Linear Algebra - Sebastian Pancratz

I = Bn−1
an−1I = Bn−2 − ABn−1
a1I = B0 − AB1
a0I = −AB0
j A n+1−j
A n + an−1A n−1 + · · · a1A + a0I = 0,
= It n + an−1It n−1 + · · · + a1It + a0I
C α ∈ End(V ) B = {v1, . . . , vn} V α(vj) ∈
〈v1, . . . , vj〉 = Uj
⎛
λ1
⎜
[α]B = ⎝
⎞
∗
⎟
⎠
0 λn
(α − λjι)Uj ⊂ Uj−i
χα(t) = (λ1 − t) · · · (λn − t)
(α − λ1ι) · · · (α − λn−1ι)(α − λnι)V
⊂(α − λ1ι) · · · (α − λn−1ι)Un−1
⊂ · · ·
mα χα
⊂(α − λ1ι)U1 = 0.
V C dim V = n
χα(t) =
k
(t − λj) aj
j=1
λ1, . . . , λk α aj λj
k
j=1 aj = n
mα(t) = k
j=1 (t − λj) ej ej 1 ≤ ej ≤ aj 1 ≤ j ≤ k
mα χα ej ≤ aj 1 ≤ j ≤ k λ α(v) = λv
v = 0 0 = mα(α)v = mα(λ)v v = 0 mα(λ) = 0 (t − λ)
mα(t)
V F α ∈ End(V )
α λ1, . . . , λk α
mα(t) = k j=1 (t − λj)