http://en.wikipedia.org/wiki/Eigenvector Eigenvalue, eigenvector and eigenspace FromWikipedia,thefreeencyclopedia (RedirectedfromEigenvector) Inmathematics,an eigenvector ofatransformation[1]isa nonnull vectorwhosedirectionisunchangedbythattransformation. Thefactorbywhichthemagnitudeisscalediscalledthe eigenvalue ofthatvector.(SeeFig.1.)Often,atransformationis completelydescribedbyitseigenvaluesandeigenvectors.An eigenspaceisasetofeigenvectorswithacommoneigenvalue. Theseconceptsplayamajorroleinseveralbranchesofbothpure andappliedmathematics—appearingprominentlyinlinearalgebra, functionalanalysis,andtoalesserextentinnonlinearsituations. Itiscommontoprefixanynaturalnameforthesolutionwitheigen Fig.1.InthissheartransformationoftheMona insteadofsayingeigenvector.Forexample,eigenfunctionifthe Lisa,thepicturewasdeformedinsuchaway eigenvectorisafunction,eigenmodeiftheeigenvectorisaharmonic thatitscentralverticalaxis(redvector)was notmodified,butthediagonalvector(blue)has mode,eigenstateiftheeigenvectorisaquantumstate,andsoon changeddirection.Hencetheredvectorisan (e.g.theeigenfaceexamplebelow).Similarlyfortheeigenvalue,e.g. eigenvectorofthetransformationandtheblue eigenfrequencyiftheeigenvalueis(ordetermines)afrequency. vectorisnot.Sincetheredvectorwasneither stretchednorcompressed,itseigenvalueis1. Allvectorsalongthesameverticallinearealso eigenvectors,withthesameeigenvalue.They formtheeigenspaceforthiseigenvalue. 1 of 14 16/Oct/06 5:01 PM http://en.wikipedia.org/wiki/Eigenvector Contents 1 History 2 Definitions 3 Examples 4 Eigenvalueequation 5 Spectraltheorem 6 Eigenvaluesandeigenvectorsofmatrices 6.1 Computingeigenvaluesandeigenvectorsofmatrices 6.1.1 Symboliccomputations 6.1.2 Numericalcomputations 6.2 Properties 6.2.1 Algebraicmultiplicity 6.2.2 Decompositiontheoremsforgeneralmatrices 6.2.3 Someotherpropertiesofeigenvalues 6.3 Conjugateeigenvector 6.4 Generalizedeigenvalueproblem 6.5 Entriesfromaring 7 Infinitedimensionalspaces 8 Applications 9 Notes 10 References 11 Externallinks History Nowadays,eigenvaluesareusuallyintroducedinthecontextofmatrixtheory.Historically,however,theyarosein thestudyofquadraticformsanddifferentialequations. Inthefirsthalfofthe18thcentury,JohannandDanielBernoulli,d'Alembert,andEulerencounteredeigenvalue problemswhenstudyingthemotionofarope,whichtheyconsideredtobeaweightlessstringloadedwithanumber ofmasses.LaplaceandLagrangecontinuedtheirworkinthesecondhalfofthecentury.Theyrealizedthatthe eigenvaluesarerelatedtothestabilityofthemotion.Theyalsousedeigenvaluemethodintheirstudyofthesolar system.[2] Eulerhadalsostudiedtherotationalmotionofarigidbodyanddiscoveredtheimportanceoftheprincipalaxes.As Lagrangerealized,theprincipalaxesaretheeigenvectorsoftheinertiamatrix.[3]Intheearly19thcentury,Cauchy sawhowtheirworkcouldbeusedtoclassifythequadricsurfaces,andgeneralizedittoarbitrarydimensions.[4] Cauchyalsocoinedthetermracinecaractéristique(characteristicroot)forwhatisnowcalledeigenvalue;histerm survivesincharacteristicequation.[5] FourierusedtheworkofLaplaceandLagrangetosolvetheheatequationbyseparationofvariablesinhisfamous 1822bookThéorieanalytiquedelachaleur.[6] SturmdevelopedFourier'sideasfurtherandhebroughtthemtothe attentionofCauchy,whocombinedthemwithhisownideasandarrivedatthefactthatsymmetricmatriceshave realeigenvalues.[4]ThiswasextendedbyHermitein1855towhatarenowcalledHermitianmatrices.[5]Aroundthe sametime,Brioschiprovedthattheeigenvaluesoforthogonalmatriceslieontheunitcircle,[4]andClebschfound thecorrespondingresultforskewsymmetricmatrices.[5]Finally,Weierstrassclarifiedanimportantaspectinthe 2 of 14 16/Oct/06 5:01 PM http://en.wikipedia.org/wiki/Eigenvector stabilitytheorystartedbyLaplacebyrealizingthatdefectivematricescancauseinstability.[4] Inthemeantime,LiouvillehadstudiedsimilareigenvalueproblemsasSturm;thedisciplinethatgrewoutoftheir workisnowcalledSturmLiouvilletheory.[7] SchwarzstudiedthefirsteigenvalueofLaplace'sequationongeneral domainstowardstheendofthe19thcentury,whilePoincaréstudiedPoisson'sequationafewyearslater.[8] Atthestartofthe20thcentury,Hilbertstudiedtheeigenvaluesofintegraloperatorsbyconsideringthemtobe infinitematrices.[9]HewasthefirsttousetheGermanwordeigentodenoteeigenvaluesandeigenvectorsin1904, thoughhemayhavebeenfollowingarelatedusagebyHelmholtz."Eigen"canbetranslatedas"own","peculiarto", "characteristic"or"individual"—emphasizinghowimportanteigenvaluesaretodefiningtheuniquenatureofa specifictransformation.Forsometime,thestandardterminEnglishwas"propervalue",butthemoredistinctive term"eigenvalue"isstandardtoday.[10] Thefirstnumericalalgorithmforcomputingeigenvaluesandeigenvectorsappearedin1929,whenVonMises publishedthepowermethod.Oneofthemostpopularmethodstoday,theQRalgorithm,wasproposed independentlybyFrancisandKublanovskayain1961.[11] Definitions Seealso:Eigenplane Transformationsofspace—suchastranslation(orshiftingtheorigin),rotation,reflection,stretching,compression, oranycombinationofthese—maybevisualizedbytheeffecttheyproduceonvectors.Vectorscanbevisualisedas arrowspointingfromonepointtoanother. Eigenvectorsoftransformationsarevectors[12]whichareeitherleftunaffectedorsimplymultipliedbya scalefactorafterthetransformation. Aneigenvector'seigenvalueisthescalefactorbywhichithasbeenmultiplied. Aneigenspaceisaspaceconsistingofalleigenvectorswhichhavethesameeigenvalue,alongwiththezero (null)vector,whichitselfisnotaneigenvector. Theprincipal eigenvectorofatransformationistheeigenvectorwiththelargestcorrespondingeigenvalue. Thegeometric multiplicityofaneigenvalueisthedimensionoftheassociatedeigenspace. Thespectrumofatransformationonfinitedimensionalvectorspacesisthesetofallitseigenvalues. Forinstance,aneigenvectorofarotationinthreedimensionsisavectorlocatedwithintheaxisaboutwhichthe rotationisperformed.Thecorrespondingeigenvalueis1andthecorrespondingeigenspacecontainsallthevectors alongtheaxis.Asthisisaonedimensionalspace,itsgeometricmultiplicityisone.Thisistheonlyeigenvalueofthe spectrum(ofthisrotation)thatisarealnumber. Examples AstheEarthrotates,everyarrowpointingoutwardfromthecenteroftheEarthalsorotates,exceptthosearrows thatlieontheaxisofrotation.ConsiderthetransformationoftheEarthafteronehourofrotation:Anarrowfromthe centeroftheEarthtotheGeographicSouthPolewouldbeaneigenvectorofthistransformation,butanarrowfrom thecenteroftheEarthtoanywhereontheequatorwouldnotbeaneigenvector.Sincethearrowpointingatthe poleisnotstretchedbytherotationoftheEarth,itseigenvalueis1. Anotherexampleisprovidedbyathinmetalsheetexpandinguniformlyaboutafixedpointinsuchawaythatthe distancesfromanypointofthesheettothefixedpointaredoubled.Thisexpansionisatransformationwith eigenvalue2.Everyvectorfromthefixedpointtoapointonthesheetisaneigenvector,andtheeigenspaceisthe setofallthesevectors. 3 of 14 16/Oct/06 5:01 PM http://en.wikipedia.org/wiki/Eigenvector However,threedimensionalgeometricspaceisnottheonlyvector space.Forexample,considerastressedropefixedatbothends,like thevibratingstringsofastringinstrument(Fig.2).Thedistancesof atomsofthevibratingropefromtheirpositionswhentheropeisat restcanbeseenasthecomponentsofavectorinaspacewithas Fig.2.Astandingwaveinaropefixedatits boundariesisanexampleofaneigenvector,or manydimensionsasthereareatomsintherope. moreprecisely,aneigenfunctionofthe transformationgivingtheacceleration.Astime Assumetheropeisacontinuousmedium.Ifoneconsidersthe passes,thestandingwaveisscaledbya equationfortheaccelerationateverypointoftherope,its sinusoidaloscillationwhosefrequencyis eigenvectors,oreigenfunctions,arethestandingwaves.The determinedbytheeigenvalue,butitsoverall standingwavescorrespondtoparticularoscillationsoftheropesuch shapeisnotmodified. thattheaccelerationoftheropeissimplyitsshapescaledbya factor—thisfactor,theeigenvalue,turnsouttobe−ω2whereωistheangularfrequencyoftheoscillation.Each componentofthevectorassociatedwiththeropeismultipliedbyatimedependentfactorsin(ωt).Ifdampingis considered,theamplitudeofthisoscillationdecreasesuntiltheropestopsoscillating,correspondingtoacomplex ω.Onecanthenassociatealifetimewiththeimaginarypartofω,andrelatetheconceptofaneigenvectortothe conceptofresonance.Withoutdamping,thefactthattheaccelerationoperator(assumingauniformdensity)is Hermitianleadstoseveralimportantproperties,suchasthatthestandingwavepatternsareorthogonalfunctions. Eigenvalue equation Mathematically,vλisaneigenvectorandλthecorrespondingeigenvalueofatransformationTiftheequation: istrue,whereT(vλ)isthevectorobtainedwhenapplyingthetransformationTtovλ. SupposeTisalineartransformation(whichmeansthat forallscalars a,b, andvectorsv,w).Considerabasisinthatvectorspace.Then,Tandvλcanberepresentedrelativetothatbasisby amatrix AT—atwodimensionalarray—andrespectivelyacolumnvectorvλ—aonedimensionalverticalarray.The eigenvalueequationinitsmatrixrepresentationiswritten wherethejuxtapositionismatrixmultiplication.SinceinthiscircumstancethetransformationTanditsmatrix representationATareequivalent,wecanoftenusejustTforthematrixrepresentationandthetransformation.This isequivalenttoasetofnlinearequations,wherenisthenumberofbasisvectorsinthebasisset.Inthisequation boththeeigenvalueλandthencomponentsofvλareunknowns. However,itissometimesunnaturalorevenimpossibletowritedowntheeigenvalueequationinamatrixform.This occursforinstancewhenthevectorspaceisinfinitedimensional,forexample,inthecaseoftheropeabove. DependingonthenatureofthetransformationTandthespacetowhichitapplies,itcanbeadvantageousto representtheeigenvalueequationasasetofdifferentialequations.IfTisadifferentialoperator,theeigenvectors arecommonlycalledeigenfunctionsofthedifferentialoperatorrepresentingT.Forexample,differentiationitselfis