Feuille de travaux dirigés - Ceremade - Université Paris

Ax=b
A=
2−140
4−151
−2 2 −2 3
0 3 −9 4
b=
8
16
3
3
.
A=
5 2 1
5−6 2
−421
b=
1
2
1
.
Ax=b
A=
1 0 6 2
8 0 −2−2
2 9 1 3
2 1 −3 10
b=
6
−2
−8
−4
.
A P AQ
U=
2100
0410
0031
0001
.
UX =I4X
4
A=
113002
465220
424134
220338

(S)Ax=b
A=
242
044
202
018
.
(S)
A
A
b(S)

Ax=b
A=
2−140
4−151
−2 2 −2 3
0 3 −9 4
b=
5
9
1
−2
.
A
A
A=
2 2 1
−264
1 0 2
, B =
1010
2451
0404
1002
, C =
aaaa
a b b b
a b c c
a b c d
,
A Mm,n(R)
2p+ 1 p aij = 0 |i−j|> p
A
n
aij = 0 |i−j|> p ⇒lij = 0 i−j > p uij = 0 j−i > p.
A=
a1c1
d2a2c2
.........
dn−1an−1cn−1
dnan
n
A L U
L=
1
l21
......
ln1
U=
u1c1
......
un−1cn−1
un
.

ui1≤i≤n lj2≤j≤n
u1=a1, li=di
ui−1
, ui=ai−lici−1,2≤i≤n.
Ax=b b Rn
A n
|aii|>
n
X
j=1
j6=i
|aij |,1≤i≤n.
n
A n A
AT
A n
A=awT
vA1,
a=a11 ∈Rv∈Rn−1w∈Rn−1A1n−1
A=10T
v
aIn−1 10T
0B awT
0In−1, B =A1−1
avwT,
0Rn−1
B A
A
n−1
X
i=1
i6=j
|vi|<|a|−|vj|,1≤j≤n−1,
n−1
X
i=1
i6=j
|(A1)ij |<|(A1)jj |−|wj|,1≤j≤n−1,
n−1
X
i=1
i6=j
|bij | ≤
n−1
X
i=1
i6=j
|(A1)ij |+|wj|
|a|
n−1
X
i=1
i6=j
|vi|,1≤j≤n−1.
B

A
A
A=
4111
1411
1141
1114
, B =
1 2 3 0
0−104
0 0 2 3
0 0 0 1
, C =
4 1 1 1
1 4 1 1
−1 1 0 0
3−300
.
n
n
∀v∈Rn,vTAv≥0,vTAv= 0 ⇒v=0.
A n A
B
A=BBT.
A A
A A
A A
LDLTA
B
A n
A
n > 1A
A=An−1l
lTann,
l∈Rn−1ann ∈RAn−1n−1
An−1
An−1
Bn−1An−1=Bn−1Bn−1T
m∈Rn−1b∈Rb > 0
B=Bn−10
mTb
A=BBT
A
B