Algèbre linéaire.

AX =Y A
X=A−1Y A−1
n Ai=ei
A AX =Y
M MA =U
UX =MY
A
A Mi,j
A=ai,j a1,16= 0 A
ai,1i
A
P1
P1
a1,1
M1
mi,i = 1
mi,1=−ai,1
a1,1
(i > 1)
mi,j = 0 (j > 1, j 6=i)
M1det(M1) = 1 M1A a1,1,0, ...0
M1A=αi,j
α2,2αi,2(i > 2) α2,2= 0
A=
3111
1311
1131
1113

A
LU
MA =U M =Mn−1Mn−2...M1
A=LU L =M−1=M−1
1...M−1 n−1
L M−1
1...M−1 n−1M−1
k
k
aj,k
ak,k
j=k+ 1...n
(λI −A) (λI −A)v= 0
λ
A λI −A
A(λI −A)P(λ)
A A
B(λ)λI −A
B(λ)∗(λI −A) = (λI −A)∗B(λ) = P(λ)∗I
A n ×n A
A
A n2
A I A An
A A
A
A2, A3, A4
I, A, A2, A3, A4
I, A...A4
A

B=
1113
1131
1311
3111
PA
PA
A
PAPA(X) = (X−λ1)×Q(X)
Q(A) = (A−λ1I)
(A−λ1I)×Q(A) = PA(A) = 0
Q(A)⊂(A−λ1I)
(X−λ1)Q(X)PA
U V
U(X)×(X−λ1) + Q(X)×V(X) = 1
U(A)×(A−λ1I) + Q(A)×V(A) = I
Q(A)⊃(A−λ1I)
Q(A)KnKer(A−
λ1I)p1λ1p1
B=
1113
1131
1311
3111
P A
P(λ) = det(λI −A) = (λ−λ1)(λ−λ2)...(λ−λn) = λn−p1λn−1−p2λn−2...−pn
s1= (A) = Σλi= Σn
i=1ai,i
sk= (Ak) = Σλk
i(k= 1..n)
p1=s1

2.p2=s2−p1.s1
k.pk=sk−p1.sk−1−... −pk−1.s1(k= 1..n)
pk(k= 1..n)
A A A2..An
B(λ)λ n −1
B(λ) = λn−1I+λn−2B1+...Bn−1
B(λ)×(λI −A) = P(λ)I
B0=I Bk=ABk−1−pkI
Bk=Ak−p1Ak−1−p2Ak−2... −pkI
pi(i= 1..n)
P(λ) = det(λI −A)
Bi(i= 1..n −1) λ
A
Ai(i= 1..n)
A1=A, p1= (A), B1=A1−p1I
A2=AB1, p2=1
2(A2), B2=A2−p2I
Ak=ABk−1, pk=1
k(Ak), Bk=Ak−pkI
Bn=An−pnI= 0
p1, p2...pnB1, B2, ...Bn
P(λ)B(λ)
Ak=Ak−p1Ak−1−... −pk−1A
Bk=Ak−p1Ak−1−... −pk−1A−pkI
P0(λ)
B(λ)B(λ)λI −A bi,j (λ)
V1(λ), ...Vn(λ)λI −Adet(λI −A)−
det(λ0I−A) = det(V1(λ)−V1(λ0), V2(λ), , ..., Vn(λ))+
det(V1(λ0), V2(λ)−V2(λ0), ..., Vn(λ))+...+det(V1(λ0), V2(λ0), ..., Vn(λ)−Vn(λ0))
P0(λ0) = lim
λ→λ0
P(λ)−P(λ0)
λ−λ0
= det(V0
1(λ0), V2(λ0), ..., Vn(λ0))+
det(V1(λ0), V 0
2(λ0), ..., Vn(λ0)) + ... + det(V1(λ0), V2(λ0), ..., V 0
n(λ0))
V0
i(λ0) = ei
det(V1(λ0), V2(λ0), ..., V 0
i(λ0), ..., Vn(λ0)) = bi,i(λ0)
P0(λ0) = Pn
i=1 bi,i(λ0) = (B(λ0)
P(λ) = λn−p1λn−1−p2λn−2... −pn
B(λ) = λn−1I+λn−2B1+... +Bn−1
P0(λ) = (B(λ)) = λn−1(I) + λn−2(B1) + ... + (Bn−1)
nλn−1−p1(n−1)λn−2−p2(n−2)λn−3...−pn−1=λn−1(I)+λn−2(B1)+
... + (Bn−1)
(Bi) = −pi(n−i)

B(λ)×(λI −A) = P(λ)I
B0=I Bi=ABi−1−piI
(Bi) = −pi(n−i) = (ABi−1)−npi
pi=(ABi−1)
i
p1= (AB0)B1=AB0−p1I
p2=(AB1)
2B2=AB1−p2I
pi=(ABi−1)
iBi=ABi−1−piI
A=
2−112
0 1 1 0
−1 1 1 1
1 1 1 0
,
p1, p2, p3p4B1, B2, B3, B4
A A−1
A
v
v0=v , vn+1 =Avn/||Avn||
v0
vn
w1, ..., wnAwi=λiwi|λ1|>|λ2|>
... > |λn|
v0=a1w1+a2w2+.... +anwn
Av0=a1λ1w1+a2λ2w2+.... +anλnwn
||Av0||−1=µ1
v1=µ1Av0
Av1=µ1A2v0=µ1(a1λ2
1w1+a2λ2 2w2+.... +anλ2
nwn)
||Av1||−1=µ2
Av2=µ2µ1(a1λ2
1w1+a2λ2 2w2+.... +anλ2
nwn)
||Avk−1||−1=µkCk=µk..µ1
vk=µkAvk−1
vk=µk..µ1Akv0=µk..µ1(a1λk
1w1+a2λk 2w2+.... +anλk
nwn)
vk=Ckλk
1(a1w1+a2(λ2
λ1
)kw2+...an(λn
λ1
)kwn)'Ckλk
1a1w1
k vkw1Avk'λ1vk