Cnc Math2 Psi 2015e1
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7/24/2019 Cnc Math2 Psi 2015e1
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SO2(R)
M2,1(R)
M2(R) I 2
M2(R)
2(R)
A ∈ M2(R)
(A)
(A)
X A
M
M2(R)
M2,1(R)
tM
M
M2(R)
M2(R)
S 2(R)
M2(R)
M2,1(R)
.2
(X, Y ) −→< X, Y >:= tXY
U
M2(R)
M2(R)
U
U =
A ∈ M2(R) ;
(A)
2
− 4
A > 0
A −→ (A) A −→ det A M2(R)
U M2(R)
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R
2
x −→ 14
x2
R
2
(A), det A
; A ∈ U
V = U ∩
a b
c d
∈ M2(R) ; b = 0
.
U M2(R)
f 1 : V −→ M2(R)
2(R)
M ∈ M2(R)
f 1(M )−1M f 1(M )
B =
α 00 β
∈ M2(R)
α = β
C (B) = {M ∈ M2(R) ; M B = B M }
U
V ∈ M2(R)
U BU −1 = V BV −1
V −1U
(M, P ) ∈ M2(R) × 2(R) D ∈ M2(R)
P −1M P = D
D
M
P
M
O2(R) = {A ∈ M2(R) ; tAA = I 2}
SO2(R) = {A ∈ O2(R) ; det A = 1}
SO2(R) =
a −bb a
∈ M2(R) ; a2 + b2 = 1
SO2(R) 2(R)
Φ : R −→ M2(R)
∀ θ ∈ R, Φ(θ) =
cos θ − sin θ
sin θ cos θ
.
Φ
Φ(R) = S O2(R)
A ∈ M2(R)
S A = {U AU −1 ; U ∈ S O2(R)}
M −→ tM
M2(R)
U −→ U −1
SO2(R)
M2(R)
U −→ U AU −1
SO2(R)
S A
α
β
α = β
B1 =
α 00 β
B2 =
β 00 α
σ : S A −→ M2(R)
σ(S A) ⊂ {B1, B2}
σ
R
{α, β }
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U
f : U −→ M2(R)
2(R)
M ∈ U
f (M )−1M f (M )
M ∈ U ∩S 2(R) C 1(M ) C 2(M ))
f (M )
C 1(M )
C 2(M )
M
(M2,1(R), < , >)
C 1(M )C 1(M )2
C 2(M )C 2(M )2
)
α(M )
g(M ) ∈ M2(R)
α(M ) C 1(M )C 1(M )2
C 2(M )C 2(M )2
)
g(M ) ∈ S O2(R
)
g : U ∩ S 2(R) −→ M2(R)
SO2(R)
g
M ∈ U ∩S 2(R)
g(M )−1M g(M )
B =
α 00 β
∈ M2(R)
α = β
S B =
U BU −1 ; U ∈ S O2(R)
U ∩ S 2(R)
h
g
S B =
U BU −1 ; U ∈ S O2(R)
M ∈ S B
h(M )−1M h(M )
B
M −→ h(M )−1M h(M )
S B
h(M )−1M h(M ) = B
M ∈ S B
M ∈ S B
h(M )−1M h(M ) = B
U ∈ SO2(R)
h(U BU −1)−1U
±I 2
ϕ : S O2(R) −→ S B × {−I 2, I 2}
ψ : S B × {−I 2, I 2} −→ SO2(R)
ϕ(U ) =
U BU −1, h(U BU −1)−1U
ψ(M, D) = h(M )D
ϕ
ψ
U −→
h(U BU −1)−1U
SO2(R)
{−2, 2}
f
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