- #1

fishshoe

- 16

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## Homework Statement

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?

## Homework 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].

## 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.