De nition22ThematricessatisfyingtheseequivalentconditionsarecalledorthogonalProofItsucestoprove123112Recallthealgebraicidentityforrealnumbersxandyxyxy2xy24Expandingasinel ID: 447620
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J.RauchMath555Fall2009ConformalMatricesAbstractWeanalysetheellipticalimageofspheresbylineartransfor-mations.Wecharacterizethosetransformationswhichpreservelengths(orthogonalmatrices)andthosethatmapspherestospheres(conformalmatrices).TheJacobianmatricesofanalyticfunctionsareconformalandorientationpreservingwherevertheyareinvertible.1Transposes.DenotethestandardscalarproductofvectorsinRnbyhx;yi=Xxiyi:SupposethatAijisannnrealmatrix.ThetransposeAtofAisde nedby(At)ij:=Aji:ThematrixofthetransposeisthematrixofA\rippedinthediagonal.Example1.1.1234t=1324:Proposition1.2.Forallvectorsxandy,nnmatricesAandB,andrealnumbers ,i.(A+B)t=At+Bt,ii.( A)t= At,iii.(AB)t=BtAt,iv.hAx;yi=hx;Atyi.2Lengthpreservinglineartransformations.Theorem2.1.IfAisarealnnmatrixthenthefollowingareequivalent.1.Forallx,kAxk=kxk.2.Forallx;y,hAx;Ayi=hx;yi.3.AisinvertibleandAt=A 1.1 De nition2.2.Thematricessatisfyingtheseequivalentconditionsarecalledorthogonal.Proof.Itsucestoprove1:)2:)3:)1:(1:)2:)Recallthealgebraicidentityforrealnumbersxandy,xy= (x+y)2 (x y)2=4:Expandingasinelementaryalgebrashowsthathx+y;x+yi=hx;x+i+hy;x+yi=hx;xi+hx;yi+hy;xi+hy;yi=hx;xi+2hx;yi+hy;yi;thelastusingthesymmetry.Similarlyhx y;x yi=hx;xi hx;yi hy;xi+hy;yi=hx;xi 2hx;yi+hy;yi;Subtractingprovesthepolarizationidentity,hx;yi= hx+y;x+yi hx y;x yi=4:Propertyi.impliesthattherighthandsideisequaltohA(x+y);A(x+y)i hA(x y);A(x y)i 4:Simplifyingthenusingthepolarizationidentityagainyields,hAx+Ay;Ax+Ayi hAx Ay;Ax Ayi 4=hAx;Ayi:Thiscompletestheproofthathx;yi=hAx;Ayi.(2)3).Using2andPropoition1.2.iv.showsthathx;yi=hAx;Ayi=hAtAx;yi:Thisshowsthatforallx,y,h(AtA I)x;yi=0:Foreachxthisshowsthat(AtA I)xisorthogonaltoallysomustvanish.Theidentity(AtA I)x=0forallxisequivalenttoAt=A 1.(3)1)Compute,hAx;Axi=hAtAx;xi=hx;xiwherethelastequalityuses3.Thisproves1. 2 3Positivesymmetricmatrices.De nition3.1.AsymmetricrealmatrixRisoneforwhichRij=Rjiforalli;j.ItisafundamentalfactthatforeverysuchmatrixthereisarealorthogonalOsothatO 1ROisadiagonalrealmatrixO 1RO=diag1;:::;n :(3.1)Ife1;e2;:::;enisthestandardorthonormalbasisforRnthenOe1;Oe2;:::;OenisaneworthonormalbasisconsistingofeigenvectorsofRwitheigenvaluesj.De nition3.2.AsymmetricrealRispositivewhenallthejarestrictlypositive.TheimagebyapositivesymmetricRoftheballofradius1centeredattheoriginisanndimensionalellipsoidwithaxesoflength2jinthedirectionsoftheeigenvectorsOej.De nition3.3.ForapositivesymmetricRasin(3.1)thesquarerootp Risde nedasp R:=Odiagp 1;:::;p n O 1:Thesquarerootissymmetric,positiveandsatis es(p R)2=R.Proof.LetD:=diagp 1;:::;p n .Computep Rt= ODO 1t=(O 1)tDtOt=ODO 1=p R;thelastusingthefactthatOt=O 1andD=Dt.Forthesquarecompute(ODO 1)2=ODO 1ODO 1=OD2O 1:ThatthisisequaltoRfollowsfromthede nitionofDand(3.1). Itisnothardtoshowthatthereisonlyonesuchpositivesquarerootsothede nitionisindependentofthechoiceofO.1 1IfAisamatrixwithA2=R,thenAR=RA=A3soAcommuteswithR.Ifinaddition,Aissymmetricthenthereisapossiblydi erentorthogonalOwhichsimultane-ouslydiagonalizesAandR.ThereforebothAandRareoftheformO(diagonal)O 1.Toproveuniquenessitsucestoshowthatapositivediagonalmatrixhasauniquepositivediagonalsquareroot.Thatiseasy.3 4Polardecomposition.IfMisinvertiblethenMMtissymmetricsince(MMt)t=(Mt)tMt=MMt:Forx=0,hMMtx;xi=hMtx;Mtxi=kMtxk2�0:ThefactthattheseexpressionsareallpositiveisequivalenttothepositivityofthematrixMMt.Theorem4.1.IfMisaninvertiblennmatrixthenthereareuniquelydeterminedpositivesymmetricRandorthogonalOsothatM=RO.OnehasR=p MMt;andO=(p MMt) 1M:(4.1)Thisiscalledthepolardecompositionofthematrix.Proof.IfM=ROwithpositiveRandorthogonalO,thenMMt=RO(RO)t=ROOtRt=R2;thelaststepbecauseOOt=IandR=Rt.Thus(4.1)istheonlypossiblepolardecomposition.Itremainstoprovethatthisuniquelydeterminedrepresentationsatis estheconditions.ItsucestoverifythatthematrixO:=p MMt 1Misorthogonal,thatisOtO=I.Compute,OtO=(p MMt 1M)tR 1M=Mtp MMt 1p MMt 1M:Nextusetheeasilyprovedfactthattheinverseofthesquarerootisthesquarerootoftheinverse,sop MMt 1p MMt 1=p (MMt) 1p (MMt) 1=(MMt) 1=(Mt) 1M 1:Insertingthisintheprecedingidentityyields,OtO=Mt(Mt) 1M 1M=I;completingtheproof. Thepolarrepresentationallowsonetodescribepreciselytheimagesofspheresandballsbylineartransformations.4 Corollary4.2.i.IfMisaninvertiblematrixthentheimagebyMoftheunitsphereisanellipsewhoseprincipalaxeshavelength2jwherethejaretheeigenvaluesofRinthepolardecompositionM=RO.Thedirectionoftheaxesarethecorrespondingeigendirections.ii.TheimageisasphereifanonlyifR=cIwithc�0ifandonlyifMtM=c2I.Proof.i.SinceOisorthogonalitmapstheunitspheretoitself.ThenRmapsittotheellipsewithaxesoneigendirectionsofRwithlength2j.ii.Theimageisasphereifandonlyifthejareallequal.Callthecommonvaluec.ThenR=cI.SquaringthisidentityshowsthatitisequivalenttoMMt=c2I. De nition4.3.Matricessatisfyingtheequivalentconditionsofii.arecalledconformal.Problem.FortheJacobiancomputedinclassJ=212 1;determinewhetheritmapscirclestocirclesortononcircularellipses.Solution.ComputeJJt=212 1221 1=5335:Sincethisisnotamultipleoftheidentity,Jmapscirclestononcircularellipses.Sothenonlinearmap,mapssmallcirclesabout(1;1)tosmallnoncircularellipses.TheeigendirectionsofJJtgivetheaxesoftheellipses.Thisexampleshowsthatusingtheresultsofthishandoutisveryeasy!Problem.Determineallconformal22matrices.Solution.WritethematrixasM=abcd:ItisconformalexactlywhenMMtisamultipleoftheidentity.ComputeMMt=abcdacbd=a2+b2ac+bdac+bdc2+d2:5 Thematrixisinvertibleandconformalifandonlyifac+bd=0;a2+b2=c2+d2:The rstconditionassertsthat(c;d)?(a;b).Since(a;b)isnonzerobyinvertibility,thisholdsifandonlyif(c;d)isamultipleof( b;a).Thesecondconditionassertsthattheyhavethesamelength.Therefore(c;d)=( b;a)andthegeneralsolutionisabba;a2+b2�0:(4.2)Example4.4.Ifb=0anda�0theconformalmatricesarea00a:The rstisatimestheidentity.Thesecondatimesre\rectioninthey-axis.Theybothmapcirclestocircles.Thesecondreversesorientation.Theorem4.5.IfMislinearandinvertiblefromR2toitselfthenthefol-lowingareequivalent.1.Misconformalandorientationpreserving.2.Thereisanonzerocomplexnumber + isothatthetheimagebyMofx+iyisequalto( +i )(x+iy).TheonlylinearconformalorientationpreservingmapsofR2toitselfaregivenbymultiplicationbycomplexnumbers.Proof.Thedeterminantofthematrix(4.2)isequalto(a2+b2).Sothetransformationisorientationpreservingexactlywhenthedeterminantispositivewhichisthecase(c;d)=( b;a).Thus,themostgeneralorientationpreservinginvertibleconformaltransformationisab ba;a2+b2�0:(4.3)Expanding( +i )(x+iy)=( x y)+i( x+ y)showsthatthematrixofthelineartranfsormationx+iy7!( +i )(x+iy)isequalto(4.3)when =aand = b.Thisprovestheresult. 6