical problem consists two (single or double-digit) numbers separated by a solidus (/). The first and second numbers identify the chapter and the problem’s sequential position in the chapter, respectively. Thus, Problem 4/15 designates the fifteenth problem in Chapter 4.

Figures in chapters are consecutively numbered starting with 1 in each chapter. The designation consists of two (single or double-digit) numbers separated by a hyphen. The first number indicates the chapter; the second number indicates the figure’s sequential position in that chapter. Thus, Fig. 4-2 designates the second figure in Chapter 4. Note that the numbers are not enclosed in parentheses and the word Figure is abbreviated to Fig.

Figures in appendices are consecutively numbered starting with 1 in each appendix. The designation is of the format Ax-n, where × is the appendix number and n is a number indicating the figure’s sequential position in that appendix. Thus, Fig. A3-1 designates the first figure in Appendix 3. Note that the word Figure is abbreviated to Fig.

Equations in the body of the text—not in the examples, problems, or appendices—are designated by two numbers separated by a hyphen. The first (single or double-digit) number indicates the chapter; the second (single or double-digit) number indicates the equation’s sequential position starting with 1 in each chapter. Equation numbers are always in parentheses. Thus, "(7-26)" designates the twenty-sixth equation in Chapter 7. On rare occasions, the second number is augmented by a lower-case letter, for example (7-19/a).

Equations in examples are designated by either a single lower-case letter or, rarely, a single character immediately followed by a one-digit number. These symbols are in alphabetic and, if applicable, numeric order, and are always in parentheses. Thus, in Example 17-9, the equations are designated by (a), (b), …, (x), whereas in Example 18-26, by (a1), (b1), …, (z1), (a2), (b2), …, (i2).

Equations in problems are designated by the symbol #, whose number starts with 1 in each problem. These symbols are always enclosed in parentheses.

Equations in the solutions of problems (in Appendix 6) are designated by asterisks whose number starts with 1 in each solution. These asterisks are always enclosed in parentheses. Thus, in the solution of Problem 4/10 (Appendix 6) there are two equations, and hence they are consecutively denoted by (*) and (**).

Equations in the proofs for selected theorems and formulas (in Appendix 7) are designated by a single one- or two-digit number in parentheses. Numeration starts with 1 for each proof. Thus, one will find the designations (1), (2), etc.

When referring to an equation, the word equation is usually omitted. Thus the sentence "The value of x = 8 is now substituted into (16-10) means The value of x = 8 is now substituted into equation (16-10). Similarly, the sentence … rearranging (6-12), we obtain … is equivalent to … rearranging equation (6-12), we obtain …."

Theorems are consecutively numbered starting with 1 in each chapter. The designation consists of two (single or double-digit) numbers separated by a hyphen. The first and second numbers identify the chapter and the theorem’s sequential position in the chapter, respectively. Thus, Theorem 7-5 is the fifth theorem in Chapter 7. Note that the numbers are not in parentheses.

Definitions are consecutively numbered starting with 1 in each chapter. The designation consists of two (single or double-digit) numbers separated by a hyphen. The first and second numbers identify the chapter and the definition’s sequential position in the chapter, respectively. Thus, Definition 10-1 is the first definition in Chapter 10. Note that the numbers are not in parentheses.

Matrices and vectors are always symbolized by bold characters. Thus B is the symbol for a matrix, but B is the symbol for a scalar quantity. Similarly, u is a matrix (or a vector), but u is a scalar. Symbols for matrices are capital or (rarely) lower-case characters, for vectors they are always lower-case letters.

A determinant is denoted by |U|, where U is any square If U is not a square matrix, then it does not have a determinant.

The dimension of any quantity z (variable or constant) is denoted by [z]. Thus, [speed] = m/s, [mass] = kg, [force] = (m·kg)/s², etc. If z is dimensionless, then [z] = 1, thus [7] = 1.

Dimensionless variables are denoted by the Greek letter πx, where x stands for a mandatory subscript. Thus, π1, π5, π21, πc are dimensionless variables, but π = 3.1415 (no subscript) is a dimensionless constant; the former have no connection to the latter, although of course [π1] = 1 and [π] = 1.

In tables where space is limited, the E-notation is used. Thus, the number 3.98 × 10−23 is written 3.98E-23, and similarly 5.4 × 10³ is written 5.4E3.

Finally, in writing this book the author has exerted some effort to avoid using clichés, modish expressions, and engineering jargon. Thus, in the following pages the word impact always means the collision of two bodies, and ongoing is completely absent. About the latter abomination, the reader may find the following anecdote to be both appropriate and amusing. A noted editor of a major New York magazine sitting in his penthouse office remarked upon reading a submitted manuscript: "If I find the word ongoing one more time, I will be downgoing and somebody will be outgoing."

CHAPTER 1

MATHEMATICAL PRELIMINARIES

By

Pál Rózsa

This chapter deals with the basic concepts of matrices, determinants and their applications in linear systems of equations. It is not the aim of this concise survey to teach mathematics, but to provide a very brief recapitulation so that the main text will be more easily understood.

More comprehensive treatment on linear systems can be found in a great many books; a selected list of such publications recommended by the writer is presented in Art. 1.5.

1.1 MATRICES AND DETERMINANTS

The array of m × n real numbers aij consisting of m rows and n columns

is called an m × n matrix with the numbers aij as its elements. The following notation will be used for a matrix:

where subscripts i and j denote the rows and columns, respectively. By interchanging the rows and columns of a matrix we get the transpose of the matrix denoted by AT. Thus we write AT = [aji]. As an example, let us consider the 3 × 4 matrix A

the transpose of which is a 4 × 3 matrix

If m = n, the matrix is called a square matrix of order n. Elements with subscripts ii of a square matrix constitute the main diagonal of the matrix. If all the elements not in the main diagonal of a matrix are zero, then the matrix is a diagonal is a diagonal matrix of order 3.

The unit matrix, or identity matrix I is defined as a diagonal matrix whose nonzero elements are all 1. For example

is an identity matrix of order 4.

If all the elements of a matrix are zero, we get the zero or null matrix, which is denoted by 0.

The m × 1 and 1 × n matrices are called column vectors and row vectors, respectively. For example,

is a column vector, and aT = [a1, a2, …, an] is a row vector.

A matrix can be split into several parts by means of horizontal and vertical lines. A matrix obtained in such a way is called a partitioned matrix. For example, let us consider matrix A partitioned as follows:

(1-1)

where A11, A12, A21, A22 are submatrices of A. Thus, by (1-1), the submatrices of A are

If a square matrix is partitioned in such a way that the submatrices along the main diagonal—i.e., submatrices with subscripts ii—are square matrices, then the given matrix is symmetrically partitioned.

A number can be assigned to any square matrix of order n. This number is called the determinant of the matrix and is denoted by |A|. We now give the formal definition of this number. The reader is advised not to be frightened off; a more palatable explanation is to follow!

Definition 1-1.

The determinant of a matrix A is

where q is the number of inversions in the permutation set j1 j2 … jn for numbers 1, 2, …, n, which are summed over all n! permutations of the first n natural numbers. (For example, the number of inversions in the permutation 35214 is 6, since 3 is in inversion with 2 and 1; 5 with 2, 1, 4; and 2 with 1).

According to this definition, any determinant of order 3 can be calculated in the following way:

Factoring out a11, a12 and a13, we can write for the above determinant

The expressions in parentheses are the determinants of the second order, namely

Therefore determinant |A| can be written as

(1-2)

Note that the elements of the first row of |A| are multiplied by the second-order determinants, which are obtained by omitting the first row and the corresponding column of |A|, and then affixing a negative sign to the second determinant. Expression (1-2) is called the expansion of the determinant by its first row. This technique can be generalized for any determinant of arbitrary order n, and, in fact, it is more useful, understandable, and much more practical to use than Definition 1-1. Thus, for the general case we proceed as follows:

First, we define the concept of cofactors. To any element aij of a determinant of order n can be assigned a subdeterminant of order n − 1, by omitting the ith row and the jth column of the determinant. Then we assign the sign (−1)i+j to it (the chessboard rule: white squares are positive, black squares are negative) and the result is the cofactor denoted by Aij. Note that the cofactor Aij already includes the appropriate sign!

Example 1-1

, what are the cofactors of its elements?

On the basis of Definition 1-1 and with the help of cofactors, the expression of a determinant can be formulated by the following theorem:

Theorem 1-1.

Any determinant |A| of order n can be expanded by its ith row (or jth column) by multiplying each element of the row (or column) by its cofactor and then summing the results. Thus,

(1-3)

(1-4)

An interesting characteristic of a determinant is that the sum of the products of the elements of any row and the cofactors of another row is zero (the same is true for columns). That is

(1-5)

The following two examples demonstrate the use of these methods.

Example 1-2

. Expanding it by the first row, we obtain

and by the second column

These two values are identical, of course. The reader may calculate the value by expanding the determinant by its third row, or by any column; the value in all cases will be 3.

Now we multiply the elements of the second row by the cofactors of the elements of the third row:

in which the second and third rows are equal.

Example 1-3

. Expanding it by the fourth column, we obtain

(a)

Next, we evaluate the above four third-order determinants by the method shown in Example 1-2. Hence

(b)

and therefore, by (a) and (b), |A| = −(−6)·(63) + (−1)·(0) − (0)·(−7) + (−3)·(35) = 273.

From Definition 1-1, the following important properties of a determinant can be derived:

• If all elements of a row are multiplied by a constant c, the determinant’s value is multiplied by c.

• If two rows are interchanged, the sign of the determinant’s value is changed.

• If a row of a determinant is a multiple of any other row, the determinant’s value is zero.

• If the multiple of a row is added to any other row, the determinant’s value does not change.

• If all the elements below (or above) the main diagonal are zero, the determinant’s value is the product of the elements of the main diagonal.

• The value of a determinant is a linear function of any of its elements (provided, of course, that all the other elements remain unchanged). That is, if z is any element, then the value of the determinant is D = k1·z + k2, where k1 and k2 are constants. In particular, if D0 and D1 are the values of the determinant if z = 0 and z = 1, respectively, then the value of the element z at which the determinant vanishes is

(1-6)

Remark: All of the above statements are true if the word row is changed to column.

1.2 OPERATIONS WITH MATRICES

Matrices A = [aij] and B = [bij] are equal if they have the same number of rows and columns, and if aij = bij for all i and j. If A is a square matrix and A = AT, then A is called a symmetric matrix. If A = −AT, then A is called a skew-symmetric is a skewsymmetric matrix.

Addition of two matrices and the multiplication of a matrix by a scalar are defined, respectively, as follows:

From these definitions, the distributive property follows:

We also have the following properties

Example 1-4

and c . Moreover,

The product of two matrices A·B are defined only if they are compatible. Matrices A and B are compatible if the number of columns of A equals the number of rows of B.

Definition 1-2.

If A is an m × p matrix and B is a p × n matrix, then the product A·B is an m × n matrix, the elements of which are the scalar products of the rows of A and the columns of B. Thus

(1-7)

Example 1-5

Let us calculate product A·B .

Since A is a 3 × 4 matrix and B is a 4 × 2 matrix, product A-B will be a 3 × 2 matrix. It is now advantageous to arrange A and B in the configuration shown in Fig. 1-1. For example, the top left element of matrix A·B is (1)·(3) + (2)·(1) + (3)·(4) + (4)·(6) = 41.

Figure 1-1 A didactically useful configuration to determine the product of matrices A and B The elements of product A·B appear at the intersections of the extensions of A’s rows and B’s columns

From Definition 1-2 it follows that the multiplication of matrices is not commutative, i.e., in general, A·B ≠ B·A. The lack of commutativity is the cause of most of the problems in matrix theory. On the other hand, this property gives the theory some of its beauty. Nevertheless, there are a few special matrices for which multiplication is commutative. Matrices with such a property are called commutative matrices. It is easy to verify that commutativity holds in the following cases:

• Any matrix is commutative with respect to the identity matrix.

• Any matrix is commutative with respect to the null matrix.

• If both A and B are diagonal matrices, then they are commutative.

By the repeated application of (1-7), it can be shown that the multiplication of three or more matrices is associative. That is

(1-8)

Moreover, the rule of distributivity holds, as well. That is

(1-9)

Hence Ap·Ap = Ap+q. The zero power of any square matrix of order n is defined to be the identity matrix of the same order. That is A⁰ = I.

The transpose of the product of matrices A and B is determined by the formula

(1-10)

Example 1-6

We shall consider the matrices in Example 1-5 and then verify relation (1-10).

(a)

By (1-10),

(b)

and we see that (a) and (b) are indeed identical.

From the definition of multiplication, the rule for the multiplication of partitioned matrices can be easily obtained. If matrices A and B are compatible, and if A is partitioned with regard to its columns in the same way as B is with regard to its rows, then product A·B can be calculated in terms of the generated submatrices.

Example 1-7

Given the following matrices and their partitioning:

(a)

Therefore

(b)

and

(c)

Thus, we write

(d)

and by direct multiplication

(e)

We see that the results of (e) and (d) are indeed identical.

We obtain special matrix products if the factors of a product are vectors. Thus, we define the inner product of two vectors

(1-11)

Obviously, the inner product produces a matrix of order 1, i.e., a scalar (this is a scalar product). Similarly, we define the outer product of two vectors, also called a dyad or dyadic product. Hence

(1-12)

If vectors a and b contain m and n elements, respectively, then the outer product a·bT is an m × n matrix.

Example 1-8

, we wish to determine aT·b and a·bT. Thus,

and

Utilizing now the outer products of vectors [see (1-12)], if in A·B, matrix A is partitioned into its columns and B into its rows, then the product obtained is the sum of the outer products, or dyads, formed by the columns of A and the rows of B. That is, the sum of dyads can always be written as the product of two matrices, where the first factor consists of the columns, the second factor the rows of the dyads. It follows that the problem of factoring a given matrix is equivalent to that of the decomposition of this matrix into dyads (i.e., outer products).

Example 1-9

Consider the matrix product

. The product written as the sum of outer products, or dyads, is

or

This result is identical to that shown in Fig. 1-1.

Let us now introduce the concept of the inverse of a matrix.

Definition 1-3.

Given the square matrix A. If X satisfies the equation A·X = I, then X is called the right inverse of A. If Y satisfies the equation Y·A = I, then Y is called the left inverse of A.

It can be shown that the right and left inverses exist if, and only if, the determinant of A is not zero. In this case X = Y.

Definition 1-4.

Matrix A is called nonsingular if |A| ≠ 0; if |A| = 0, then A is singular.

Definition 1-5.

The transpose of the matrix formed by the cofactors Aij of a matrix A = [aij] is called the adjoint of A and is denoted by adj A; i.e.,

(1-13)

The formulas (1-3), (1-4), and (1-5) imply that

(1-14)

(1-15)

Assuming now that A is nonsingular, we obtain from (1-14) and (1-15)

(1-16)

and

(1-17)

Based on the existence of the inverse of a matrix, we now state the following theorem:

Theorem 1-2.

If A is nonsingular, then its inverse is

(1-18)

Proof. Postmultiplication both sides of (1-17) by A−1 yields

This proves the theorem.

The inverse of a matrix product is equal to the product of the inverses taking the factors in reverse order. That is

(1-19)

To show this, by (1-8) we write

therefore (B−1A−1) = I·(A·B)−1 = (A·B)−1, which was to be demonstrated.

Example 1-10

Find the inverse of the matrix

By Definition 1-5, the adjoint matrix of A is

Since the determinant of A is |A| = −3, therefore, by (1-18), the inverse of A is

Example 1-11

Find the determinant of the matrix

Expanding the determinant by the first row, we get

Thus, matrix A is singular and hence its inverse does not exist. The adjoint of A is

and formula (1-14) gives

Now we introduce the concept of a permutation matrix.

Definition 1-6.

If the columns of an identity matrix are permuted, a permutation matrix is obtained. Accordingly,

(1-20)

where ej is the jth column vector of the identity matrix and the indices j1, j2, …, jn form a permutation of the natural numbers 1, 2, …, n. It follows that postmultiplication of an arbitrary matrix A by P results in a permutation of the columns of A.

Example 1-12

Consider

In this case permutation matrix P was generated from the fourth-order identity matrix I since

the first column of I became the second column of P,

the second column of I became the third column of P,

the third column of I became the fourth column of P,

the fourth column of I became the first column of P.

Therefore

the first column of A became the second column of A·P,

the second column of A became the third column of A·P,

the third column of A became the fourth column of A·P,

the fourth column of A became the first column of A·P.

In a similar way, the premultiplication of matrix A by PT will produce the same permutations of the rows of A.

Example 1-13

Consider

Here permutation matrix PT was generated from the fourth-order identity matrix I since

the first row of I became the second row of PT,

the second row of I became the third row of PT

the third row of I became the fourth row of PT,

the fourth row of I became the first row of PT.

Therefore

the first row of A became the second row of PT·A,

the second row of A became the third row of PT·A,

the third row of A became the fourth row of PT·A,

the fourth row of A became the first row of PT·A.

We now define the orthogonality of a matrix.

Definition 1-7.

If the inverse of matrix Q is equal to its transpose, i.e.,

(1-21)

then Q is called an orthogonal matrix.

It can be shown that every permutation matrix is orthogonal, i.e., PT = P−1.

Example 1-14

Consider permutation . Its inverse and its transpose . Therefore its transpose and inverse are identical and hence, by Definition 1-7, the matrix is orthogonal.

Example 1-15

Prove that matrix

is orthogonal.

We could first determine A−1, and then AT, to see whether they are identical. If they were, then by (1-21) A would be proven to be orthogonal. But to obtain the inverse of A is labor intensive. So to alleviate our burden, we consider that by (1-21) we can write

Thus we only have to come up with AT, which is easy. Accordingly,

Thus A is indeed orthogonal.

1.3 THE RANK OF A MATRIX

In a sense rank serves as a measure of the singularity of a matrix. To define and determine a matrix’s rank we shall use the notion of a minor. The determinant of a square submatrix of a given matrix is called a minor is a minor of matrix G. The former is associated with the element 4, the latter with the element 7. We have the following definition:

Definition 1-8.

The rank of matrix A is the order of its nonvanishing minor of highest order.

The rank of matrix A shall be denoted by R(A). Thus, R(A) = r means that all minors of order greater than r are equal to zero, but there is at least one minor of order r that is not zero.

It is now obvious that the outer product of two vectors (i.e., a dyad) gives a matrix of rank 1. Thus we write

(1-22)

Example 1-16

Recall the dyad in . Since here the rows (columns) are multiples of any other row (column), therefore any second-order minor (i.e., subdeterminant) must be zero, and hence the rank must be less than 2. On the other hand, since the elements (i.e., minors of order 1) are different from zero, the rank is 1 (it is sufficient if there is only one element different from zero). Therefore

In connection with this property of a dyad, another definition of the rank can be given; it is related to the sum of the dyads by which a matrix can be expressed.

Definition 1-9.

If

(1-23)

where r is the minimum number of dyads by which A can be expressed, then r is the rank of A.

We now present an algorithm to find the rank of any matrix by the minimal dyadic decomposition of that matrix. In order to make this process easy to understand, we use a numerical example.

Example 1-17

. What is its rank?

Step 1. Select any nonzero element of A1. Say we select the top left element 3, which is then marked, as above. We call this element the pivot.

Step 2. Generate matrix A2 such that

Note that the denominator of the fraction (just before the pivot’s column vector) is the pivot itself (in this case 3). If A2 happens to be a null matrix, then the process terminates and the rank of A1 is 1, which is then the largest subscript of a nonzero matrix. However, in our case here, A2 is not zero, and so we continue with Step 3.

Step 3. Select any nonzero element of A2. Say we select the top left nonzero element −3, which is then marked as above. Again, we call this element the pivot.

Step 4. Generate matrix A3

Since A3 is still not zero, the process continues.

Step 5. Select another pivot, , which is marked. With this we now have

Thus we got a null matrix, and hence in this example the largest subscript designating a nonzero matrix is 3; it follows that the rank of A1 is 3.

The above process can be condensed into the following compact protocol. We wish to find the rank of an arbitrary matrix A1.

Step 1. Set n = 1.

Step 2. Select a pivot "p" in