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Homework Help: Nilpotent operator or not given characteristic polynomial?

  1. Mar 4, 2012 #1
    1. The problem statement, all variables and given/known data
    Suppose [itex]T:V \rightarrow V [/itex] has characteristic polynomial [itex] p_{T}(t) = (-1)^{n}t^n[/itex].
    (a) Are all such operators nilpotent? Prove or give a counterexample.
    (b) Does the nature of the ground field [itex]\textbf{F}[/itex] matter in answering this question?

    2. Relevant equations
    Nilpotent operators have a characteristic polynomial of the form in the problem statement, and [itex]\lambda=0[/itex] is the only eigenvalue over any field [itex]\textbf{F}[/itex].

    3. The attempt at a solution
    I originally thought that any linear transformation with the given characteristic polynomial would therefore have a block upper or lower triangular form with zeros on the diagonals, and therefore be nilpotent. But I'm confused by part (b), and the more I think about it, I'm not sure how to rule out that another more complex matrix representation of a non-nilpotent transformation might have the same form. And I have no idea how the choice of the field affects it. The very fact that they asked part (b) makes me think it does depend on the field, but I can't figure out why.
  2. jcsd
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