Résumé Vecteur propre d`une application linéaire Diagonalisation d

g x g(x)
x
g E E x E
g λ
g(x) = λx.
λ x g
(e1, .., en)g
g
(e1, .., en)
g(λ1, ..., λn)g
λ10· · · 0
0λ2
0
0· · · 0λn
G n
Rn(e1, .., en)ein
i g
G
(e0
1, .., e0
n)Rn
g G0
P(e1, .., en) (e0
1, .., e0
n)
G0=P−1GP.
G=P G0P−1.

G
g λ
x λ
gλ(x) = g(x)−λx x gλ(x) = 0
x Ker(gλ)
g λ Ker(gλ)
g λ G −λIn
Ker(gλ)gλ
gλG−λIn
λ G −λIn
G−λInG λ
G−λInQ(λ)n λ
n
n
G
λiQ
e0
in
k(k+ 1)
e0
k+1 =µ1e0 1+...µke0 k
g(e0
k+1) = λk+1e0 k+1 =λk+1µ1e0 1+... +λk+1µke0 k
g(e0
k+1) = g(µ1e0 1+... +µke0 k) = µ1g(e0 1) + ... +µkg(e0 k) = λ1µ1e0 1+... +λkµke0 k
k i = 1 k λk+1µi=
λiµiµiλk+1 =λii
n
G

λ Ker(gλ)
G
n
Ker(gλ)
Ker(gλ)
A(a1, ..., ap)
An(an
1, ..., an
p)
G G =P DP −1D
Gn=P DP −1P DP −1 · · · P DP −1=P DnP−1
Dn