分析詳解放大器ch02

OperationswithPropertiesOperationswithPropertiesofMatrixTheInverseofaElementaryApplicationsofMatrix2.1Operationswith2.1OperationswithDenotation:mna1nMamMDenotation:mna1nMamMa2nAaaaa3nMamnLain第i個列向量(rowvector):riaic1Lc第j個行向量(columnvectorc2McComm.,3A253ccc1123Rowr1A253ccc1123Rowr1 r25Columnccc123Comm.,4EqualityofTwomatricesA=[aij]andB=EqualityofTwomatricesA=[aij]andB=[bij]areequaliftheythesamesizem=bijfor1imand1jExampleA2B1,C3,DAandBarenotequalbecausetheyareofdifferentBandCarenotAandDareequalifandonlyifx=Comm.,5MatrixIfA=[aij]andB=[bij]arematricesofMatrixIfA=[aij]andB=[bij]arematricesofsizemn,A+B=[aij+ThesumoftwomatricesofdifferentsizesisExample31123210 01 0 3333 Comm.,6ScalarIfA=[aij]isanmnmatrixandcisacA=Example4ScalarIfA=[aij]isanmnmatrixandcisacA=Example4122010303220112AB1123646031313330313A3323220 32 03032042B1Comm.,7MatrixIfA=[aij]andB=[bij]areofthesamesize,A–MatrixIfA=[aij]andB=[bij]areofthesamesize,A–representsthesumofAand(1)B.ThatA–B=A+Example40303220112A12B103126460312 0 6403213AB3316 2 Comm.,8MatrixIfA=[aij]isanMatrixIfA=[aij]isanmnmatrixandB=[bij]isannpmatrix,thetheproductABisanmpmatrixAB=n aai2b2LkComm.,9AB[aij[bij]nSizeof c1pabbLLb1cccaaa2AB[aij[bij]nSizeof c1pabbLLb1cccaaa2nM22pMMMMMMciMMMMMbb3pLLMbnpMMMMMLLamnLLLncijkai1b1jai2b2LComm., ai ainb2jb3jExample4:FindingtheProductofTwoFindtheproductABExample4:FindingtheProductofTwoFindtheproductAB,32120BA45AB(4)(3)1610(4)(2)(2)(1)Comm.,TheoperationofmatrixadditionisTheoperationofTheoperationofmatrixadditionisTheoperationofmatrixmultiplicationisnotAandBCalculateA+BandB+CalculateABandBComm.,Example5:MatrixExample5:MatrixComm.,SystemsofLinear線性方程式系統(tǒng)之矩陣形a11x1a12La1nxnSystemsofLinear線性方程式系統(tǒng)之矩陣形a11x1a12La1nxnLa2nxnMa21a22ASystemofLinearxL xm1m maAx1 b1a2nx2b2M MMatrixMMx bamnn m=xbComm.,Example6:SolvingaSystemofLinearSolvethematrixequationAx=0,xExample6:SolvingaSystemofLinearSolvethematrixequationAx=0,x13 A,x and02x3Asasystemoflinear 00Augmented310012Comm.,Reducedrow-echelon70471Solutionx1=t,x2=(4/7)t,=tReducedrow-echelon70471Solutionx1=t,x2=(4/7)t,=t,tisanyreal1177x4411 tts4,t,s7x 7 x3 1t Comm.,PartitionedPartitioningthea14AAaaaa24A aa34a14r1APartitionedPartitioningthea14AAaaaa24A aa34a14r1Araaa242a34 a14 Acaacc123424a34Comm.,LinearAxx1 a2nx2b2MMMMLinearAxx1 a2nx2b2MMMMxbaamnnnma11x1a12La1nb1a21x1a22b2L MM bxxLxm1 n mm a11 b1Lxa21b212nMMMM bm1m2mnmAxx1a1x2a2LxnanComm.,Thea11a1nbxa21Thea11a1nbxa21LxxLx12n MMM m1m2mniscalledalinearcombinationofthecolumnmatricesa2,…,anwithcoefficientsx1,x2,…,Comm.,Example7:SolvingaSystemsofLinearThelinearx1Example7:SolvingaSystemsofLinearThelinearx12x23x34x15x27x18x2Matrix3x1258 6x2x14x25x36 9x3UsingGaussianelimination,thissystemhasaninfinitenumberofsolutions,oneofwhichisx1=1,x2=1,x3=Comm.,bcanbeexpressedasalinearcombinationbcanbeexpressedasalinearcombinationoftheof31415(1)Comm.,2.2PropertiesofMatrix2.2PropertiesofMatrixPropertiesMatrixAdditionandScalarTheoremIfA,B,andPropertiesMatrixAdditionandScalarTheoremIfA,B,andCaremnmatricesandcanddarescalars,A+B=B+Commutativepropertyofaddition(加法交換律A+(B+C)=(A+B)+Associativepropertyofaddition(加法結(jié)合律(cd)A=c(dA)1A=Ac(A+B)=cA+(分配律Distributive(c+d)A=cA+(分配律Comm.,Comm.,Comm.,ZeroIfAisZeroIfAisanmnmatrixandOmnisanmnmatrixconsistingentirelyofzeros,thenA+Omn=ThematrixOmniscalledazeroThematrixOmnistheadditiveidentity加法單位矩陣forthesetofallmnmatricesComm.,PropertiesofZeroTheoremPropertiesofZeroTheoremIfAisanmnmatrixandcisascalar,A+Omn=A+(A)=IfcA=Omn,thenc=0orA=AistheadditiveinverseofComm.,Example2:SolvingaMatrixSolveforXintheequation3X+A=BExample2:SolvingaMatrixSolveforXintheequation3X+A=BA3X+A–A=B–A3X=B–(1/3)3X=(1/3)(B–A)X=(1/3)(B–61214X3 3222 3Comm.,PropertiesofMatrixTheoremIfPropertiesofMatrixTheoremIfA,B,andCarematricesandcisascalar,Ａ(BC)=Associativepropertyofmultiplication(乘法結(jié)合律Ａ(B+C)=AB+Distributiveproperty(分配律(A+B)C=AC+Distributiveproperty(分配律c(AB)=(cA)B=乘法交換律不成立消去法不成立IfAC=BC,itisnotnecessarilytruethatA=Comm.,Comm.,Comm.,Example3:MatrixMultiplicationIsAssociativeFindthematrixproductABCbygroupingthefactorsfirstas(AB)CExample3:MatrixMultiplicationIsAssociativeFindthematrixproductABCbygroupingthefactorsfirstas(AB)CandthenasA(BC),andshowthatthesameresultis10142013212CAB1321Comm.,Example4:NoncommutativityofMatrixShowthatABandBAarenotequalforExample4:NoncommutativityofMatrixShowthatABandBAarenotequalfortheAB111AB BA3 ABComm.,Example5:AnExampleinWhichCancellationIsNotValidShowthatAC=BCforthefollowingExample5:AnExampleinWhichCancellationIsNotValidShowthatAC=BCforthefollowingABC12AC1 4 2BC 2 ThusAC=BC,andyetAComm.,IdentityTheidentitymatrixisasquarematrixthathas1’sonmaindiagonalIdentityTheidentitymatrixisasquarematrixthathas1’sonmaindiagonaland0’s010M0001M0LThematrixInistheidentityformatrixmultiplicationThematrixIniscalledtheidentitymatrixofordern010I1II3Comm.,PropertiesoftheIdentityTheoremPropertiesoftheIdentityTheoremIfAisamatrixofsizemn,AIn=ImA=IfAisasquarematrixofordern,thenAIn=InAAComm.,PropertiesoftheIdentityExample6:MultiplicationbyanIdentity201PropertiesoftheIdentityExample6:MultiplicationbyanIdentity201233044101002 20 1 114 4Comm.,RepeatedMultiplicationofSquareIfAisasquareRepeatedMultiplicationofSquareIfAisasquarematrixandjandkarepositiveintegers,=A2==AjAk=(Aj)k==Comm.,Example7:RepeatedMultiplicationofaSquareFindA3fortheExample7:RepeatedMultiplicationofaSquareFindA3fortheA 2121 0313Comm.,TransposeofaIfAisthemnmatrixgivenAMaMMTransposeofaIfAisthemnmatrixgivenAMaMMamnmthenthetranspose,denotedbyAT,isthenmmatrixgivenam1aam2MMamnComm., AsquarematrixAsquarematrixAiscalledsymmetricifA=AlsoifAaijisasymmetricmatrix對稱矩陣foralliaij=Comm.,Example8:TheTransposeofaA2Example8:TheTransposeofaA236900114BC789001258210456210D021110124,1 TComm.,PropertiesofTheoremIfAandPropertiesofTheoremIfAandBarematricesandcisascalar,(AT)T=(A+B)T=AT+(cA)T==Comm.,(AB)T=LLLLc1paLLbbb1cccaaa2nM2p2MMMMMciMMMCMMbb3pLLMbnpMMMAMMLLamnL(AB)T=LLLLc1paLLbbb1cccaaa2nM2p2MMMMMciMMMCMMbb3pLLMbnpMMMAMMLLamnLLLBccaabn2cm2am2MMMMMaacLLcmj12MMMMMbMbMMMLLamnLbcLcLc223jthrow,ithComm., b1 2 3 aaiMa ai ainb2jb3jExample9:FindingtheTransposeofaShowthatExample9:FindingtheTransposeofaShowthat(AB)TandBTATareequalforthefollowing231B110103232A0Comm.,Example10:TheProductofaMatrixandItsForthe312Example10:TheProductofaMatrixandItsForthe3121A FindtheproductAATandshowthatitis52542311320T201Because(AAT)T=AAT,soAATisComm.,2.3TheInverseof2.3TheInverseofaAnInverseofaMatrixAnInverseofaMatrix反矩陣AnnnmatrixAisinvertible(ornonsingular)ifthereexistsannnmatrixBsuchthatAB=BA=whereInistheidentitymatrixoforderThematrixBiscalledthe(multiplicative)inverseofAmatrixdoesnothaveaninverseiscallednoninvertible(orsingular)TheinverseofAisdenotedbyComm.,UniquenessofanInverseUniquenessofanInverseTheoremIfAisaninvertiblematrix,thenitsinverseisAssumethatBandCaretheinversesofAandBABIC(AB)CI(CA)BCIBBConsequentlyB=C,anditfollowsthattheinverseofamatrixisComm.,Example1:TheInverseofaShowExample1:TheInverseofaShowthatBistheinverseofA,21A2111BComm.,FindingtheInverseofaExampleFindtheinverseoftheATofindtheinverseofA,trytosolvetheFindingtheInverseofaExampleFindtheinverseoftheATofindtheinverseofA,trytosolvethematrixequationAX=4x224x1211x22x123x221001 x123Xx221Comm.,UsingGauss-JordanEliminationtoFindtheInverseMatrixExample 10 UsingGauss-JordanEliminationtoFindtheInverseMatrixExample 10 M1M0444MMM0M1110M1 410411MM01 141A1X1Comm.,FindingtheInverseofaFindingtheInverseofaMatrixbyGauss-JordanLetAbeasquarematrixoforder[A|[A|I][I|CheckthatAA–1=I=Comm.,Example3:FindingtheInverseofaExample3:FindingtheInverseofaFindtheinverseofthefollowing020131A1Comm.,Example4:ASingularShowthattheExample4:ASingularShowthatthefollowingmatrixhasno022231A3Comm.,Example5:FindingtheInverseofa2Example5:FindingtheInverseofa22Ifpossible,findtheinverseofthefollowing1212 AB32Comm.,PropertiesofTheoremIfAisanPropertiesofTheoremIfAisaninvertiblematrix,kisapositiveinteger,andcisascalar,then=(Ak)–1=A–1=·(cA)–1=(1/c)A–1,c(AT)–1=Comm.,Comm.,Comm.,Example6:TheInverseoftheSquareofaExample6:TheInverseoftheSquareofaMatrixComputeA–2intwodifferentwaysandshowthattheresultsareequalA===Comm.,TheInverseofaTheoremTheInverseofaTheoremIfAandBareinvertiblematricesofordern,thenABinvertibleand=(AB)(B–1A–1)=A(BB–1)A–1==(AI)A–1==Comm.,CancellationTheoremIfCisCancellationTheoremIfCisaninvertiblematrix,thenthefollowingpropertiesIfAC=BC,thenA=RightcancellationIfCA=CB,thenA=LeftcancellationIfCisnotinvertible,thencancellationisnotusuallyComm.,SystemsofEquationswithSystemsofEquationswithUniqueTheorem2.11:SystemsofEquationswithUniqueIfAisaninvertiblematrix,thenthesystemoflinearAx=bhasauniquesolutiongivenx=Ax=bA–1Ax=A–1bIx=A–1bx=Comm.,Example8:SolvingaSystemofEquationsExample8:SolvingaSystemofEquationsUsinganInverseMatrixUseaninversematrixtosolvethefollowing2x3yz2x3yz43x3yz2x4yz2x3yz03x3yz2x4yz3x3yz12x4yzComm.,2.4Elementary2.4ElementaryElementaryAnnElementaryAnnnmatrixiscalledanelementarymatrixifitcanbeobtainedfromtheidentitymatrixInbyasingleelementaryrowoperationComm.,Example1:ElementaryMatricesandNonelementaryMatricesWhichofthefollowingmatricesareelementary?Forthosethatare,describethecorrespondingelementaryrowoperationExample1:ElementaryMatricesandNonelementaryMatricesWhichofthefollowingmatricesareelementary?Forthosethatare,describethecorrespondingelementaryrowoperation001010000003001000010001100010201Comm.,Example2:ElementaryMatricesandElementaryRowOperations001220210Example2:ElementaryMatricesandElementaryRowOperations00122021042210004611 10001200100131 61 103063121001 401111533111101001035 Comm.,RepresentingElementaryRowRepresentingElementaryRowTheoremLetEbetheelementarymatrixobtainedbyperforminganelementaryrowoperationonIm.IfthatsomeelementaryrowoperationisperformedonanmnmatrixA,thentheresultingmatrixisgivenbytheproductComm.,Example3:UsingElementaryFindasequenceofelementarymatricesthatcanbeusedtowritethefollowingmatrixinrow-echelonformAExample3:UsingElementaryFindasequenceofelementarymatricesthatcanbeusedtowritethefollowingmatrixinrow-echelonformA1302R11103020321002E1A R3+(2)R1110001003203215A2E20 Comm.,(2)R30010(2)R3001010 5 5032031A3E3A20110B=2Comm.,RowLetAandRowLetAandBbemnmatrices.MatrixBisrow-equivalentAifthereexistsafinitenumberofelementarymatricesE1,E2,…,EksuchthatB=Comm.,ElementaryMatricesAreTheoremIfEisanelementarymatrix,thenE–1existsandisanelementarymatrixElementaryMatricesAreTheoremIfEisanelementarymatrix,thenE–1existsandisanelementarymatrix100010010100(R(RE)E)11211210(RR(R(a)RREE0123132313 01001aR3R3(aR3R3(01Comm.,PropertyofInvertibleTheoremAsquarematrixPropertyofInvertibleTheoremAsquarematrixAisinvertibleifandonlyifitcanbewrittenastheproductofelementarymatrices先假設(shè)AA為可逆假設(shè)A為可則AxO只有顯然解。(Theorem2.11)[A|O][I|O]EkE2E1A=A=E1–1·EComm.,Example4:WritingaMatrixastheProductofElementaryMatricesFindasequenceofelementarymatriceswhoseproduct2A38E1 E2Example4:WritingaMatrixastheProductofElementaryMatricesFindasequenceofelementarymatriceswhoseproduct2A38E1 E21A021 08E3012E102E4E3E2E1A1E1