《矩阵分析与应用》课件.ppt

1、l清华大学l张贤达教授l矩阵分析与应用l学习矩阵理论的很好的一部书1562010 09 266A314:(FIT)3-117()FIT1-113():62794875(),62794138()Email:zxd-;2561(2)1.4Ax=b1.51.6Moore-Penrose1.7KroneckerAXB=C1.8Hadamard356 1.4bn1 =Annxn1x =A1bA1b =A1AxA1A1A=Ix=A1bAA1AA1bAA1=IAAA1 =A1A =IAAx=bn nn nA1Ax =A1A456AA Cnn(1)A(2)A1(3)rank(A)=n(4)A(5)A(6)de

2、t(A)=0b(7)Ax=b(8)Ax=0(x=0)|A|=556A1:(1)A1A=AA1=I(2)A1(3)11|A|(4)(5)(A1)1=A(6)A1AH(AH)1 =(A1)HAH =(A1)H656(7)AH=A,(A1)H=A1(8)(A)1=(A1)(9)AB(AB)1=B1A1(10)A=diag(a1,a2,am)(11)A1=diag(a 1 1,a 2 1,a m1)A A1=ATA A1=AHA(A+xy )756(Sherman-Morrison)An nxyn 1(A+xyH)H 1=A1A1xyHA11+yHA1x(Wood-bury):(A+UBV)1=A1

3、A1UB(B+BV A1UB)1BV A1=A1 A1U(I+BV A1U)1BV A1=A1 A1U(B1+V A1U)1V A1B=I(A UV)1=A1+A1U(I V A1U)1V A1856Duncan-Guttman:(AUD1V)1=A1+A1U(D V A1U)1V A11.7.14 -(1.7.17)=956(1)AAUV1=A1+A1U(D V A1U)1V A1A1U(D V A1U)1(D V A1U)1D(2)A(D V A1U)1V A1DAUVD1(A UD1V)1A1U(D V A1U)1D1V(A UD1V)1(D V A1U)11056(3)ADAUVD1=

4、(A UD1V)1(A UD1V)1UD1(D V A1U)1V A1(D V A1U)1AUVD1=(A UD1V)1(V DU1A)1(U AV 1D)1(D V A1U)11156HermitianHermitianRmrH mrmmRm 1Rm+1=R m1+1R m1+1=R m10H m0m0+1mbmbH mbH mbm1bm=R m1rmm=m rH mR m1rm=m+rH mbm1256R m1+1=Qm+1=QmqH mqmmR m1+1Qm+1=RmrH mrmmQmqH mqmm=Im0H m0m1QmqmmA1=12 L1=1 1 313561.5LAAILA=I(

5、1)L2 2 111011 251 2 4 L1A1 =IA1L1 =IL1A14 8A2=5 7 L2=5 1 R=1 0 ,R=0 0 ,1456(2)L2 3 7680217205,L2=003 72 5,L2A2=I(3)(A2L2=I)LA3=123 11 1 1 121 3AR=I1(RA=I)1556LA=ILA()mnA CmnAR =IRA()nA CmnmA1656mnArank(A)=nL=(AHA)1AHAmnArank(A)=mR=AH(AAH)1A1756Amnxn1=ym1Ayx1+x2=43x1+3x2=9Ax =yA,yAmnrank(A,y)=rank(A)

6、()()1856m nAmnrank(A)k n)Moore-PenroseAmn (m n)AH(AAH)1Moore-Penrose(4)LAmn =InLnmMoore-Penrose(1),(2),(4)2956(5)AR =ImMoore-Penrose(1),(2),(3)A(6)(1)m nAMoore-PenroseMoore-PenroseA=(AHA)AHA=AH(AAH)Moore-Penrose:pp.86-883056Moore-Penrose1.AAHXH=AAHAY =AHXHYA=XAYp.9031562.KLA =KLAmn(KmpLpn)AmnG=LH(KH

7、ALH)1KHMoore-Penrose32563.Moore-PenroseA 1=a 1=(aH1a1)1aH1k=2,3,ndk=A k1akkbk=A k=(1+dH kdk)1dH kA k1,dH kdk=1(ak Ak1dk),dHdk=1A k1 dkbkbkA =33564.Moore-PenroserB=AATC1=IAmn123Ci+1=1 itr(CiB)I CiB,i=1,2,r 14rtr(CiB)CiATCi+1B=Otr(CiB)=03456GAx=yGGyAGA=A(AG)#=AGG AGAG=G(GA)#=GAMoore-Penrose 2unvecm,n(

8、a)=Amn=.am(n1)+2.35561.7KroneckerAXB=C?ABm n p qXCn p m qLX+XN=Y?a1am+1am(n1)+1 a .amam+2.a2mamn3656a11 .vec(A)=.amnrvec(A)=a11,a1n,am1,amn3756unvecm,n(a)=Amn=vec(Amn)=arvec(A)=(vec(AT)T,vec(AT)=(rvec(A)T(commutation matrix)Kmnvec(A)=vec(AT)000=00000 000,K42=0 0000000003856K2410 0 0 0 00 000 10 00 0

9、1 00 00 00 00 10 00 01 00 000000100100000 0 0 0 00 110 0 0 0 00 0 0 0 0 01 00 00 10 00 00 10 00 00 00 01 000100000001000 0 0 0 00 1000K42vec(A)=000 21a a 0a31 a210a41a22=vec(AT)0a12a 0a22 a320a32a41395610 0 0 0 00 000 01 00 00 10 00 00 10 00 00 00 01 000100000001000 0 0 0 00 1a11a42a11a42 12 31 21A

10、B=aijB=.A Bleft=Abij=.4056m nAp qBKroneckerA Ba11Ba12Ba1nB a Bam1Ba22B.am2Ba2nB amnBAp qBKroneckerm nA BAb11Ab12Ab1qAb21Ab22.Abp1Abp2AbpqAb2q 4156(direct product)KroneckerKronecker(tensor product)Kronecker(1)AmnBpqA B =B AKronecker(2)Amn,Bnk,Clp,DpqAB CD=(A C)(B D)4256(3)Amn,Bpq,CpqA (B C)=A B A C(B

11、 C)A=B A C A(4)ABAB(A B)=A BAB(A B)1=A1 B1pp.108-1102 AN B=.4356Nm rANAi,i =1,2,NN lBKroneckerKroneckerA1 b1 A2 b AN bNbiBi44561.10.2LX+XN=Y(1),nnL NX:(2)LXIn+InXN=YIn L+NT In vec(X)=vec(Y)(3)4556L NLai=iaibT i N=ibT i(4)(5)1inIn L+NT In (bj ai)=(j+i)(bj ai)(6)L Naibiuivini=1ni=1aiuT i =InbivT i =In

12、(7)(8)In L+N InIn L+N In4656i+j=0,1 i,j n,6Ti,j(bj ai)(j+i)vT j uT i=i,j(bj ai)vT j uT i=In2(9)T1=i,j(bj ai)(j+i)vT j uT i47563vec(X)=i,j(bj ai)(j+i)vT j uT i vec(Y)=i,jbjvT j aiuT i(j+i)vec(Y)X=i,jaiuT i Y vjbT ji+j4856HaarKroneckerHadamard386Kronecker249561.8 Hadamardm mAn nBAB(m+n)(m+n)A B=AOnmOm

13、nBHadamardmnA=aij mnB =bij HadamardABm nAB=aijbij5056HadamardSchur(elementwise product)Hadamardm mA,B()HadamardAB()Hadamard(1)A,Bm nA(A(A(AB=BB)T=ATB)H=AHB)=AABTBHB5156m n(2)Omnm nHadamardAm nAOmn=OmnA=Omn(3)cc(AB)=(cA)B=A(cB)(4)Amm=aijImHadamardm mAIm=ImA=diag(A)=diag(a11,amm)5256(5)A,B,C,Dm nA(BC)

14、=(AB)C=ABC(A B)C=AC BC(A+B)(C+D)=AC+AD+BC+BD(6)A,CmmB,Dnn(A B)(C D)=(AC)(BD)5356(7)A,B,Cm ntr AT(BC)=tr (ATBT)C(8)A,B,Dm mD=(DA)(BD)=D(A B)D(9)m mA,B()HadamardAB()5456x(n)1ARMA(p,q)a1,apl 1a1Rx(q+l1)+apRx(q+lp)=Rx(q+l),Rx()=Ex(n)x(n )l=1,pa1,apRx()=1NNn=1x(n)x(n )a1,ap,?55562m nBA n m(AB)1BA,ABA B?56561.35,1.36,1.38,1.70,1.76,1.77,1.83,1.85,1.972