kronecker product(矩阵张量乘)

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1、KroneckerProductsTomLycheUniversityofOsloNorwayKroneckerProducts–p.1/22Example:PoissonProblemu=0@2u@2u¢u=+u=0-Du=fu=0@2x@2ym2N;h=1=(m+1)u=0vj;k¼u(jh;kh)4hu(jh,kh)1,32,33,33hfj,k1,22,23,22h1,12,13,1h00h2h3h4hvingridj,kKroneckerProducts–p.2/22DiscreteEquation00g(t¡h)¡2g(t)+g(t+h)g(t)¼h2

2、vj,k+1vvvj-1,kj,kj+1,kvj,k-1Fromdifferentialequationforj;k=1;2;:::;m(¡v+2v¡v)+(¡v+2v¡v)=h2fj¡1;kj;kj+1;kj;k¡1j;kj;k+1j;k(1)Fromboundaryconditionsvj;k=0hvisj=0;m+1ellerk=0;m+1(2)KroneckerProducts–p.3/22MatrixEquationJV+VJ=h2F"#"#"#"#"#2¡1v11v12v11v122¡11f11f12+=¡12v21v22v21v22¡129f21f2

3、2J:=Jm;mm=tridiagm(¡1;2;¡1)2R2323v11¢¢¢v1mf11¢¢¢f1m6....7m;m6....7m;mV:=4..52RF:=4..52Rvm1¢¢¢vmmfm1¢¢¢fmm2(1);(2),JV+VJ=hF¡¢XmXmJV+VJ=Jv+vJ=l.h.sin2jkjiikjiik(1)=h(F)jki=1i=1KroneckerProducts–p.4/22ConvertJV+VJ=h2FtoAx=b1,32,33,37891,22,23,2>4561,12,13,1123vingridxingriidj,k-v-xj,k+1i

4、+m-vj-1,k4v-v-xi-14x-xj,kj+1,kii+1-->-v-xj,k-1i-m4xi¡xi¡1¡xi+1¡xi¡m¡xi+m=biKroneckerProducts–p.5/22LinearsystemAx=bForm=3orn=9wehavethefollowinglinearsystem:4x21¡x2¡x4=hf11¡x214x2¡x3¡x5=hf21¡x224x3¡x6=hf31¡x214x4¡x5¡x7=hf12¡x22¡x44x5¡x6¡x8=hf22¡x23¡x54x6¡x9=hf32¡x244x7¡x8=hf13¡x25¡x74

5、x8¡x9=hf23¡x26¡x84x9=hf33KroneckerProducts–p.6/22BlockstructureofA234¡10¡100000676¡14¡10¡1000076760¡1400¡1000767676¡1004¡10¡100767A=60¡10¡14¡10¡10767600¡10¡1400¡1767676000¡1004¡1076740000¡10¡14¡1500000¡10¡1423J+2I¡I067A=4¡IJ+2I¡I50¡IJ+2IKroneckerProducts–p.7/22KroneckerProductDefinitio

6、n1.Foranypositiveintegersp;q;r;swedefinetheKroneckerp;qr;spr;qsproductoftwomatricesA2RandB2RasamatrixC2Rgiveninblockformas23Ab1;1Ab1;2¢¢¢Ab1;s676Ab2;1Ab2;2¢¢¢Ab2;s7C=6...7:6...74...5Abr;1Abr;2¢¢¢Abr;sWedenotetheKroneckerproductofAandBbyC=A•B.Definedforrectangularmatricesofanydimension#o

7、frows(columns)=productof#ofrows(columns)inAandBKroneckerProducts–p.8/22KroneckerProductExample"#"#2¡110J=;I=¡1201232¡100"#67J06¡12007J•I==670J4002¡1500¡122320¡10"#672I¡I6020¡17I•J==67¡I2I4¡102050¡102KroneckerProducts–p.9/22PoissonMatrix=J•I+I•J2323J2I¡I06767A=4J5+4¡I2I¡I5=J•I+I•JJ0¡I2

8、Ir;rs