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TSI2
no6
α, β, γ KA
α2αβ αγ
αβ β2βγ
αγ βγ γ2
fR3A
Ker fR3
f
f
n∈Nαn∈K
α, β γ An=αnA
H
N
Ker f
α2αβ αγ
αβ β2βγ
αγ βγ γ2
x
y
z
=
0
0
0
α(αx +βy +γz) = 0
β(αx +βy +γz) = 0
γ(αx +βy +γz) = 0
(α, β, γ)6= (0,0,0) αx +βy +γz = 0
R3
Ker fR3αx +βy +γz = 0
Ker f2 0
2λ
C
0 + 0 + λ= Tr A=α2+β2+γ2
A α2+β2+γ2
Spec(f) = 0,0, α2+β2+γ2
0 Ker f
2
f

P
P−1AP =D=
0 0 0
0 0 0
0 0 α2+β2+γ2
Dn=
0 0 0
0 0 0
0 0 (α2+β2+γ2)n
An=P DnP−1
An=P
0 0 0
0 0 0
0 0 (α2+β2+γ2)n
P−1
= (α2+β2+γ2)n−1P
0 0 0
0 0 0
0 0 α2+β2+γ2
P−1
D
n∈NAn= (α2+β2+γ2)n−1A
A∈ Mn(K)
χAT=χA
A AT
A I +A+A2
H
N
A∈ Mn(K)
χAT(x) = det(xI −AT) = det (xI −AT)T
= det xIT−(AT)T= det(xI −A) = χA(x)
χAT=χA

TSI2
A P
D D =P−1AP
DT=D= (P−1AP )T=PTAT(P−1)T
Q= (P−1)TD=Q−1ATQ
AT
A AT
A P
D D =P−1AP
A=P DP −1I=P IP −1A2=P D2P−1
I+A+A2=P IP −1+P DP −1+P D2P−1=P(I+D+D2)P−1
I+D+D2
I+A+A2
A I +A+A2
α∈RA=cos α−sin α
sin αcos αB=cos αsin α sin α−cos α∈ M2(K)
K=CA
K=RA
B
H
N
α∈RA=cos α−sin α
sin αcos αB=cos αsin α sin α−cos α∈ M2(K)
K=C
χA(x) = x−cos αsin α
−sin α x −cos α=x2−2xcos α+ 1
∆ = 4(cos2α−1) = −4 sin2α
α= 0 [π]A±I

α6= 0 [π] ∆ 6= 0
Spec(A) = cos α+isin α, cos α−isin α=eiα, e−iα
CA
K=CA
K=Rα6= 0 [π] ∆ <0A
K=RA α 6= 0 [π]
K=R C
χB(x) = x−cos α−sin α
−sin α x + cos α=x2−1=(x−1)(x+ 1)
1−1R
K=R C A
A=cos θ2 sin θ
1
2sin θcos θχAA
χA(A)=0
Ann>1
H
N
A=cos θ2 sin θ
1
2sin θcos θ
χA(x) =
x−cos θ−2 sin θ
−1
2sin θ x −cos θ=x2−2xcos θ+ cos2θ−sin2θ
χA(x) = x2−2xcos θ+ cos2θ−sin2θ
χA(A) = A2−2A+ cos θ+ (cos2θ−sin2θ)I
=1 4 cos θsin θ
cos θsin θ1−2 cos2θ4 sin θcos θ cos θsin θ2 cos2θ
+cos2θ−sin2θ0
0 cos2θ−sin2θ

TSI2
χA(A) = 0
θ= 0 [2π]A=I An=I
θ=π[2π]A=−I An=I n An=−I n
θ6= 0 [π]XnχA(X)
Xn=χA(X)Q(X) + aX +b(1)
χA
∆ = 4 cos2θ−4(cos2θ−sin2θ) = (2 sin θ)2
χAα1= cos θ+ sin θ α2= cos θ−sin θ
(cos θ+ sin θ)n=a(cos θ+ sin θ) + b
(cos θ−sin θ)n=a(cos θ−sin θ) + b
a=(cos θ+ sin θ)n−(cos θ−sin θ)n
2 sin θ
b=−(cos θ−sin θ)(cos θ+ sin θ)n+ (cos θ+ sin θ)(cos θ−sin θ)n
2 sin θ.
An=χA(A)Q(A) + aA +b=aA +bI
An= 1
2[(cos θ+ sin θ)n+ (cos θ−sin θ)n] (cos θ+ sin θ)n−(cos θ−sin θ)n
1
4[(cos θ+ sin θ)n−(cos θ−sin θ)n]1 2[(cos θ+ sin θ)n+ (cos θ−sin θ)n]!
A2
θ= 0 [π]