(完整版)张贤达教授课件-清华之第2讲

A† = (AHA)†AH
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A† = AH(AAH)†
Moore-Penrose 88
:
pp.86-
Moore-Penrose 1.
AAHXH = A AHAY = AH
XH Y
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A† = XAY
p.90
2. KL
A = KL
Km×p
Lp×n
Am×n )
G = LH(KHALH)−1KH
00 00 00 1 0 K24 = 0 1 0 0 0 0 0 0 ,K42 =
00 00 01 0 0 00 1 0 0 0 0 0
00 01 00 0 0
00 0 0 0 0 1 0
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00 00 01 0 0
00 01 00 0 0
0 0 00 00 0 1
0 0 00 00 0 1
1 0 0 0 0 0 0 0 a11
1
(2)
1.4
1.5
Ax = b
1.6 Moore-Penrose
1.7 Kronecker
AXB = C
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1.8 Hadamard
§ 1.4
A−1b A−1
bn×1 = An×nxn×1 A−1b = A−1Ax
x=
A−1A = I
x = A−1b
A−1
A Ax =
AA−1b
Ax = b
Am×n Moore-Penrose
(
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3. Moore-Penrose
A† 1 = a† 1 = (aH1a1)−1aH1 k = 2,3,··· ,n
dk = A† k−1ak
bk =
(1 + dH kdk)−1dH kA† k−1, dH kdk = −1
(ak − Ak−1dk)†,
,L2
=
0 3 7 , ··· 02 5
L2A2 = I (3)
( A2L2 = I) L
A3 = 1 3 1 ⇒
21 5 11
−1 1
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R = −1 0 , R = 0 0 , ···
3 −1
2 −1
AR = I ( RA = I)
LA = I
LA
()
mn
A ∈ Cm×n
AR = I
RA
(AG)H = AG
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(4) GA Hermitian
(GA)H = GA
Moore-Penrose
:
(1) (2)
G = A†
A
(reflexive generalized inverse)
(1) (2) (3)
A†
A
(normalized generalized inverse)
(1) (2) (4)
(1)
Am×n Bp×q
A⊗B =
B⊗A
Kronecker
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(2)
Am×n,Bn×k,Cl×p,Dp×q
AB ⊗ CD = (A ⊗ C)(B ⊗ D)
rvec(A) = (vec(AT))T, vec(AT) = (rvec(A))T
(commutation matrix)
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Kmnvec(A) = vec(AT)
1 0 00 00 0 0
1 0 00 00 0 0
00 10 00 0 0
00 00 10 0 0
00 00 10 0 0
01 0 0 0 0 0 0
(
)
(
)
m×n min(m,n) m × n
A
mn
rank(A) = k <
n×m
A−
A
A−A AA−
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AA−
A− A−y
A x = A−y
AA−Ax = Ax
Ax = y AA−Ax = AA−y Ax = y ⇒ x =
AA−Ax = AA−y
x
AA−A = A m×n A
(1)
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0H m 1
Qm qm αm
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§1.5
L
A
A
I LA =
I
(1)
L
2 −2 −1
−1 −2 5
A1 = 1 1 −2 ⇒ L1 = −1 −1 3
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1 0 −1
−1 −2 4
L1A1 = I
L1
A1
A1L1 = I
(2) 48
A2 = −2 3
L
5
7
−7608
2 17
⇒2
0 L25=
AH(AAH)−1
Moore-Penrose
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(4) LAm×n = In Moore-Penrose (1),(2),(4)
Ln×m
(5) AR = Im Moore-Penrose (1),(2),(3)
(6)
A−
Moore-Penrose
(1)
m × n A Moore-Penrose
(3)
|A −1|
=
1 |A|
(4)
(5) (A−1)−1 = A
(6) A−1
AH
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(AH)−1 = (A−1)H
A−H = (A−1)H
(7) AH = A, (A−1)H = A−1 (8) (A∗)−1 = (A−1)∗ (9) A B
(AB)−1 = B−1A−1
(10) A = diag(a1,a2,··· ,am)
()
mn
A ∈ Cm×n
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A
m>n
A
n
L = (AHA)−1AH
A
rank(A) =
m<n
A
m
rank(A) =
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R = AH(AAH)−1 A
Am×nxn×1 = ym×1 A y
x1 + x2 = 4 3x1 + 3x2 = 9
Ax = y 17 56
[A,y]
Am×n
rank([A,y]) = rank(A)
+
1 βm
bmbH mbm bH m 1
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bm = −R− m1rm βm = ρm − rH mR− m1rm = ρm + rH mbm
R− m1 +=1 Qm+1 = Qm qm qH m αm
R− m1 +1Qm+=1 Rm rm rH m ρm
Qm qm = Im 0m
qH m αm
A−1 = diag(a− 1 1,a− 2 1,··· ,a− m1)
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(11) A
A
⇐⇒ A−1 = AT
A
⇐⇒ A−1 = AH
n×n
(Sherman-Morrison ) A
xy
n×1
(A + xyH)
(A
+
xy
H)
−1
=
A −1
−
A−1xyHA−1 1 + yHA−1x
(Wood-
A†
A
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(weak generalized inverse)
A† A Moore-
Penrose
Moore-Penrose
(1) n × n
An×n
A−1
Moore-Penrose
(2) m × n (AHA)−1AH
Am×n (m > n) Moore-Penrose
(3)m × n Am×n (m < n)
0 0 0 0 1 0 0 0 a21
0 1 0 0 0 0 0 0 a31
K42vec(A) =
00 00 01 0 0 00 10 00 0 0
a41 a12
0 0 0 0 0 0 1 0 a22
00 01 00 0 0
a32
0 0 0 0 0 0 0 1 a42
a11 a
12
a21
a22 =
a
RA = AH(AAH)−1A = (RA)H
m×n
A n×m
A†
AA† A†A
Im×m In×n 25 56
AA† = (AA†)H (1) ∼ (3)
A†A = (A†A)H
(3)
A
m×n
GA
G
(
Moore-Penrose )
(1) AGA = A
(2) GAG = G
(3) AG Hermitian
bury ):
(A + UBV )−1 = A−1 − A−1UB(B + BV A−1UB)−1BV A−1
= A−1 − A−1U(I + BV A−1U)−1BV A−1
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= A−1 − A−1U(B−1 + V A−1U)−1V A−1
B=I (A − UV )−1 = A−1 + A−1U(I − V A−1U)−1V A−1
(2) A D
A U −1
(A − UD−1V )−1
−A−1U(D − V A−1U)−1
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VD
= −D−1V (A − UD−1V )−1
(D − V A−1U)−1
(3) A D
A U −1
(A − UD−1V )−1
VD
= −(D − V A−1U)−1V A−1
−(A − UD−1V )−1UD−1 (D − V A−1U)−1
dk Hdk = −1
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A† k =
A† k−1 − dkbk bk
4. Moore-Penrose
Am×n 1
r B = AAT
2 C1 = I 3
Ci+1 = 1 itr(CiB)I − CiB,
i = 1,2,··· ,r − 1
4
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A† = r tr(CiB)
CiAT
Ci+1B = O tr(CiB) = 0
unvecm,n(a) = Am×n =
a .2.
am+2 ..
···
am(n−1)+2 ..
am a2m ···
amn
a11 .
vec(A) = .
. amn
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rvec(A) = [a11,··· ,a1n,··· ,am1,··· ,amn]
unvecm,n(a) = Am×n =⇒ vec(Am×n) = a
A U −1 (A − UD−1V )−1
VD
= (U − AV −1D)−1
−(V − DU−1A)−1 (D − V A−1U)−1
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Hermitian Hermitian
Rm+1 =
Rm rm rH m ρm
Rm −1 R− m1 +1
R− m1 +=1
R− m1 0m 0H m 0
n×m
AA−A = A A−
1
A−1
AA−1A = A
2
L ALA = AI = A
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3
R ARA = IA = A
x = Gy
Ax = y
Leabharlann BaiduAGA = A,
(GA)# = GA
GA
1
G 21 56
2
Am×n
m
Ax = y
xo = AH(AAH)−1y
G Ax = y
AGA = A,
1
x ˆ = Gy (AG)# = AG
31
a32
a41
a42
= vec(AT)
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m × n A p × q B Kronecker
A⊗B
A⊗B = [aijB] =
a11B a12B ···
a21B a22B ···
..
..
a1nB
a2nB ..
am1B am2B ··· amnB
m × n A p × q B Kronecker
Ax Ax − y
2
Ax = y
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x ˆ = Gy + (I − GA)z,z
3
Ax = y
A
xo = (AHA)−1AHy
§1.6 Moore-Penrose
A
A−
A
A A−
A†
A A−
A
AA−A =
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Ax = y A†
x = A†y xy
x = A†y
A†A
A†AA†y
A A†
G Gy
Ax = y G
AGA = A
(AG)# = AG
GAG = G
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(GA)# = GA
G A Moore-Penrose
§1.7 Kronecker
AXB = C ?
AB XC
m×n p×q n×p m×q
LX + XN = Y ?
a1 am+1 ··· am(n−1)+1
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Duncan-Guttman
:
(A−UD−1V )−1 = A−1+A−1U(D −V A−1U)−1V A−1
(1.7.17)
1.7.14 -
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(1) A
A U −1 A−1 + A−1U(D − V A−1U)−1V A−1
= VD
−(D − V A−1U)−1V A−1
−A−1U(D − V A−1U)−1 (D − V A−1U)−1
x = A†y ⇒ Ax = y
Ax = y
A†AA†y
A†y = A†AA†y
y
A†AA† = A†
A†Ax = A†Ax =
(2)
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m×n A A†
m×n
A
(AHA)−1AH
LA = In×n
L= AL =
Im×m
AL = A(AHA)−1AH = (AL)H R=
AH(AAH)−1
AR = Im×m
A⊗B
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Ab11 Ab12 ··· Ab1q
[A⊗B]left = [Abij] =
Ab21 Ab22 ··· .. ..
Ab2q ..
Abp1 Abp2 ··· Abpq
Kronecker
(direct product)
(tensor product)
Kronecker
Kronecker
AA−1 = I
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A
n×n
A AA−1 = A−1A = I
n×n A−1
A−1 A
A ∈ Cn×n
(1) A (2) A−1
(3) rank(A) = n (4) A (5) A (6) det(A) = 0
(7) Ax = b (8) Ax = 0
(
b x=0 )
A
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A−1
:
(1) A−1A = AA−1 = I (2) A−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
2010 09 26 · 6A314
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: (FIT) 3-117 ( ) FIT 1-113 ( ) : 62794875 ( ), 62794138 ( )
Email: zxd-dau@tsinghua.edu.cn; fmhan@tsinghua.edu.cn