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 # SpringerScience+BusinessMediaB.V.2010 Nopartofthisworkmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformorbyany means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permissionfromthePublisher,withtheexceptionofanymaterialsuppliedspecificallyforthepurpose ofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthework. Coverdesign:eStudioCalamarS.L.,Germany Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) 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