P(u1)P(u2)
u1
x
y
z
=
x+ 2y−z
y−z
x−y
=:
X
Y
Z
u2
1
x
y
z
=u1
X
Y
Z
=
X+ 2Y−Z
Y−Z
X−Y
=
(x+ 2y−z) + 2(y−z)−(x−y)
(y−z)−(x−y)
(x+ 2y−(y−z)
=
5y−3z
−x+ 2y−z
x+y
=:
A
B
C
u3
1
x
y
z
=u1
A
B
C
=
A+ 2B−C
B−C
A−B
=
(5y−3z) + 2(−x+ 2y−z)−(x+y)
(−x+ 2y−z)−(x+y)
(5y−3z)−(−x+ 2y−z)
=
−3x+ 8y−5z
−2x+y−z
x+ 3y−2z
P(u1)
x
y
z
=
−3x+ 8y−5z
−2x+y−z
x+ 3y−2z
−
5y−3z
−x+ 2y−z
x+y
−2
x+ 2y−z
y−z
x−y
+
x
y
z
=
−4x−y
−x−2y+ 2z
−2x+ 4y−z
P(u2)E
nB
u∈L(E)7→ MatB(u)∈ Mn(R)
L(E)
P
MatB(P(u)) = P(MatB(u)) .
u2
x
y
z
=
0
2x+y−z
x−y
=
0 0 0
2 1 −1
1−1 0
x
y
z
M=
0 0 0
2 1 −1
1−1 0
M2=
0 0 0
2 1 −1
1−1 0
0 0 0
2 1 −1
1−1 0
=
000
1 2 −1
−2−1 1
M3=
0 0 0
2 1 −1
1−1 0
000
1 2 −1
−2−1 1
=
000
3 3 −2
−1−2 1
P(M) =
000
3 3 −2
−1−2 1
−
000
1 2 −1
−2−1 1
−2
0 0 0
2 1 −1
1−1 0
+
1 0 0
0 1 0
0 0 1
=
1 0 0
−2 0 1
−1 1 1
P(u2)
x
y
z
=
x
−2x+z
−x+y+z
A
A2=
−1000
0−1 0 0
0 0 1 −2
0 0 0 1
A3=
1 2 0 0
−1−1 0 0
0 0 1 −3
0 0 0 1
P(A) =
−3−4 0 0
2 1 0 0
0 0 1 −4
0 0 0 1
P=X5+X4−2X3+ 3X−1
(a0, a1, a2, a3, a4, a5, ...)=(−1,3,0,−2,1,1,0, ...).
PpPi
(−1,0,1,0, ...),(3,−2,1,0, ...),
Pp=X2−1, Pi= 3 −2X+X2.
P(X) = X
n≥0
a2nX2n+X
n≥0
a2n+1X2n+1 =X
n≥0
a2n(X2)n+XX
n≥0
a2n+1(X2)n=Pp(X2) + XPi(X2).
A2=λ1λ20
0λ1λ2=λ1λ2Id2.
∀n≥0, A2n= (λ1λ2)nId2, A2n+1 = (λ1λ2)nA.
P(A) = Pi(λ1λ2)A+Pp(λ1λ2)Id2=Pp(λ1λ2)λ1Pi(λ1λ2)
λ2Pi(λ1λ2)Pp(λ1λ2).
A
A=0A1
A20,
A1A2
A2=A1A20
0A2A1, A3=0A1A2A1
A2A1A20.
A2n=(A1A2)n0
0 (A2A1)n, A2n+1 =0A1(A2A1)n
A2(A1A2)n0.
P(A) = Pp(A1A2)A1Pi(A2A1)
A2Pi(A1A2)Pp(A2A1).
A1=−1−2
1 1 , A2=1−1
1 0 ,
A1A2=−3 1
2−1, A2A1=−2−3
−1−2.
P=X3+ 3X Pp= 0 Pi=X+ 3
Pi(A1A2) = 0 1
2 2, Pi(A2A1) = 1−3
−1 1 ,
A2Pi(A1A2) = −2−1
0 1 , A1Pi(A2A1) = 1 1
0−2.
P(A) =
0 0 1 1
0 0 0 −2
−2−1 0 0
0 1 0 0
.
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