8 Polynôme minimal d`une transformation linéaire - ANMC

pA(λ)
A, B ∈Mn×n(K)AB =BA
P∈Mn×n(K)P−1AP P −1BP
A B
A∈Mn×n(K)A
pA(λ)pA(λ) = Πr
i=1(λ−λi)mi
Pn
i=1 mi=n
A∈Mn×n(C)
p(t)∈K(t)A∈Mn×n(K)
V K B V
p(t)∈K(t)f∈L(V, V )
[p(f)]B,B =p([f]B,B ).
p(A)=0
A∈Mn×n(K)
A∈Mn×n(K)mA(t)
p(t)A∈Mn×n(K)
g(t)A g(t)p(t)
A
A, B ∈Mn×n(K)∃P B =
P−1AP mA(t) = mB(t)

A∈Mm×n(R)R
det A= det A>A∈Mn×n(K)
A·A=A·A= det A·In
det A6= 0 A
A∈Mn×n(K)pA(λ)
A pA(A)=0 f∈L(V, V )
A∈Mn×n(K)pA(λ)
mA(λ)
pA(λ)=Πr
i=1(λ−λi)mimi=m(λi)mA(λ) =
Πr
i=1(λ−λi)ki1≤ki≤mii= 1, . . . , r