Question

Cayley hamilton theorem ptoof by schur triangularisation

Answer 1

The Cayley-Hamilton theorem says that every square matrix can satisfy its own characteristic equation,
p(λ)=0
p(λ)=0
, or
p(A)=0
p(A)=0
.
The question is to show how the Cayley-Hamilton theorem follows from Schur's triangularization theorem.
If
σ(A)={
λ
1
,
λ
2
…,
λ
k
}
σ(A)={λ1,λ2…,λk}
, with
λ
i
λi
repeated
a
i
ai
times, then there is a unitary
U
U
such that
U∗AU=T=
⎛
⎝
⎜
⎜
⎜
⎜
⎜
T
1
⋆
T
2
⋯
⋯
⋱
⋆
⋆
⋮
T
k
⎞
⎠
⎟
⎟
⎟
⎟
⎟
U∗AU=T=(T1⋆⋯⋆T2⋯⋆⋱⋮Tk)
, where
T
i
=
⎛
⎝
⎜
⎜
⎜
⎜
⎜
λ
i
⋆
λ
i
⋯
⋯
⋱
⋆
⋆
⋮
λ
i
⎞
⎠
⎟
⎟
⎟
⎟
⎟
a
i
×
a
i
Ti=(λi⋆⋯⋆λi⋯⋆⋱⋮λi)ai×ai
.Moreover,
(
T
i
−
λ
i
I
)
a
i
=0,so(T−
λ
i
I
)
a
i
(Ti−λiI)ai=0,so(T−λiI)ai
has the form
(
T
i
−
λ
i
I
)
a
i
=
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⋆
⋯
⋱
⋆
⋮
0
⋯
⋯
⋱
⋆
⋮
⋆
⋮
⋆
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
←
i
th
row of blocks
(Ti−λiI)ai=(⋆⋯⋆⋯⋆⋱⋮⋮0⋯⋆⋱⋮⋆)←ith row of blocks