《矩阵分析与应用》课件

1、清华大学张贤达教授矩阵分析与应用学习矩阵理论的很好的一部书156201009 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)

2、rank(A) = n(4) A(5) A(6) det(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(Sher

3、man-Morrison)An nxyn 1(A + xyH)H 1= A1A1xyHA11 + yHA1x(Wood-bury):(A + UBV )1 = A1 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(

4、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= (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

5、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(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(

6、 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)()()1856m nAmnrank(A)k n)Moore-PenroseAmn (m n)AH(AAH)1Moore-Penrose(4)LAmn = InLnmMoore-Penrose(1),(2),(4)2956(5)AR = ImMoo

7、re-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(KHALH)1KHMoore-Penrose32563.Moore-PenroseA 1 = a 1 = (aH1a1)1aH1k = 2,3, ,ndk = A k1akkbk =A k =(1 + dH kdk)1dH

8、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(a) = Amn = .am(n1)+2. .35561.7KroneckerAXB = C ?ABm np qXCn pm qLX

9、 + 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 00 000 000 10 00 01 00 00 00 00 10 00 01 00 0

10、00000100100000 00 000 110 00 000 0 0 0 0 01 00 00 10 00 00 10 00 00 00 01 000100000001000 00 000 1000K42vec(A) = 000 21a a 0a31 a210a41a22 = = vec(AT)0a12a 0a22 a320a32a41395610 00 000 000 01 00 00 10 00 00 10 00 00 00 01 000100000001000 00 000 1a11a42a11a42 12 31 21AB = aijB = . . .ABleft = Abij =

11、. . .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 C) A = B

12、 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 Naibiuivin

13、i=1ni=1aiuT i = InbivT i = In(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 + j4856HaarKroneckerHadamard386Kro

14、necker249561.8Hadamardm mAn nBAB(m+n)(m+n)A B =AOnmOmnBHadamardmnA = 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) =(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)