kronecker-product(矩阵张量乘)

KroneckerProductsTomLycheUniversityofOsloNorwayKroneckerProducts–p.1/22Example:PoissonProblem-Δu=fu=0u=0u=0u=0¢u=@2u@2x+@2u@2ym2N;h=1=(m+1)vj;k¼u(jh;kh)0h2h3h4h2hh03h4hj,kfu(jh,kh)vingridj,k1,11,21,32,12,22,33,13,23,3KroneckerProducts–p.2/22DiscreteEquationg00(t)¼g(t¡h)¡2g(t)+g(t+h)h2j,kj+1,kj,k+1j-1,kj,k-1vvvvvFromdifferentialequationforj;k=1;2;:::;m(¡vj¡1;k+2vj;k¡vj+1;k)+(¡vj;k¡1+2vj;k¡vj;k+1)=h2fj;k(1)Fromboundaryconditionsvj;k=0hvisj=0;m+1ellerk=0;m+1(2)KroneckerProducts–p.3/22MatrixEquationJV+VJ=h2F2¡1¡12#v11v12v21v22#+v11v12v21v22#2¡1¡12#=19f11f12f21f22#J:=Jm=tridiagm(¡1;2;¡1)2Rm;mV:=264v11¢¢¢v1m......vm1¢¢¢vmm3752Rm;mF:=264f11¢¢¢f1m......fm1¢¢¢fmm3752Rm;m(1);(2),JV+VJ=h2F¡JV+VJ¢jk=mXi=1Jjivik+mXi=1vjiJik=l.h.sin(1)=h2(F)jkKroneckerProducts–p.4/22ConvertJV+VJ=h2FtoAx=bvingridj,k1,11,21,32,12,22,33,13,23,3147258369xingridij,kj+1,kj,k+1j-1,kj,k-1ii+1i+mi-1i-m4x-x-x-x-x-vvv-v-v-4--4xi¡xi¡1¡xi+1¡xi¡m¡xi+m=biKroneckerProducts–p.5/22LinearsystemAx=bForm=3orn=9wehavethefollowinglinearsystem:4x1¡x2¡x4=h2f11¡x14x2¡x3¡x5=h2f21¡x24x3¡x6=h2f31¡x14x4¡x5¡x7=h2f12¡x2¡x44x5¡x6¡x8=h2f22¡x3¡x54x6¡x9=h2f32¡x44x7¡x8=h2f13¡x5¡x74x8¡x9=h2f23¡x6¡x84x9=h2f33KroneckerProducts–p.6/22BlockstructureofAA=266666666666666644¡10¡100000¡14¡10¡100000¡1400¡1000¡1004¡10¡1000¡10¡14¡10¡1000¡10¡1400¡1000¡1004¡100000¡10¡14¡100000¡10¡1437777777777777775A=264J+2I¡I0¡IJ+2I¡I0¡IJ+2I375KroneckerProducts–p.7/22KroneckerProductDefinition1.Foranypositiveintegersp;q;r;swedefinetheKroneckerproductoftwomatricesA2Rp;qandB2Rr;sasamatrixC2Rpr;qsgiveninblockformasC=266664Ab1;1Ab1;2¢¢¢Ab1;sAb2;1Ab2;2¢¢¢Ab2;s.........Abr;1Abr;2¢¢¢Abr;s377775:WedenotetheKroneckerproductofAandBbyC=A­B.Definedforrectangularmatricesofanydimension#ofrows(columns)=productof#ofrows(columns)inAandBKroneckerProducts–p.8/22KroneckerProductExampleJ=2¡1¡12#;I=1001#J­I=J00J#=266642¡100¡1200002¡100¡1237775I­J=2I¡I¡I2I#=2666420¡10020¡1¡10200¡10237775KroneckerProducts–p.9/22PoissonMatrix=J­I+I­JA=264JJJ375+2642I¡I0¡I2I¡I0¡I2I375=J­I+I­JDefinition2.Letforpositiveintegersr;s;k,A2Rr;r,B2Rs;sandIkbetheidentitymatrixoforderk.ThesumA­Is+Ir­BisknownastheKroneckersumofAandB.ThePoissonmatrixistheKroneckersumofJwithitself.KroneckerProducts–p.10/22Matrixequation$KroneckerGivenA2Rr;r,B2Rs;s,F2Rr;s.FindV2Rr;ssuchthatAVBT=FForB2Rm;ndefinevec(B):=266664b1b2...bn3777752Rmn;bj=266664b1jb2j...bmj377775jthcolumnLemma3.AVBT=F,(A­B)vec(V)=vec(F)(3)AV+VBT=F,(A­Is+Ir­B)vec(V)=vec(F):(4)KroneckerProducts–p.11/22ProofWepartitionV,F,andBTbycolumnsasV=(v1;:::;vs),F=(f1;:::;fs)andBT=(b1;:::;bs).Thenwehave(A­B)vec(V)=vec(F),264Ab11¢¢¢Ab1s......Abs1¢¢¢Abss375264v1...vs375=264f1...fs375,AXjbijvj=fi;i=1;:::;m,[AVb1;:::;AVbs]=F,AVBT=F:Thisproves(3)KroneckerProducts–p.12/22ProofContinued(A­Is+Ir­B)vec(V)=vec(F),(AVITs+IrVBT)=F,AV+VBT=F:¤ThisgiveaslickwaytoderivethePoissonmatrix.RecallAV+VBT=F,(A­Is+Ir­B)vec(V)=vec(F)Therefore,JV+VJ=h2F,(J­I+I­J)vec(V)=h2vec(F)KroneckerProducts–p.13/22PropertiesofKroneckerProductsTheusualarithmeticruleshold.Notehoweverthat(A­B)T=AT­BT(A­B)¡1=A¡1­B¡1ifAandBarenonsingularA­B6=B­A(A­B)(C­D)=(AC)­(BD)(mixedproductrule)ThemixedproductruleisvalidforanymatricesaslongastheproductsACandBDaredefined.KroneckerProducts–p.14/22ProofMixedProductRuleIfB2Rr;tandD2Rt;sforsomeintegersr;tthen264Ab1;1¢¢¢Ab1;t......Abr;1¢¢¢Abr;t375264Cd1;1¢¢¢Cd1;s......Cdt;1¢¢¢Cdt;s375=264E1;1¢¢¢E1;s......Er;1¢¢¢Er;s375;whereforalli;jEi;j=tXk=1bi;kdk;jAC=(AC)(BD)i;j=((AC)­(BD))i;j:whereinthelastformulai;jreferstotheij-blockintheKroneckerproduct.¤KroneckerProducts–p.15/22EigenvaluesandEigenvectorsTheKroneckerproductoftwovectorsu2Rpandv2Rrisavectoru­v2Rprgivenbyu­v=£uTv1;:::;uTvr¤TSupposenowA2Rr;randB2Rs;sandAui=¸iui;i=1;:::;r;Bvj=¹jvj;j=1;:::;s;thenfori=1;:::;r;j=1;:::;s(A­B)(ui­vj)=¸i¹j(ui­vj);(5)(A­Is+Ir­B)(ui­vj)=(¸i+¹j)(ui­vj):(6)ThustheeigenvaluesofaKroneckerproduct(sum)aretheproducts(sums)oftheeigenvaluesofthefactors.TheeigenvectorsofaKroneckerproduct(sum)aretheproductsoftheeigenvectorsofthefactors.KroneckerProducts–p.16/22ProofofEigen-formulaeThisfollowsdirectlyfromthemixedproductrule.For(5)(A­B)(ui­vj)=(Aui)­(Bvj)=(¸iui)­(¹jvj)=(¸i¹j)(ui­vj):From(5)(A­Is)(ui­vj)=¸i(ui­vj);and(Ir­B)(ui­vj)=¹j(ui­vj)Theresultnowfollowsbysummingtheserelations.¤KroneckerProducts–p.17/22ThePoissonMatrixA=J­I+I­JNeedeigenvaluesandeigenvectorsoftheJmatrix.Leth=1=(m+1).1.WehaveJsj=¸jsjforj=1;:::;m,wheresj=(sin(j¼h);sin(2j¼h);:::;sin(mj¼h))T;(7)¸j=4sin2(j¼h2):(8)2.TheeigenvaluesaredistinctandtheeigenvectorsareorthogonalsTjsk=12h±j;k;j;k=1;:::;m:(9)KroneckerProducts–p.18/22Eigenvalues/-vectorsA=J­I+I­J1.WehaveAxj;k=¸j;kxj;kforj;k=1;:::;m,wherexj;k=sj­sk;(10)sj=(sin(j¼h);sin(2j¼h);:::;sin(mj¼h))T;(11)¸j;k=4sin2(j¼h2)+4sin2(k¼h2):(12)2.TheeigenvectorsareorthogonalxTj;kxp;q=14h2±j;p±k;q;j;k;p;q=1;:::;m:(13)3.AissymmetricAT=(J­I+I­J)T=JT­IT+IT­JT=A4.Aispositivedefinite(positiveeigenvalues)KroneckerProducts–p.19/22A=J­I+I­Jform=2s1=sin(¼3)sin(2¼3)#=p3211#;s2=sin(2¼3)sin(4¼3)#=p321¡1#¸1=4sin2(¼6)=1;¸2=4sin2(2¼6)=3Axij=¹ijxij;¹ij=¸i+¸j;xij=si­sji;j=1;2¹11=2;¹