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Yair Shapira Linear Algebra and Group Theory for Physicists and Engineers Second Edition Yair Shapira Linear Algebra and Group Theory for Physicists and Engineers Second Edition YairShapira DepartmentofComputerScience Technion,IsraelInstituteofTechnology Haifa,Israel ISBN978-3-031-22421-8 ISBN978-3-031-22422-5 (eBook) https://doi.org/10.1007/978-3-031-22422-5 MathematicsSubjectClassification:15-01,20-01,00A06 1stedition:©SpringerNatureSwitzerlandAG2019 2ndedition:©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SwitzerlandAG2023 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered companySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface What’s new in this edition? Every matrix has a Jordan form (Chap.16). To show this,we’lldesignnewvectors:generalizedeigenvectors.Intermsofthesevectors, thematrixlooksquitesimple.Onthemaindiagonal,wehavetheeigenvalues,one byone.Onthesuperdiagonaljustaboveit,wehave1’sor0’s.Alltherestiszero. Isn’tthissimple?Itisalsouseful.ForHermitianmatrix,forexample,thisgivesthe eigenbasis. Todesignthis,wealsouseafundamentaltheoreminnumbertheory:theChinese remaindertheorem.ThiswillhelpdesigntheJordandecomposition(Chap.17).This isabitdifferentfromtheJordanform:itisgivenintermsofanewpolynomialin the original matrix. To show how useful this is, we’ll design the Jordan form in a specialcaseofinterest(Chap.18). This edition also contains more applications. It shows how to linearize the Einsteinequationsingeneralrelativity(Chap.19).Infiniteelements,youcanthen use the Newton iteration, to solve the nonlinear system of PDEs. There are some numericalresultsinelectromagnetics(Tables9.1–9.4). Linear algebra and group theory are closely related. This book introduces both at the same time. Why? Because they go hand in hand. This is particularly good for (undergraduate) students in physics, chemistry, engineering, CS, and (applied) math. This is a new (interdisciplinary) approach: math is no longer isolated, but envelopedwithpracticalapplicationsinappliedscienceandengineering. Thelinear-algebra partintroduces bothvectorsandmatricesfromscratch,with alotofexamplesintwoandthreedimensions:2×2Lorentzmatrices,and3×3 rotation and inertia matrices. This prepares the reader for the group-theory stuff: 2 × 2 Moebius and Pauli matrices, 3 × 3 projective matrices, and more. This way, the reader gets ready for higher dimensions as well: big Fourier and Markov matrices,quantum-mechanicaloperators,andstiffnessandmassmatricesin(high- order)finiteelements. Oncematricesareready,theyareusedtomirror(orrepresent)usefulgroups.This is how linear algebra is used in group theory. This makes groups ever so concrete andaccessible. v vi Preface Why learn linear and modern algebra at the same time? Because they can completeandsupporteachother.Indeed,intheapplications,wealsoworktheother wayaround:grouptheorypavesthewaytolinearalgebra,touncovertheelectronic structureintheatom. HowtoUsetheBook inAcademicCourses? Thebookcouldbeusedasatextbookinthree(undergraduate)mathcourses: • Linearalgebraforphysicistsandengineers(Chaps.1–4and14–18) • Grouptheoryanditsgeometricalapplications(Chaps.5–7and14–15) • Numericalanalysis:finiteelementsandtheirapplications(Chaps.8–15and19) Indeed,PartIintroduceslinearalgebra,withapplicationsinphysicsandCS.PartII, ontheotherhand,introducesgrouptheory,withapplicationsinprojectivegeometry. PartsIII–IVintroducehigh-orderfiniteelements,inaregularmeshin3-D.PartV assemblesthestiffnessandmassmatricesinquantumchemistry.PartVIdesignsthe Jordanformofamatrix,andtheeigenbasisofaHermitianmatrix.Finally,PartVII linearizestheEinsteinequationsinnumericalrelativity. The book is nearly self-contained: the only prerequisite is elementary calculus, whichcouldbeattendedatthesametime.Thereareplentyofexamplesandfigures, tomakethematerialmorevividandvisual. Each chapter ends with a lot of relevant exercises, with solutions or at least guidelines. This way, the reader gets to see how the theory develops step by step, exercisebyexercise. Roadmaps:How toReadthe Book? Howtoreadthebook?Hereareafewoptions:(Figs.1,2,and3): • Physicists,chemists,andengineerscould – Start from Chaps.1–2 about linear algebra, with applications in geometrical mechanics – SkiptoChaps.4–5,tousesmallmatricesinspecialrelativityandgrouptheory – Usebiggermatricesinquantummechanicsaswell(Chaps.7and14–15) – ProceedtotheJordanform(Chap.16) – Concludewithadvancedapplicationsingeneralrelativity(Chap.19) • Computerscientists,ontheotherhand,could – StartfromChaps.1–2aboutlinearalgebra – ProceedtoChap.3,whichusesaMarkovmatrixtodesignasearchengine Preface vii Fig.1 Howcouldaphysicist/chemist/engineerreadthebook? Fig.2 Howcouldacomputerscientistreadthebook? viii Preface Fig.3 Howcouldanumericalanalystora(applied)mathematicianreadthebook? – Skip to Chaps.5–6 about group theory and its applications in computer graphics – ConcludewithChaps.16–17abouttheJordanform • Numericalanalystsand(applied)mathematicians,ontheotherhand,could – StartfromChaps.1–2aboutlinearalgebra – Skip to Chaps.8–15 about finite elements and regular meshes, with applica- tionsinquantumchemistryandsplines – ProceedtoChaps.16–18abouttheJordanform – ConcludewithChap.19aboutnumericalrelativity And a little remark: how to pronounce the title of the book? This is a bit tricky: write “physics,” but say “phyZics.” This is a phonetic law: use the easiest way to pronounce, no matter how the word is written. Likewise, write “tensor” and “isomorphism,”butsay“tenZor”and“iZomorphiZm,”andsoon. Haifa,Israel YairShapira Contents PartI IntroductiontoLinearAlgebra 1 VectorsandMatrices....................................................... 3 1.1 VectorsinTwoandThreeDimensions.............................. 4 1.1.1 Two-DimensionalVectors ................................ 4 1.1.2 AddingVectors............................................ 4 1.1.3 ScalarTimesVector....................................... 5 1.1.4 Three-DimensionalVectors............................... 6 1.2 VectorsinHigherDimensions....................................... 7 1.2.1 MultidimensionalVectors................................. 7 1.2.2 AssociativeLaw........................................... 7 1.2.3 TheOrigin................................................. 8 1.2.4 MultiplicationandItsLaws............................... 8 1.2.5 DistributiveLaws.......................................... 8 1.3 ComplexNumbersandVectors...................................... 9 1.3.1 ComplexNumbers ........................................ 9 1.3.2 ComplexVectors.......................................... 11 1.4 RectangularMatrix................................................... 11 1.4.1 Matrices.................................................... 11 1.4.2 AddingMatrices........................................... 13 1.4.3 ScalarTimesMatrix....................................... 13 1.4.4 MatrixTimesVector ...................................... 14 1.4.5 Matrix-Times-Matrix...................................... 15 1.4.6 DistributiveandAssociativeLaws ....................... 16 1.4.7 TheTransposeMatrix..................................... 17 1.5 SquareMatrix ........................................................ 18 1.5.1 SymmetricSquareMatrix................................. 18 1.5.2 TheIdentityMatrix ....................................... 19 1.5.3 TheInverseMatrixasaMapping ........................ 20 1.5.4 InverseandTranspose..................................... 21 1.6 ComplexMatrixandItsHermitianAdjoint......................... 22 ix x Contents 1.6.1 TheHermitianAdjoint.................................... 22 1.6.2 Hermitian(Self-Adjoint)Matrix ......................... 23 1.7 InnerProductandNorm ............................................. 23 1.7.1 Inner(Scalar)Product..................................... 23 1.7.2 Bilinearity ................................................. 24 1.7.3 Skew-Symmetry........................................... 24 1.7.4 Norm....................................................... 25 1.7.5 Normalization ............................................. 25 1.7.6 OtherNorms............................................... 25 1.7.7 InnerProductandtheHermitianAdjoint ................ 26 1.7.8 InnerProductandaHermitianMatrix ................... 26 1.8 OrthogonalandUnitaryMatrix...................................... 27 1.8.1 InnerProductofColumnVectors......................... 27 1.8.2 OrthogonalandOrthonormalColumnVectors .......... 28 1.8.3 ProjectionMatrixandItsNullSpace..................... 29 1.8.4 UnitaryandOrthogonalMatrix........................... 30 1.9 EigenvaluesandEigenvectors ....................................... 31 1.9.1 EigenvectorsandTheirEigenvalues...................... 31 1.9.2 SingularMatrixandItsNullSpace....................... 31 1.9.3 EigenvaluesoftheHermitianAdjoint.................... 32 1.9.4 EigenvaluesofaHermitianMatrix....................... 33 1.9.5 EigenvectorsofaHermitianMatrix...................... 33 1.10 TheSineTransform.................................................. 34 1.10.1 DiscreteSineWaves....................................... 34 1.10.2 OrthogonalityoftheDiscreteSineWaves ............... 35 1.10.3 TheSineTransform....................................... 37 1.10.4 Diagonalization............................................ 37 1.10.5 SineDecomposition....................................... 38 1.10.6 MultiscaleDecomposition................................ 38 1.11 TheCosineTransform ............................................... 39 1.11.1 DiscreteCosineWaves.................................... 39 1.11.2 OrthogonalityoftheDiscreteCosineWaves ............ 39 1.11.3 TheCosineTransform .................................... 41 1.11.4 Diagonalization............................................ 41 1.11.5 CosineDecomposition.................................... 41 1.12 Positive(Semi)definiteMatrix....................................... 42 1.12.1 PositiveSemidefiniteMatrix.............................. 42 1.12.2 PositiveDefiniteMatrix................................... 42 1.13 Exercises:GeneralizedEigenvalues................................. 43 1.13.1 TheCauchy–SchwarzInequality......................... 43 1.13.2 TheTriangleInequality................................... 44 1.13.3 GeneralizedEigenvalues.................................. 44 1.13.4 RootofUnityandFourierTransform.................... 46