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TSI2
no5
(a, b)∈R2
M(a, b) = 
a−a b
−a a b
b b 0
.
M(a, b) (a, b)∈R2E
A B R3
EM3(R)
M(a, b)
M(a, b)
M(a, b) = P DP T
D=
2a0 0
0b√2 0
0 0 −b√2
P=1
2
√2 1 1
−√2 1 1
0√2−√2
.
E
M(a, b)E
A=1
2
−1 1 √2
1−1√2
√2√2 0
ψR3A
R3
λ, a b N =λI3+M(a, b)
U=
x
y
z
V=
x0
y0
z0
R3φ(U, V ) = UTNV
λ, a b φ
R3
1
φ(U, V )φR
Z=PTU P z1, z2z3
Z Z =
z1
z2
z3
φ(U, U) = ZT(λI3+D)Z=λ+ 2az2
1+λ+b√2z2 2+λ−b√2z2 3
λ > max −2a, |b|√2φR3

A=
1−1 0
−110
0 0 0
B=
0 0 1
0 0 1
1 1 0
∀a, b ∈R, M(a, b) = aA +bB
E= Vect(A, B)
EM3(R)A B
(A, B)M3(R)
(A, B)Edim E= 2
M(a, b)
∀a, b ∈R, M(a, b)
P P TM(a, b)P
M(a, b)
χM(a,b)(x) =
x−a a −b
a x −a−b
−b−b x
=
x−2a a −b
2a−x x −a−b
0−b x
(C1←C1−C2)
=
x−2a a −b
0x−2b
0−b x
(L2←L2+L1)
χM(a,b)(x) = (x−2a)(x2−2b2) = (x−2a)(x−√2b)(x+√2b)
Sp M(a, b)=2a, √2b, −√2b
2a6=±√2b
√2b6=−√2bb6=±√2a
b6= 0
M(a, b)
E2a
(M(a, b)−2aI)
x
y
z
=
0
0
0
⇐⇒ −ax −ay +bz = 0
bx +by −2az = 0
⇐⇒ (b2−2a2)z= 0 (bL1+aL2)
bx +by −2az = 0
⇐⇒
b6=±2a
b6=0
z= 0
x=−y
E2a= Vect 
1
−1
0

TSI2
E√2b
(M(a, b)−√2bI)
x
y
z
=
0
0
0
⇐⇒ 
((a−b√2)x−ay +bz = 0
−ax + (a−b√2)y+bz = 0
bx +by −√2bz = 0
⇐⇒ 
(a√2−b)x+ (b−a√2)y= 0 (L3+√2L1)
(b−a√2)x+ (√2a−b)y= 0 (L3+√2L2)
bx +by −√2bz = 0
⇐⇒
b6=±2a
b6=0
x=y
z=√2x
E√2b= Vect 
1
1
√2
E−√2b
(M(a, b) + √2bI)
x
y
z
=
0
0
0
⇐⇒ 
(a+b√2)x−ay +bz = 0
−ax + (a+b√2)y+bz = 0
bx +by +√2bz = 0
⇐⇒ 
(−a√2−b)x+ (b+a√2)y= 0 (L3−√2L1)
(b+a√2)x+ (−a√2−b)y= 0 (L3−√2L2)
bx +by +√2bz = 0
⇐⇒
b6=±2a
b6=0
x=y
z=−√2x
E−√2b= Vect 
1
1
−√2
X1=1
√2
1
−1
0
;X2=1
2
1
1
√2
;X3=1
2
1
1
−√2
X1, X2X3
2a, √2b−√2b
P X1, X2X3
M(a, b) = P DP TD=
2a0 0
0b√2 0
0 0 −b√2
P=1
2
√2 1 1
−√2 1 1
0√2−√2
.
E
M(a, b)M(a, b)TM(a, b) = I
M(a, b)TM(a, b) = M(a, b)2=
2a2+b2−2a2+b20
−2a2+b22a2+b20
0 0 2a2+b2

b2−2a2= 0
b2+ 2a2= 1 2b2= 1 L1+L2 4a2= 1 L2−L1
a=±1
2
b=±1
√2
M(a, b)a=±1
2b=±1
√2
A=1
2
−1 1 √2
1−1√2
√2√2 0
=M(−1/2,1/√2)
A ψ
det A=1
8
−1 1 √2
1−1√2
√2√2 0
=1
8
0 0 2√2
1−1√2
√2√2 0
(L1←L1+L2)
=1
8×2√2×2√2=1
ψ w ω
ψ1 = √2b b = 1/√2
w= (1,1,√2)
θRw w
Tr ψ= 2 cos θ+ 1 = Tr A=−1
cos θ=−1θ=π
ψR(1,1,√2)
A
Rw
R3
λ, a b N =λI3+M(a, b)
U=
x
y
z
V=
x0
y0
z0
R3φ(U, V ) = UTNV
R
UT∈ M1,3(R)NV ∈ M3,1(R)
φ(U, V ) = UT×(NV )∈ M1,1(R) = R
φR
φ(U, V ) = φ(U, V )T=UTNV T=VTNTUTT
=VTNU =φ(V, U)
NT=λIT
3+M(a, b)T=λI3+M(a, b) = N

TSI2
φ
φ(λU +µU0, V ) = (λU +µU0)TNV = (λUT+µU0T)NV
=λUTNV +µU0TNV =λφ(U, V ) + µφ(U0, V )
φ
Z=PTU U =P Z P T=P−1
φ(U, U) = φ(P Z, P Z) = (P Z)TN(P Z) = ZT(PTN P )Z
PTNP =PTλI3+M(a, b)P=λP TP+PTM(a, b)P=λI3+D
φ(U, U) = ZT(λI3+D)Z= (λ+ 2a)z2
1+ (λ+b√2)z2 2+ (λ−b√2)z2 3.
λ > max −2a, |b|√2= max(−2a, b√2,−b√2)
λ+ 2a > 0 ; λ−b√2>0 ; λ+b√2>0
φ(U, U)>0
φ(U, U) = 0 =⇒λ+ 2az2
1=λ+b√2z2 2=λ−b√2z2 3= 0
=⇒z1=z2=z3= 0 =⇒Z= 0 =⇒U= 0.
φ
λ > max −2a, |b|√2φR3
λ6max −2a, |b|√2λ6−2a λ + 2a60
U=P
1
0
0
6=
0
0
0
φ(U, U) = λ+ 2a60φ
λ6max −2a, |b|√2φR3