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About the invariance of similar linear operators and their minimal polynomial

  1. Jun 20, 2009 #1
    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 mτ(x) = mσ(x), i.e., the minimal polynomial is an invariant under similarity of operators.

    I wonder how it's proved.

    Thanks for any help!
     
  2. jcsd
  3. Jun 20, 2009 #2
    Use dummy variables and factor them wisely, here is a taste

    [itex] \alpha = \alpha\varphi\varphi^{-1}, \alpha\in\mathbb{F},\varphi\in\mathbb{V}[/itex]
     
  4. Jun 20, 2009 #3
    Last edited by a moderator: Apr 24, 2017
  5. Jun 21, 2009 #4
    Because I am too cool!

    just kidding, it says alpha = alpha*phi*phi^(-1) and alpha in F and phi in V.
     
  6. Jun 22, 2009 #5
    I still don't understand.
    For example, suppose mτ(x) = s1+s2τ+s3τ2 = 0 and τ = φσφ-1,

    then mτ(x) = s1+s2φσφ-1+s3φ2σ2-1)2 =0,

    multiply (φ2)-1 on the left and φ2 on the right of the both sides of mτ(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 ≠φ2σ2-1)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 mτ(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)
     
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