About the invariance of similar linear operators and their minimal polynomial

[sanctifier]

About the invariance of similar linear operators and their minimal polynomial

Notations:
F denotes a field
V denotes a vector space over F
L(V) denotes a vector space whose members are linear operators from V to V itself and its field is F, then L(V) is an algebra where multiplication is composition of functions.
τ, σ, φ denote distinct linear operators contained in L(V)
m_?(x) denotes the minimal polynomial of the linear operator "?"
～ denotes the similarity of the left and right operand, e.g., if τ ～ σ, then τ = φσφ^-1

Question:
If τ ～ σ are similar linear operators on V, then m_τ(x) = m_σ(x), i.e., the minimal polynomial is an invariant under similarity of operators.

I wonder how it's proved.

Thanks for any help!

 

[trambolin]

Use dummy variables and factor them wisely, here is a taste

$\alpha = \alpha\varphi\varphi^{-1}, \alpha\in\mathbb{F},\varphi\in\mathbb{V}$

 

[sanctifier]

Why the image in your reply is black? I mean that equation , I can't see it clearly.

I take the screen shot, but I don't know how to display the image on this forum, so I paste it here:
http://lh6.ggpht.com/_n_dfO1lVN1o/Sj2JssUWSYI/AAAAAAAAACI/bnUlb9LE52E/s128/1-1.JPG"

 

[trambolin]

Because I am too cool!

just kidding, it says alpha = alpha*phi*phi^(-1) and alpha in F and phi in V.

 

[sanctifier]

I still don't understand.
For example, suppose m_τ(x) = s₁+s₂τ+s₃τ² = 0 and τ = φσφ^-1,

then m_τ(x) = s₁+s₂φσφ^-1+s₃φ²σ²(φ^-1)² =0,

multiply (φ²)^-1 on the left and φ² on the right of the both sides of m_τ(x)

we will get s₁+s₂φ^-1σφ+s₃σ² = 0,
this doesn't equal to s₁+s₂σ+s₃σ² = 0, does it?

Ah, yes, now I know where I made a mistake, τ² ≠φ²σ²(φ^-1)²

since τ² = (φσφ^-1)² = φσφ^-1φσφ^-1
= φσ²φ^-1,

then multiply φ^-1 on the left and φ on the right of of the both sides of the equation m_τ(x) = s₁+s₂φσφ^-1+s₃φσ²φ^-1=0

we will get s₁+s₂σ+s₃σ² = 0

The property that τ² = φσ²φ^-1 I've explored before reaching this theorem, it seems I need more practice.

Thanks for reply, trambolin.

There are some other questions I posted in "Linear & Abstract Algebra" with no replies yet, maybe you can help me again, thanks a lot!

(This is not my mother language, please forgive the grammar mistakes I've made)