Linear Algebra - Sebastian Pancratz

V C α ∈ End(V )
B V [α]B
B = {v1, . . . , vn} α(vj) ∈ 〈v1, . . . , vj〉 j = 1, . . . , n
n n = 1 n > 1
V C λ ∈ C α − λι
U = Im(α−λι) V U
α(U) ⊂ U
α(U) = α((α − λι)(V )) = (α − λι)(αV ) ≤ (α − λι)(V ) = U.
α ′ = α| U : U → U dim U < dim V
B ′ = {v1, . . . , vk} U A ′ = [α ′ ]B ′
B = {v1, . . . , vk, . . . , vn} V [α]B
[α]B =
A ′ ∗
.
0 λI
1 ≤ j ≤ k α(vj) = α ′ (vj) ∈ U k
j > k (α − λι)(vj) ∈ U U α(vj) = λvj + u u ∈ U
k j u B ′
λ (j, j)
v1 α α(v1) = λv1 U
〈v1〉 V v ∈ V v = λvv1+u u ∈ U λv ∈ F
π(v) = u V U u ∈ U ˜α : U → U ˜α(u) = π(α(u))
˜α ∈ End(U) v2, . . . , vn U ˜α(vj) ∈ 〈v2, . . . , vj〉
2 ≤ j ≤ n α(vj) = λ α(vj)vj + ˜α(vj) ∈ 〈v1, . . . , vj〉 α(v1) ∈ 〈v1〉
C
R ±I R 2
V F α ∈
End(V ) B V [α]B
χα χα F
[α]B =
⎛
⎜
⎝
a11
∗
0 ann
χα(t) = (a11 − t) · · · (ann − t) aij ∈ F
dimF V λ F U =
(α − λι)(V ) α(U) ≤ U V α ′ = α| U ∈ End(U) B ′ U
B V
[α]B =
⎞
⎟
⎠
[α ′
]B ′ ∗
.
0 λI
χα(t) = χα ′(t)χλI(t) χα ′ F
B ′ [α ′ ]B ′