Advanced Modern Engineering Mathematics

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Advanced Modern

Engineering Mathematics

Glyn James fourth edition

Advanced Modern Engineering

Mathematics

Fourth Edition

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Advanced Modern Engineering

Mathematics

Fourth Edition

Glyn James Coventry University

and

David Burley University of Sheffield

Dick Clements University of Bristol

Phil Dyke University of Plymouth

John Searl University of Edinburgh

Nigel Steele Coventry University

Jerry Wright AT&T

Essex CM20 2JE

England

and Associated Companies throughout the world

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© Pearson Education Limited 1993, 2011

The rights of Glyn James, David Burley, Dick Clements, Phil Dyke, John Searl,

Nigel Steele and Jerry Wright to be identified as authors of this work have been asserted

by them in accordance with the Copyright, Designs and Patents Act 1988.

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ISBN: 978-0-273-71923-6

British Library Cataloguing-in-Publication Data

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

Library of Congress Cataloging-in-Publication Data

Advanced modern engineering mathematics / Glyn James [et al.] –

Contents

1.5 Numerical methods 30

2.1 Introduction 116

2.3 Numerical solution of first-order ordinary differential

Chapter 2 Numerical Solution of Ordinary Differential Equations 115

3.2 Derivatives of a scalar point function 199

5.2.3 Existence of the Laplace transform 353

5.5.11 Relationship between Heaviside step and impulse functions 418

5.7 Solution of state-space equations 450

6.6 Discrete linear systems: characterization 509

7.2.4 Even and odd functions 573

8.2 The Fourier transform 638

8.8.3 Identification and isolation of the

8.9 Engineering application: direct design of digital filters

9.7 Integral solutions 815

10.5 Engineering application: chemical processing plant 896

Chapter 11 Applied Probability and Statistics 905

11.6.2 Contingency tables 949

Throughout the course of history, engineering and mathematics have developed inparallel All branches of engineering depend on mathematics for their description andthere has been a steady flow of ideas and problems from engineering that has stimulatedand sometimes initiated branches of mathematics Thus it is vital that engineering stu-dents receive a thorough grounding in mathematics, with the treatment related to theirinterests and problems As with the previous editions, this has been the motivation for

the production of this fourth edition – a companion text to the fourth edition of Modern Engineering Mathematics, this being designed to provide a first-level core studies

course in mathematics for undergraduate programmes in all engineering disciplines.Building on the foundations laid in the companion text, this book gives an extensivetreatment of some of the more advanced areas of mathematics that have applications invarious fields of engineering, particularly as tools for computer-based system model-ling, analysis and design Feedback, from users of the previous editions, on subjectcontent has been highly positive indicating that it is sufficiently broad to provide thenecessary second-level, or optional, studies for most engineering programmes, where

in each case a selection of the material may be made Whilst designed primarily for use

by engineering students, it is believed that the book is also suitable for use by students

of applied mathematics and the physical sciences

Although the pace of the book is at a somewhat more advanced level than the panion text, the philosophy of learning by doing is retained with continuing emphasis

com-on the development of students’ ability to use mathematics with understanding to solveengineering problems Recognizing the increasing importance of mathematical model-ling in engineering practice, many of the worked examples and exercises incorporatemathematical models that are designed both to provide relevance and to reinforce therole of mathematics in various branches of engineering In addition, each chapter con-tains specific sections on engineering applications, and these form an ideal frameworkfor individual, or group, study assignments, thereby helping to reinforce the skills ofmathematical modelling, which are seen as essential if engineers are to tackle theincreasingly complex systems they are being called upon to analyse and design Theimportance of numerical methods in problem solving is also recognized, and its treat-ment is integrated with the analytical work throughout the book

Much of the feedback from users relates to the role and use of software packages,particularly symbolic algebra packages Without making it an essential requirement theauthors have attempted to highlight throughout the text situations where the user couldmake effective use of software This also applies to exercises and, indeed, a limitednumber have been introduced for which the use of such a package is essential Whilstany appropriate piece of software can be used, the authors recommend the use ofMATLAB and /or MAPLE In this new edition more copious reference to the use of these

two packages is made throughout the text, with commands or codes introduced andillustrated When indicated, students are strongly recommended to use these packages

to check their solutions to exercises This is not only to help develop proficiency in theiruse, but also to enable students to appreciate the necessity of having a sound knowledge

of the underpinning mathematics if such packages are to be used effectively Throughoutthe book two icons are used:

• An open screen indicates that the use of a software package would be useful(e.g for checking solutions) but not essential

• A closed screen indicates that the use of a software package is essential orhighly desirable

As indicated earlier, feedback on content from users of previous editions has beenfavourable, and consequently no new chapter has been introduced However, inresponse to feedback the order of presentation of chapters has been changed, with aview to making it more logical and appealing to users This re-ordering has necessitatedsome redistribution of material both within and across some of the chapters Anothernew feature is the introduction of the use of colour It is hoped that this will make the textmore accessible and student-friendly Also, in response to feedback individual chaptershave been reviewed and updated accordingly The most significant changes are:

• Chapter 1 Matrix Analysis: Inclusion of new sections on ‘Singular value position’ and ‘Lyapunov stability analysis’

decom-• Chapter 5 Laplace transform: Following re-ordering of chapters a more unifiedand extended treatment of transfer functions/transfer matrices for continuous-time state-space models has been included

• Chapter 6 Z-transforms: Inclusion of a new section on ‘Discretization ofcontinuous-time state-space models’

• Chapter 8 Fourier transform: Inclusion of a new section on ‘Direct design ofdigital filters and windows’

• Chapter 9 Partial differential equations: The treatment of first order equationshas been extended and a new section on ‘Integral solution’ included

• Chapter 10 Optimization: Inclusion of a new section on ‘Least squares’

A comprehensive Solutions Manual is available free of charge to lecturers adopting thistextbook It will also be available for download via the Web at: www.pearsoned.co.ck/james

Acknowledgements

The authoring team is extremely grateful to all the reviewers and users of the text whohave provided valuable comments on previous editions of this book Most of this hasbeen highly constructive and very much appreciated The team has continued to enjoythe full support of a very enthusiastic production team at Pearson Education and wishes

to thank all those concerned Finally I would like to thank my wife, Dolan, for her fullsupport throughout the preparation of this text and its previous editions

Glyn James

Coventry University July 2010

About the Authors

at Coventry University in 2001 and is now Emeritus Professor in Mathematics at theUniversity He graduated from the University College of Wales, Cardiff in the late 1950s,obtaining first class honours degrees in both Mathematics and Chemistry He obtained

a PhD in Engineering Science in 1971 as an external student of the University of Warwick

He has been employed at Coventry since 1964 and held the position of the Head ofMathematics Department prior to his appointment as Dean in 1992 His research interestsare in control theory and its applications to industrial problems He also has a keeninterest in mathematical education, particularly in relation to the teaching of engineer-ing mathematics and mathematical modelling He was co-chairman of the EuropeanMathematics Working Group established by the European Society for EngineeringEducation (SEFI) in 1982, a past chairman of the Education Committee of the Institute

of Mathematics and its Applications (IMA), and a member of the Royal Society matics Education Subcommittee In 1995 he was chairman of the Working Group thatproduced the report ‘Mathematics Matters in Engineering’ on behalf of the professionalbodies in engineering and mathematics within the UK He is also a member of theeditorial/advisory board of three international journals He has published numerouspapers and is co-editor of five books on various aspects of mathematical modelling He

Mathe-is a past Vice-President of the IMA and has also served a period as Honorary Secretary

of the Institute He is a Chartered Mathematician and a Fellow of the IMA

mathe-matics from King’s College, University of London in 1955 and obtained his PhD inmathematical physics After working in the University of Glasgow, he spent most of hisacademic career in the University of Sheffield, being Head of Department for six years

He has long experience of teaching engineering students and has been particularlyinterested in encouraging students to construct mathematical models in physical andbiological contexts to enhance their learning His research work has ranged throughstatistical mechanics, optimization and fluid mechanics He has particular interest in theflow of molten glass in a variety of situations and the application of results in the glassindustry Currently he is involved in a large project concerning heat transfer problems

in the deep burial of nuclear waste

at Bristol University He read for the Mathematical Tripos, matriculating at Christ’sCollege, Cambridge in 1966 He went on to take a PGCE at Leicester University School

of Education (1969 –70) before returning to Cambridge to research a PhD in AeronauticalEngineering (1970 –73) In 1973 he was appointed Lecturer in Engineering Mathematics

at Bristol University and has taught mathematics to engineering students ever since,

becoming successively Senior Lecturer, Reader and Professorial Teaching Fellow He hasundertaken research in a wide range of engineering topics but is particularly interested

in mathematical modelling and in new approaches to the teaching of mathematics to

engineering students He has published numerous papers and one previous book, ematical Modelling: A Case Study Approach He is a Chartered Engineer, a Chartered

Math-Mathematician, a member of the Royal Aeronautical Society, a Fellow of the Institute

of Mathematics and Its Applications, an Associate Fellow of the Royal Institute ofNavigation, and a Fellow of the Higher Education Academy He retired from full time work

in 2007 but continues to teach and pursue his research interests on a part time basis

Head of School of Mathematics and Statistics for 18 years then Head of School ofComputing, Communications and Electronics for four years but he now devotes histime to teaching and research After graduating with a first in mathematics he gained

a PhD in coastal engineering modelling He has over 35 years’ experience teachingundergraduates, most of this teaching to engineering students He has run an interna-tional research group since 1981 applying mathematics to coastal engineering and shal-low sea dynamics Apart from contributing to these engineering mathematics books, hehas written seven textbooks on mathematics and marine science, and still enjoys trying

to solve environmental problems using simple mathematical models

University of Edinburgh before his recent retirement As well as lecturing on ical education, he taught service courses for engineers and scientists His most recentresearch concerned the development of learning environments that make for the effectivelearning of mathematics for 16–20 year olds As an applied mathematician who workedcollaboratively with (among others) engineers, physicists, biologists and pharmacologists,

mathemat-he is keen to develop tmathemat-he problem-solving skills of students and to provide tmathemat-hem withopportunities to display their mathematical knowledge within a variety of practical con-texts These contexts develop the extended reasoning needed in all fields of engineering

2004 He has had a career-long interest in engineering mathematics and its teaching,particularly to electrical and control engineers Since retirement he has been EmeritusProfessor of Mathematics at Coventry, combining this with the duties of HonorarySecretary of the Institute of Mathematics and its Applications Having responsibility forthe Institute’s education matters he became heavily involved with a highly successfulproject aimed at encouraging more people to study for mathematics and other ‘maths-rich’courses (for example Engineering) at University He also assisted in the development

of the mathematics content for the advanced Engineering Diploma, working to ensurethat students were properly prepared for the study of Engineering in Higher Education

Jersey, USA He graduated in Engineering (BSc and PhD at the University of Southampton)and in Mathematics (MSc at the University of London) and worked at the National PhysicalLaboratory before moving to the University of Bristol in 1978 There he acquired wideexperience in the teaching of mathematics to students of engineering, and became SeniorLecturer in Engineering Mathematics He held a Royal Society Industrial Fellowshipfor 1994, and is a Fellow of the Institute of Mathematics and its Applications In 1996 hemoved to AT&T Labs (formerly part of Bell labs) to continue his research in spokenlanguage understanding, human/computer dialog systems, and data mining

Publisher’s

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We are grateful to the following for permission to reproduce copyright material:

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Woodhead Publishing Ltd (Chapman, N, Goodhall, D, Steele, N)

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we would appreciate any information that would enable us to do so

In this chapter we turn our attention again to matrices, first considered in Chapter 5

of Modern Engineering Mathematics, and their applications in engineering At the

outset of the chapter we review the basic results of matrix algebra and briefly introducevector spaces

As the reader will be aware, matrices are arrays of real or complex numbers, and have

a special, but not exclusive, relationship with systems of linear equations An (incorrect)initial impression often formed by users of mathematics is that mathematicians havesomething of an obsession with these systems and their solution However, such systemsoccur quite naturally in the process of numerical solution of ordinary differential equa-tions used to model everyday engineering processes In Chapter 9 we shall see that theyalso occur in numerical methods for the solution of partial differential equations, forexample those modelling the flow of a fluid or the transfer of heat Systems of linear

first-order differential equations with constant coefficients are at the core of the

of such systems can conveniently be performed in the state-space representation, withthis form assuming a particular importance in the case of multivariable systems

In all these areas it is convenient to use a matrix representation for the systems underconsideration, since this allows the system model to be manipulated following the rules

of matrix algebra A particularly valuable type of manipulation is simplification in some

sense Such a simplification process is an example of a system transformation, carriedout by the process of matrix multiplication At the heart of many transformations are

the eigenvalues and eigenvectors of a square matrix In addition to providing the means

by which simplifying transformations can be deduced, system eigenvalues provide vitalinformation on system stability, fundamental frequencies, speed of decay and long-termsystem behaviour For this reason, we devote a substantial amount of space to theprocess of their calculation, both by hand and by numerical means when necessary Ourtreatment of numerical methods is intended to be purely indicative rather than complete,because a comprehensive matrix algebra computational tool kit, such as MATLAB, isnow part of the essential armoury of all serious users of mathematics

In addition to developing the use of matrix algebra techniques, we also demonstratethe techniques and applications of matrix analysis, focusing on the state-space system modelwidely used in control and systems engineering Here we encounter the idea of a function

of a matrix, in particular the matrix exponential, and we see again the role of theeigenvalues in its calculation This edition also includes a section on singular valuedecomposition and the pseudo inverse, together with a brief section on Lyapunov stability

of linear systems using quadratic forms

Review of matrix algebra

This section contains a summary of the definitions and properties associated with matrices

and determinants A full account can be found in chapters of Modern Engineering Mathematics or elsewhere It is assumed that readers, prior to embarking on this chapter,

have a fairly thorough understanding of the material summarized in this section

1.1

1.2

1.2.1 Definitions

(a) An array of real numbers

is called an m × n matrix with m rows and n columns The a ij is referred to as the

then A is called a square matrix of order n If the matrix has one column or one

row then it is called a column vector or a row vector respectively.

( b) In a square matrix A of order n the diagonal containing the elements a11, a22, ,

a nn is called the principal or leading diagonal The sum of the elements in this diagonal is called the trace of A, that is

(c) A diagonal matrix is a square matrix that has its only non-zero elements along the leading diagonal A special case of a diagonal matrix is the unit or identity matrix I

for which a11= a22= = a nn= 1

(d) A zero or null matrix 0 is a matrix with every element zero.

(e) The transposed matrix AT is the matrix A with rows and columns interchanged,

its i, jth element being a ji

(f ) A square matrix A is called a symmetric matrix if AT= A It is called skew symmetric if AT= −A

1.2.2 Basic operations on matrices

In what follows the matrices A, B and C are assumed to have the i, jth elements a ij , b ij and c ij respectively

(i) commutative law: A+ B= B+ A

(ii) associative law: (A+ B) + C= A+ (B+ C)(iii) distributive law: λ(A+ B) = λA+ λB, λ scalar

Matrix multiplication

If A is an m × p matrix and B a p × n matrix then we define the product C= AB as the

m × n matrix with elements

(ii) Associative law: A(BC) = (AB)C

(iii) If λ is a scalar then(λA)B= A(λB) = λAB

(iv) Distributive law over addition:

(A+ B)C= AC+ BC

A(B+ C) = AB+ AC

Note the importance of maintaining order of multiplication

(v) If A is an m × n matrix and if I m and I n are the unit matrices of order m and n

respectively then

I m A= AI n= A

Properties of the transpose

If AT is the transposed matrix of A then(i) (A+ B)T= AT+ BT

1.2.3 Determinants

The determinant of a square n × n matrix A is denoted by det A or |A|

If we take a determinant and delete row i and column j then the determinant

remaining is called the minor M ij of the i, jth element In general we can take any row

A minor multiplied by the appropriate sign is called the cofactor A ij of the i, jth element

so A ij= (−1)i +j M ij and thus

Some useful properties

(i) |AT| = |A|

(ii) |AB| = |A||B|

(iii) A square matrix A is said to be non-singular if |A| ≠ 0 and singular if |A| = 0

1.2.4 Adjoint and inverse matrices

(ii) | adj A| = |A|n−1, n being the order of A

(iii) adj (AB) = (adj B)(adj A)

(i) If A is non-singular then |A| ≠ 0 and A−1= (adj A)/|A|

(ii) If A is singular then |A| = 0 and A−1 does not exist

(iii) (AB)−1= B−1A−1

All the basic matrix operations may be implemented in MATLAB and MAPLEusing simple commands In MATLAB a matrix is entered as an array, with rowelements separated by spaces (or commas) and each row of elements separated by asemicolon(;), or the return key to go to a new line Thus, for example,

C=A+B, C=A-B, C=A*B

The trace of the matrix A is determined by the command trace(A), and itsdeterminant by det(A)

Multiplication of a matrix Aby a scalar is carried out using the command *, whileraising Ato a given power is carried out using the command ^ Thus, for example,

3A2 is determined using the command C=3*A^2.The transpose of a real matrix A is determined using the apostrophe ’ key; that

is C=A’ (to accommodate complex matrices the command C=A.’ should be used) The inverse of A is determined by C=inv(A)

For matrices involving algebraic quantities, or when exact arithmetic is desirableuse of the Symbolic Math Toolbox is required; in which matrices must be expressed

in symbolic form using the sym command The command A=sym(A) generates thesymbolic form of A For example, for the matrix

1.2.5 Linear equations

In this section we reiterate some definitive statements about the solution of the system

of simultaneous linear equations

C:=A+B; and C:=A–B;

Multiplication of a matrix A by a scalar k is implemented using the command k*A;

so, for example, (2A + 3B) is implemented by

2*A+3*B;

The product AB of two matrices is implemented by either of the following twocommands:

A.B; or Multiply(A,B);

(Note: A*B will not work)

The transpose, trace, determinant, adjoint and inverse of a matrix A are returnedusing, respectively, the commands:

or, in matrix notation,

If b= 0 and |A|= 0 then we have infinitely many solutions

Case (iv) is one of the most important, since from it we can deduce the importantresult that the homogeneous equation A x= 0 has a non-trivial solution if and only

if |A|= 0

Provided that a solution to (1.1) exists it may be determined in MATLAB using thecommand x=A\b For example, the system of simultaneous equations

x+y+z= 6, x+ 2y+ 3z = 14, x + 4y + 9z = 36

may be written in the matrix form

Entering A and b and using the command x = A\b provides the answer x = 1, y = 2, z = 3.

x

=

61436

b

1.2.6 Rank of a matrix

The most commonly used definition of the rank, rank A, of a matrix A is that it is the order

of the largest square submatrix of A with a non-zero determinant, a square submatrixbeing formed by deleting rows and columns to form a square matrix Unfortunately it

is not always easy to compute the rank using this definition and an alternative definition,which provides a constructive approach to calculating the rank, is often adopted First,using elementary row operations, the matrix A is reduced to echelon form

in which all the entries below the line are zero, and the leading element, marked *, ineach row above the line is non-zero The number of non-zero rows in the echelon form

is equal to rank A

When considering the solution of equations (1.1) we saw that provided the determinant

of the matrix A was not zero we could obtain explicit solutions in terms of the inverse matrix.However, when we looked at cases with zero determinant the results were much less clear.The idea of the rank of a matrix helps to make these results more precise Defining the

we can state the results of cases (iii) and (iv) of Section 1.2.5 more clearly as follows:

In MAPLE the commands

with(LinearAlgebra):

soln:=LinearSolve(A,b);

will solve the set of linear equations A x = b for the unknown x when A , b given.

Thus for the above set of equations the commands

If A and (A:b) have different rank then we have no solution to (1.1) If the twomatrices have the same rank then a solution exists, and furthermore the solutionwill contain a number of free parameters equal to (n− rank A)

Vector spaces

Vectors and matrices form part of a more extensive formal structure called a vector space

The theory of vector spaces underpins many modern approaches to numerical methodsand the approximate solution of many of the equations that arise in engineering analysis

In this section we shall, very briefly, introduce some of the basic ideas of vector spacesnecessary for later work in this chapter

Definition

A real vector space V is a set of objects called vectors together with rules for additionand multiplication by real numbers For any three vectors a,b and c in V and any realnumbers α and β the sum a+b and the product αa also belong to V and satisfy thefollowing axioms:

In MATLAB the rank of the matrix A is generated using the command rank(A).For example, if

the commands

A=[-1 2 2; 0 0 1; -1 2 0];

rank(A)

generateans=2

In MAPLE the command is also rank(A).

It is clear that the real numbers form a vector space The properties given are also

satisfied by vectors and by m × n matrices so vectors and matrices also form vector spaces The space of all quadratics a + bx + cx2 forms a vector space, as can be estab-lished by checking the axioms, (a) – (h) Many other common sets of objects also formvector spaces If we can obtain useful information from the general structure then thiswill be of considerable use in specific cases

1.3.1 Linear independence

The idea of linear dependence is a general one for any vector space The vector x is said

to be linearly dependent on x1, x2, , x m if it can be written as

x = α1x1+ α2x2+ + αm x m

for some scalars α1, , αm The set of vectors y1, y2, , y m is said to be linearly

β1y1+ β2y2+ + βm y m= 0implies that β1= β2= = βm= 0

Let us now take a linearly independent set of vectors x1, x2, , x m in V and

con-struct a set consisting of all vectors of the form

form a linearly independent set and describe S(e1, e2) geometrically

is only satisfied if α = β = 0, and hence e1 and e2 are linearly independent

S(e1, e2) is the set of all vectors of the form , which is just the (x1, x2)

plane and is a subset of the three-dimensional Euclidean space

Example 1.1

e1

100

010

=

0 αe1+βe2

αβ0

αβ0

If we can find a set B of linearly independent vectors x1, x2, , x n in V such that

S(x1, x2, , x n) = V

then B is called a basis of the vector space V Such a basis forms a crucial part of the theory, since every vector x in V can be written uniquely as

x = α1x1+ α2x2+ + αn x n The definition of B implies that x must take this form To establish uniqueness, let us assume that we can also write x as

x = β1x1+ β2x2+ + βn x n

Then, on subtracting,

0 = (α1− β1)x1+ + (αn− βn )x n and since x1, , x n are linearly independent, the only solution is α1= β1, α2= β2, ;

hence the two expressions for x are the same.

It can also be shown that any other basis for V must also contain n vectors and that any n + 1 vectors must be linearly dependent Such a vector space is said to have

A + B(x − 1) + Cx(x − 1) We note that this space is three-dimensional.

1.3.2 Transformations between bases

Since any basis of a particular space contains the same number of vectors, we can look

at transformations from one basis to another We shall consider a three-dimensional

space, but the results are equally valid in any number of dimensions Let e1, e2, e3 and

e1′, e2′, e′3 be two bases of a space From the definition of a basis, the vectors e1′, e2′ and e′3

can be written in terms of e1, e2 and e3 as

(1.2)

e1

100

010

001

=

d1

100

110

111

Taking a typical vector x in V, which can be written both as

Thus changing from one basis to another is equivalent to transforming the coordinates

by multiplication by a matrix, and we thus have another interpretation of matrices

Successive transformations to a third basis will just give x′ = B x″, and hence the

composite transformation is x = (AB )x″ and is obtained through the standard matrixrules

For convenience of working it is usual to take mutually orthogonal vectors as abasis, so that and = δij, where δij is the Kronecker delta

Using (1.2) and multiplying out these orthogonality relations, we have

The eigenvalue problem

A problem that leads to a concept of crucial importance in many branches of

math-ematics and its applications is that of seeking non-trivial solutions x ≠ 0 to the matrixequation

A x = λxThis is referred to as the eigenvalue problem; values of the scalar λ for which non-

trivial solutions exist are called eigenvalues and the corresponding solutions x ≠ 0 are

called the eigenvectors Such problems arise naturally in many branches of engineering.

For example, in vibrations the eigenvalues and eigenvectors describe the frequency andmode of vibration respectively, while in mechanics they represent principal stressesand the principal axes of stress in bodies subjected to external forces In Section 1.11,and later in Section 5.7.1, we shall see that eigenvalues also play an important role inthe stability analysis of dynamical systems

For continuity some of the introductory material on eigenvalues and eigenvectors,

contained in Chapter 5 of Modern Engineering Mathematics, is first revisited.

1.4

Which of the following sets form a basis for a

three-dimensional Euclidean space?

Under this, how does the vector

x = x1e1+ x2e2+ x3e3 transform and what

is the geometrical interpretation? What lines transform into scalar multiples of themselves?

Show that the set of all cubic polynomials forms a vector space Which of the following sets of functions are bases of that space?

(a) {1, x, x2, x3 } (b) {1 − x, 1 + x, 1 − x3

, 1 + x3 } (c) {1 − x, 1 + x, x2

(1 − x), x2

(1 + x)}

(d) {x(1 − x), x(1 + x), 1 − x3

, 1 + x3 } (e) {1 + 2x, 2x + 3x2, 3x2+ 4x3, 4x3 + 1}

Describe the vector space

S(x + 2x3

, 2x − 3x5

, x + x3 ) What is its dimension?

1 2 3

1 0 1

1 2 3

3 2 5 1

0

0

1 1 0

2 1 0 2

1 0 0

0 1 0

0 0 1

=

2 -

=

1 1 0

2 -

=

1

−1 0

e3 ′

0 0 1

=

3

4

1.4.1 The characteristic equation

The set of simultaneous equations

where A is an n × n matrix and x = [x1 x2 x n]T is an n × 1 column vector can

be written in the form

where I is the identity matrix The matrix equation (1.5) represents simply a set ofhomogeneous equations, and we know that a non-trivial solution exists if

Here c( λ) is the expansion of the determinant and is a polynomial of degree n in λ,

called the characteristic polynomial of A Thus

c(λ) = λn + c n−1λn−1+ c n−2λn−2+ + c1λ + c0

and the equation c(λ) = 0 is called the characteristic equation of A We note that thisequation can be obtained just as well by evaluating |A− λI| = 0; however, the form(1.6) is preferred for the definition of the characteristic equation, since the coefficient

of λn is then always +1

In many areas of engineering, particularly in those involving vibration or the control

of processes, the determination of those values of λ for which (1.5) has a non-trivial

solution (that is, a solution for which x ≠ 0) is of vital importance These values of

λ are precisely the values that satisfy the characteristic equation, and are called the

eigenvalues of A

Find the characteristic equation for the matrix

Solution By (1.6), the characteristic equation for A is the cubic equation

Expanding the determinant along the first column gives