Fiche TD Algèbre 2 : Réduction des endomorphismes & Systèmes différentiels

1er
◦
3A1A2
A1RA2
A1=
2−5−3
−1−2−3
3 15 12
;A2=
411
241
014
.
R
M1=
110
012
0−1−1
;M2=
411
141
114
et M3=
3−2 2
1 0 2
1−1 3
.
M=
3−1 1
−131
2 2 2
et Nm=
0m m2
m−10m
m−2m−10
,(m∈R∗).
M Nmnieme n≥1
M2(R)
A=3−4
2−3
f:M2(R)→ M2(R)M7→ f(M) = M.A −A.M
f
A2=I2f3−4f= 0 f
R
E=R3B={e1, e2, e3}f
E B
A=
−241
−241
002
.
Pf(X)f f
A Qf(X)f
J2Jnn≥2Ann∈N

(S) : x0(t)=2x(t)−y(t)
y0(t) = 3x(t)−2y(t)
(S) : 
−x(t) + y(t) + z(t) = x0(t)
x(t)−y(t) + z(t) = y0(t)
x(t) + y(t)−z(t) = z0(t)
x y z t
M3(R)
A=
−2 3 −3
1−3 1
1−4 2
.
A Ann∈N
(S) : 
−2x(t)+3y(t)−3z(t) = x0(t)
x(t)−3y(t) + z(t) = y0(t)
x(t)−4y(t)+2z(t) = z0(t)
x y z t
(S) : x0(t)=4x(t)−3y(t) + t+ 12
y0(t) = 6x(t)−5y(t)+3t+ 19
(Sh) (S)
(S)
(S)
(S)
x(0) = 1 y(0) = 2
(S) : x0(t) = x(t)+8y(t) + et
y0(t) = 2x(t) + y(t) + e−3t