Table Of ContentArakM.Mathaiand HansJ.Haubold
LinearAlgebra
DeGruyterTextbook
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Arak M. Mathai and Hans J. Haubold
Linear Algebra
|
A Course for Physicists and Engineers
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MathematicsSubjectClassification2010
15-01,15A03,15A04,15A05,15A09,15A15,15A16,15A18,15A21,15A63
Authors
Prof.DrArakM.Mathai Prof.DrHansJ.Haubold
McGillUniversity UnitedNationsOfficeforOuterSpaceAffairs
DepartmentofMathematicsandStatistics ViennaInternationalCentre
805SherbrookeSt.West P.O.Box500
Montreal,QCH3A2K6 1400Vienna
Canada Austria
[email protected] [email protected]
ISBN978-3-11-056235-4
e-ISBN(PDF)978-3-11-056250-7
e-ISBN(EPUB)978-3-11-056259-0
ThisworkislicensedundertheCreativeCommonsAttribution-NonCommercial-NoDerivatives4.0
License.Fordetailsgotohttp://creativecommons.org/licenses/by-nc-nd/4.0/.
LibraryofCongressCataloging-in-PublicationData
ACIPcatalogrecordforthisbookhasbeenappliedforattheLibraryofCongress.
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2017ArakM.Mathai,HansJ.Haubold,publishedbyWalterdeGruyterGmbH,Berlin/Boston.
Thebookispublishedwithopenaccessatwww.degruyter.com.
Typesetting:VTeXUAB,Lithuania
Printingandbinding:CPIbooksGmbH,Leck
Coverimage:Pasieka,Alfred/SciencePhotoLibrary
♾Printedonacid-freepaper
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Basicpropertiesofvectors,matrices,determinants,eigenvaluesandeigenvectorsare
discussed.Then,applicationsofmatricesanddeterminantstovariousareasofsta-
tisticalproblemssuchasprincipalcomponentsanalysis,modelbuilding,regression
analysis,canonicalcorrelationanalysis,designofexperimentsetc.areexamined.Ap-
plications of vector/matrix derivatives in the simplification of Taylor expansions of
functionsofmanyrealscalarvariablesareconsidered.Jacobiansofmatrixtransfor-
mationsofreal-valuedscalar functionsofmatrixargument, maxima/minimaprob-
lems,optimizationsoflinearforms,quadraticforms,bilinearformswithlinearand
quadraticconstraintsareexamined.Matrixsequencesandseries,convergenceofma-
trix series etc. and applications in physical sciences, chemical sciences, social sci-
ences,input-analysis,linearprogrammingproblem,non-linearleastsquaresanddy-
namicprogrammingproblemsetc.arestudiedinthisbook.
Eachtopicismotivatedbyreal-lifesituationsandeachconceptisillustratedwith
examplesandcounterexamples.Thebookisclass-testedsince1999.Itiswrittenwith
theexperienceofteachingfiftyyearsinvariousuniversitiesaroundtheworld.Thefirst
threeModulesoftheCentreforMathematicalandStatisticalSciences(CMSS)arecom-
binedtomakethisbook.TheseModulesareusedforintensiveundergraduatemathe-
maticstrainingcampsofCMSS.Eachcampisa10-dayintensivetrainingcoursewith
40hoursoflecturesand40hoursofproblem-solvingsessions.Thirtysuchcampsare
alreadyconductedbyCMSS.Onlyhighschoollevelmathematicsisassumed.Thebook
iswrittenasaself-studymaterial.Eachtopicisbroughtfromfundamentalstothese-
niorundergraduatetograduatelevel.Usualdoubtsofthestudentsonvarioustopics
areansweredinthebook.
Since 2004, the material in this book was made available to UN-affiliated Re-
gionalCentresforSpaceScienceandTechnologyEducation,locatedinIndia,China,
Morocco, Nigeria, Jordan, Brazil, and Mexico (http://www.unoosa.org/oosa/en/
ourwork/psa/regional-centres/index.html).
Since 1988 the material was taken into account for the development of educa-
tion curricula in the fields of remote sensing and geographic information systems,
satellitemeteorologyandglobalclimate,satellitecommunications,spaceandatmo-
spheric science, and global navigation satellite systems (http://www.unoosa.org/
oosa/en/ourwork/psa/regional-centres/study_curricula.html).
Assuchthematerialwasconsideredtobeaprerequisiteforapplications,teach-
ing,andresearchinspacescienceandtechnology.Itwasalsoaprerequisiteforthe
nine-monthspost-graduatecoursesinthefivedisciplinesofspacescienceandtech-
nology,offeredbytheRegionalCentresonanannualbasistoparticipantsfromall194
MemberStatesoftheUnitedNations.
Since1991,wheneversuitableattheresearchlevel,thematerialinthisbookwas
utilizedinlecturesinaseriesofannualworkshopsandfollow-upprojectsoftheso-
OpenAccess.©2017ArakM.Mathai,HansJ.Haubold,publishedbyDeGruyter. Thisworkislicensed
undertheCreativeCommonsAttribution-NonCommercial-NoDerivatives4.0License.
https://doi.org/10.1515/9783110562507-201
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VI |
calledBasicSpaceScienceInitiativeoftheUnitedNations(http://www.unoosa.org/
oosa/en/ourwork/psa/bssi/index.html).
Assuchthematerialwasconsideredaprerequisiteforteachingandresearchin
astronomyandphysics.
RIPPLESIGHTINGThecosmicdanceoftwoblackholeswarpedspacetimeasthe
pair spiraled inward and merged, creating gravitational waves (illustration below).
AdvanceLaserInterferometerGravitational-WaveObservatory(LIGO)detectedthese
ripples,producedbyblackholeseightand14timesthemassofthesun,onDecem-
ber26,2015.Einstein’stheoryofgeneralrelativitywas100yearsoldin2015.Ithasbeen
veryimportantinapplicationssuchasGPS(GNSS),andtremendouslysuccessfulin
understandingastrophysicalsystemslikeblackholes.Gravitationalwaves,whichare
ripplesinthefabricofspaceandtimeproducedbyviolenteventsinthedistantuni-
verse–forexample,bythecollisionoftwoblackholesorbythecoresofsupernova
explosions–werepredictedbyAlbertEinsteinin1916asaconsequenceofhisgeneral
theoryofrelativity.Gravitationalwavesareemittedbyacceleratingmassesmuchin
thesamewayelectromagneticwavesareproducedbyacceleratingcharges,suchas
radiowavesradiatedbyelectronsacceleratinginantennas.AstheytraveltoEarth,
these ripples in the space–time fabric carry information about their violent origins
andaboutthenatureofgravitythatcannotbeobtainedbytraditionalastronomical
observationsusinglight.Gravitationalwaveshavenowbeendetecteddirectly.Scien-
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| VII
tistsdo,however,havegreatconfidencethattheyexistbecausetheirinfluenceona
binarypulsarsystem(twoneutronstarsorbitingeachother)hasbeenmeasuredac-
curatelyandisinexcellentagreementwiththepredictions.Directlydetectinggravi-
tationalwaveshasconfirmedEinstein’spredictioninanewregimeofextremerela-
tivisticconditions,andopenapromisingnewwindowintosomeofthemostviolent
andcataclysmiceventsinthecosmos.TheGNSSeducationcurriculaprovidesoppor-
tunitiestoteachnavigationanddoresearchinastrophysics(basicspacescience).The
developmentoftheeducationcurricula(illustratedabove)startedin1988atUNHead-
quartersinNewYork,thespecificGNSScurriculumemanatedonlyin1999afterthe
UNISPACEIIIConference,heldatandhostedbytheUnitedNationsatVienna.
Usuallystudentsfromotherareas,otherthanmathematics,areintimidatedby
seeingtheoremsandproofs.Hencenosuchphraseas“theorem”isusedinthebook.
Mainresultsarecalled“results”andarewritteninboldsothatthematerialwillbe
user-friendly.
Thisbookcanbeusedasatextbookforabeginningundergraduatelevelcourse
onvectors,matricesanddeterminants,andtheirapplications,forstudentsfromall
disciplines.
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Preface
Thebasicmaterialinthisbookoriginatedfromacoursegivenbythefirstauthorat
theUniversityofTexasatElPasoin1998–1999academicyear.Studentsfrommath-
ematics, engineering, biology, economics, physics and chemistry were in the class.
Thetextbookassignedtothecoursedidnotsatisfythestudentsfromanyofthedis-
ciplines,includingmathematics.HenceDrMathaistarteddevelopingacoursefrom
fundamentals, assuming no background, with lots of examples and counter exam-
ples taken from day to day life. All sections of the students enjoyed the course. Dr
Mathaigavecoursesoncalculusandlinearalgebraandforbothofthesecourseshe
developedhisownmaterialsincloseinteractionwithstudents.TheElPasoexperi-
mentwasinitiallyforonesemesteronlybut,duetothepopularity,extendedtomore
semesters.
During2000to2006thesenotesweredevelopedintoCMSSModulesandbased
ontheseModules,occasionalcoursesweregivenforteachersandstudentsatvarious
levelsinKerala,India,asperrequestsfromteachers.From2007onwardCMSSbecame
aDepartmentofScienceandTechnology,GovernmentofIndiacentreformathemat-
icalandstatisticalsciences.Modulesinotherareaswerealsodevelopedduringthis
period,andby2014,tenModulesweredeveloped.
AsaLifeMemberofCMSS,thesecondauthorisanactiveparticipantofallpro-
gramsatCMSS,includingtheundergraduatemathematicstrainingcamps,Ph.Dtrain-
ingetc.andheisalsoafrequentvisitortoCMSStoparticipateinandcontributeto
variousactivities.
Chapter1isdevotedtoallbasicpropertiesofvectorsasorderedsetofrealnum-
bers,Eachdefinitionismotivatedbyreal-lifeexamples.Afterintroducingmajorprop-
ertiesofvectorswiththerealelements,vectorsinthecomplexdomainareconsidered
andmorerigorousdefinitionsareintroduced.Chapter1endswithGram–Schmidtor-
thogonalizationprocess.
Chapter2dealswithmatrices.Again,alldefinitionsandpropertiesareintroduced
fromreal-lifesituations.Rolesofelementarymatricesandelementaryoperationsin
solvinglinearequations,checkingconsistencyoflinearsystems,checkinglinearde-
pendenceofvectors,evaluatingrankofamatrix,canonicalreductionsofquadratic
and bilinear forms, triangularizations and diagonalizations of matrices, computing
inversesofmatricesetc.arehighlighted.
Chapter3dealswithdeterminants.Anaxiomaticdefinitionisintroduced.Various
typesofexpansionsofdeterminantsaregiven.Roleofelementarymatricesinevalu-
atingdeterminantsishighlighted.ThischaptermeltsintoChapter4oneigenvalues
andeigenvectorsandtheirproperties.
Chapters 5 and 6 are on applications of matrices and determinants to various
disciplines.Applicationstomaxima/minimaproblems,constrainedmaxima/minima,
optimizationoflinear,quadraticandbilinearforms,withlinearandquadraticcon-
OpenAccess.©2017ArakM.Mathai,HansJ.Haubold,publishedbyDeGruyter. Thisworkislicensed
undertheCreativeCommonsAttribution-NonCommercial-NoDerivatives4.0License.
https://doi.org/10.1515/9783110562507-202
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X | Preface
straintsareconsidered.Foreachoptimization,atleastonepracticalproceduresuch
asprincipalcomponentsanalysis,canonicalcorrelationanalysis,regressionanalysis
etc.isillustrated.SomeadditionaltopicsarealsodevelopedinChapter6.Matrixpoly-
nomials,matrixsequencesandseries,convergence,normsofmatrices,singularvalue
decompositionofmatrices,simultaneousreductionofmatricestodiagonalformsetc.
arealsodiscussedinChapter6.
A.M.Mathai
14thMarch2017 H.J.Haubold
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