Exercice. On considère la matrice 1. Déterminer les valeurs propres

A=
1 0 −1
010
−1 2 1
.
A
−P(X) = 
1−X0−1
0 1 −X0
−1 2 1 −X
= (1−X)
1−X−1
−1 1 −X
=−X(X−1)(X−2)
A0,1 2
A
λ= 0 (x, y, z)∈ker A
x−z= 0
y= 0
−x+2y+z= 0
(x, y, z) = (x, 0, x) = x(1,0,1)
((1,0,1)) ker A
λ= 1 (x, y, z)∈ker(A−I)
−z= 0
0 = 0
−x+2y= 0
(x, y, z) = (2y, y, 0) =
y(2,1,0) ((2,1,0)) ker(A−I)
λ= 2 (x, y, z)∈ker(A−2I)
−x−z= 0
−y= 0
−x+2y−z= 0
(x, y, z) = (x, 0,−x) =
x(1,0,−1) ((1,0,−1)) ker(A−2I)
AR
AR
((1,0,1),(2,1,0),(1,0,−1)) Mat((1,0,0),(0,1,0),(0,0,1))
((1,0,1),(2,1,0),(1,0,−1))(id) =
P=
1 2 1
0 1 0
1 0 −1
P−1AP =
0 0 2
0 1 0
0 0 2
XnX(X−1)(X−2) n < 3
XnXnX(X−1)(X−2)
n≥3Xn=X(X−1)(X−2)Q(X) + aX2+bX +c X = 0,1,2
c= 0
a+b+c= 1
4a+2b+c= 2n
a= 2n−1−1b= 2 −2n−1Xn
X(X−1)(X−2) (2n−1−1)X2+ (2 −2n−1)X n ≥1, n = 0 1
AnA A2.
Xn=X(X−1)(X−2)Q(X) + aX2+bX An=A(A−I)(A−2I)Q(A) + aA2+bA
A A(A−I)(A−2I) = 0
An= (2n−1−1)A2+ (2 −2n−1)A n ≥1A0=I n = 0
J=
0100
0010
0001
1000
.
J2, J3J4. j J (e1, e2, e3, e4)
j(e1) = e4, j(e2) = e1, j((e3) = e2j(e4) = e3j2(e1) = e3. . . J2=
0 0 1 0
0 0 0 1
1 0 0 0
0 1 0 0
J3=
0001
1000
0100
0010
J4=I

J. J4=I X4−1
P∈C[X] deg(P)≤3P(J)=0 P= 0 P(X) = a0+a1X+
a2X2+a3X3P(J) = a0I+a1J+a2J2+a3J3=
a0a1a2a3
a3a0a1a2
a2a3a0a1
a1a2a3a0
P
a0=a1=a2=a3= 0 P
J. J
3 4
X4−1 1
J
J X4−1=(X2−1)(X2+ 1) =
(X−1)(X+ 1)(X−i)(X+i)CJC R
R
J.
4 1,−1, i, −i
C1−1
R