《矩阵分析与应用》课件.ppt

文档介绍：清华大学
张贤达教授
《矩阵分析与应用》
学****矩阵理论的很好的一部书
1
56
2010
09 26
·
6A314
:
(FIT)
3-117 (
)
FIT
1-113 (
)
: 62794875m −1
Rm+1 =
R− m1 +1
R− m1 +1
=
R− m1
0H m
0m
0
+
1
βm
bmbH m
bH m
bm
1
bm = −R− m1rm
βm = ρm − rH mR− m1rm = ρm + rH mbm
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56
R− m1 +1
= Qm+1 =
Qm
qH m
qm
αm
R− m1 +1Qm+1
=
Rm
rH m
rm
ρm
Qm
qH m
qm
αm
=
Im
0H m
0m
1
Qm
qm
αm
A1 = 1
−2 ⇒ L1 = −1 −1 3
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56
§
L
A
A
I
LA =
I
(1)
L


2 −2 −1
1
1
0
−1


−1 −2
5
−1 −2 4

  
L1A1 = I
A1L1 = I
L1
A1
4 8
A2 =  5 −7 ⇒ L2 =
5 1 
R = −1 0 ,
R =  0 0 , ···
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56
(2)
L
−2 3
 
 
7
68
0
2
17
2
0
5
,L2 =
0
0
3 7
2 5
, ···
L2A2 = I
(3)
( A2L2 = I)
L
A3 =
⇒
1
2
3 1
1 1
 

−1 1
2
−1

 
3
AR = I
−1
( RA = I)
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56
LA = I
L
A
(
)
m
n
A ∈ Cm×n
AR = I
R
A
(
)
n
A ∈ Cm×n
m
A
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56
m>n
A
rank(A) =
n
L = (AHA)−1AH
A
m<n
A
rank(A) =
m
R = AH(AAH)−1
A
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56
Am×nxn×1 = ym×1
A
y
x1 + x2 = 4
3x1 + 3x2 = 9
Ax = y
[A,y]
Am×n
rank([A,y]) = rank(A)
(
)
(
)
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56
m × n
A
m
n
rank(A)
k
<
min(m,n)
m × n
n × m
=
A−
A
A−A
AA−
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56
Ax = y
AA−
AA−Ax = AA−y
A−
A−y
A
x = A−y
Ax = y ⇒ x =
AA−Ax = AA−y
AA−Ax = Ax
AA−A = A
x
(1)
A
m × n
AA−A = A
n × m
A−
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56
1
2
3
A−1
L
R
AA−1A = A
ALA = AI = A
ARA = IA = A
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56
x = Gy
AGA = A,
Ax = y
(GA)# = GA
G A
1
G
2
Am×n
m
Ax = y
xo = AH(AAH)−1y
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56
x ˆ = Gy
(AG)# = AG
G
Ax = y
AGA = A,
1
Ax
Ax − y
2
Ax = y
x ˆ = Gy + (I − GA)z,z
3
Ax = y
A
xo = (AHA)−1AHy
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56
§
Moore-Penrose
AA−A =
A
A
A
A−
A−