Matrix operations for engineers and scientists: An essential guide in linear algebra

327 Pages·2010·2.15 MB·English

by  Alan Jeffrey (auth.)

Matrix Operations for Engineers and Scientists
.
Alan Jeffrey
Matrix Operations for
Engineers and Scientists
An Essential Guide in Linear Algebra
Prof.Dr.AlanJeffrey{
16BruceBldg.
UniversityofNewcastle
NE17RUNewcastleuponTyne
UnitedKingdom
ISBN978-90-481-9273-1 e-ISBN978-90-481-9274-8
DOI10.1007/978-90-481-9274-8
SpringerDordrechtHeidelbergLondonNewYork
LibraryofCongressControlNumber:2010932003
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Preface
ThisbookisbasedonmanycoursesgivenbytheauthortoEnglishandAmerican
undergraduatestudentsinengineeringandtheappliedsciences.Thebookseparates
naturallyintotwodistinctparts,althoughthesearenotshownaspartsoneandtwo.
The first part, represented by Chapters 1–4 and a large part of Chapter 5, gives a
straightforwardaccountoftopicsfromthetheoryofmatricesthatformpartofevery
basic mathematics course given to undergraduate students in engineering and the
appliedsciences.However,thepresentationofthebasicmaterialgiveninthisbook
is in greater detail than is usually found in such courses. The only unusual topics
appearinginthefirstpartofthebookareinChapter3.Thesearetheinclusionofthe
techniqueofleast-squaresfittingofpolynomialstoexperimentaldata,andtheway
matrices enter into a finite difference approximation for the numerical solution of
the Laplace equation. The least-squares fitting of polynomials has been included
becauseitisusefulandprovides a simpleapplication ofmatrices,while the finite
differenceapproximationfortheLaplaceequationshowshowmatricesplayavital
part in the numerical solution of this important partial differential equation. This
last application also demonstrates one of the ways in which very large matrix
equations can be generated when seeking the numerical solution of certain types
ofproblem.
The last part of Chapter 5 forms the start of the second part of the book, and
containsvarious important topics which, although belonging tothe subject matter
of the chapter, are not discussed in courses as often as they deserve. Chapter 6
describes a matrix approach to the study of systems of ordinary differential equa-
tions and, although this approach is straightforward and found in courses for
mathematics majors, it is still a relatively new topic in courses for engineers and
applied scientists. In particular, the chapter shows how to use matrices when
solving the homogeneous and nonhomogeneous systems of linear constant coeffi-
cient differential equations that model so many physical situations. It makes full
use of the diagonalization of matrices when seeking solutions of systems of
differentialequations,anditalsoshowshowtheLaplacetransformcanbeapplied
to matrix systems of differential equations. The chapter also provides motivation
v
vi Preface
for the concept of the matrix exponential, which is then applied to differential
equations.
Chapter7usesmatricesasatypicalmodelwhenexplainingthenotionofvector
spaces that are the key to understanding many applications of mathematics. This
enables the basic ideas of vector spaces to be introduced at an early stage in an
undergraduate course. Chapter 8 develops the important and useful concept of a
linear transformation and provides motivation by using matrices when applying
lineartransformationstothegeometryoftheplane.Theseapplicationsillustratethe
generalideasoflineartransformationsintermsofsimpleandfamiliargeometrical
operations like stretching, rotating and reflecting shapes while, at the same time,
relating them directly to the study of matrices. Although these applications are
elementary, they are nevertheless useful, because while they can be combined to
make more complicated transformations, they also serve as a foundation for the
techniquesusedinapplicationsasdiverseassolidmechanics,crystallographyand
computergraphics.
Thisbookcanbeusedasatextforacourse,tosupplementanexistingcourse,for
private study, or to refresh and extend the reader’s knowledge of the theory of
matrices.Allchaptersareprovidedwithclearanddetailedillustrativeexamplesas
each new idea is introduced, so, for example, attention is drawn to the fact that a
twicerepeatedeigenvaluedoesnotnecessarilyhaveassociatedwithittwolinearly
independent eigenvectors, and it is then shown how this influences the nature of
solutionsofsystemsofdifferentialequations.ApartfromChapter6ondifferential
equations,nosystematicattempthasbeenmadetodescribethenumerousapplica-
tionsofmatricesthatarepossible.Nevertheless,becauseoftheintendedreadership
of the book, where appropriate a few relevant applications have been included.
Some of these applications have already been mentioned, but others illustrate the
waymatricescanbeusedtosolvelinearsecond-orderdifferenceequationslikethe
onethatgeneratestheFibonaccisequenceand,becauseoftheimportanceoftwo-
pointboundary-valueproblemsinapplicationsofdifferentialequations,itisshown
howmatricesenterintothenumericalsolutionofsomeoftheseproblems.
Throughoutthebook,workedexamplesarenumerousandtheyaresupplemen-
tedbyexercisesetsattheendofeachchapter.Solutionsforalloftheexercisesare
given attheendofthe book,always provided with adequate detailifamethod of
solution is notcompletely obvious. Detailed explanations of new ideas have been
given throughout the book, because the author’s experience has shown that an
inadequate explanation when a topic is first encountered can cause unnecessary
difficulties for a student at later stages of study when matrix methods need to be
applied.
Thereadyavailabilityofcomputeralgebrasoftwaremakesthemanipulationof
matricesasimplematterand,inreallifeapplications,suchsoftwareshouldbeused
whenever possible and, indeed, for complicated and large problems its use is
essential. However, the use of such software tools when learning about matrices,
before having first understood the underlying theory by working well-chosen
examples by hand with the help of a hand-held calculator, is likely to limit the
Preface vii
reader’sabilitytomakefulluseofmatrices whenthetimecomestoapplymatrix
methodstonewproblems.
The efficient ways software manipulates matrices when performing numerical
operations,likefindingtherankofamatrix,itseigenvaluesandeigenvectors,and
accelerating computations while maintaining high accuracy, depend for their suc-
cess on the use of sophisticated numerical techniques. Of necessity, the approach
usedinsuchsoftwarewilldifferfromthewaythesameoperationsaredescribedin
this book, where only straightforward and direct methods are given, and the
necessary numerical calculations in examples and exercises have been reduced to
a minimum. For example, to simplify the numerical calculations involved when
workingwitheigenvaluesandeigenvectors,theworkedexamplesandexercisesets
dealing with this topic have been constructed in such a way that, whenever a
characteristic equation occurs, its roots can be found by inspection. This allows
theanalysistoproceedwithouttheinterruptionthatwouldotherwisebecausedifa
numericalroot-findingtechniqueforpolynomialshadfirsttobeexplainedandthen
used.
It is hoped readers will find the book helpful when working with matrices and
when applying linear algebra, and that it will encourage them to apply matrix
methods to the wide range of problems that are often solved less efficiently and
conciselybyothermeans.
UniversityofNewcastle AlanJeffrey
MatricesandLinearAlgebraforEngineeringandScience
.
Contents
1 MatricesandLinearSystemsofEquations ............................. 1
1.1 SystemsofAlgebraicEquations ....................................... 1
1.2 SuffixandMatrixNotation ............................................ 3
1.3 Equality,AdditionandScalingofMatrices ........................... 4
1.4 SomeSpecialMatricesandtheTransposeOperation ................. 6
2 Determinants,andLinearIndependence .............................. 13
2.1 IntroductiontoDeterminantsandSystemsofEquations ............ 13
2.2 AFirstLookatLinearDependenceandIndependence .............. 15
2.3 PropertiesofDeterminantsandtheLaplaceExpansionTheorem ... 16
2.4 GaussianEliminationandDeterminants ............................. 25
2.5 HomogeneousSystemsofEquationsandaTestforLinear
Independence ......................................................... 28
2.6 DeterminantsandEigenvalues:AFirstLook ........................ 30
3 MatrixMultiplication,theInverseMatrixandPartitioning ......... 35
3.1 TheInnerProduct,OrthogonalityandtheNorm ..................... 35
3.1.1 ADigressiononNorms ........................................ 36
3.2 MatrixMultiplication ................................................. 37
3.3 QuadraticForms ...................................................... 42
3.4 TheInverseMatrix ................................................... 45
3.5 OrthogonalMatrices .................................................. 50
3.6 AMatrixProofofCramer’sRule .................................... 52
3.7 PartitioningofMatrices ............................................... 54
3.8 MatricesandLeast-SquaresCurveFitting ........................... 59
3.9 MatricesandtheLaplaceEquation ................................... 64
4 SystemsofLinearAlgebraicEquations ................................ 75
4.1 TheAugmentedMatrixandElementaryRowOperations ........... 75
4.2 TheEchelonandReducedEchelonFormsofaMatrix .............. 78
ix