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Let $K$ be a field of characteristic zero and Let $A$ be an $n \times n$ matrix over $K$.Prove the following facts:

(a), $N$ is nilpotent iff Trace$(N^m)=0$ for all $0<m \leq n$ iff Trace$(N^m)=0$ for all $m \in \mathbb{N_{+}}$

(b), If $N$ is nilpotent then $N$ is similar to a strict upper triangular matrix.

Additional Query: Is above true if $K$ is is not assumed to be algebraically closed?(Then one can't apply Jordan normal form anymore)

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1 Answer 1

up vote 7 down vote accepted

The existence of a Jordan normal form of a (finite dimensional) vector space endomorphism $\phi$ does not depend on the fact that ground the field is algebraically closed, just that the minimal polynomial of $\phi$ splits over the ground field. Since an endomorphism is nilpotent if and only if its minimal polynomial is of the form $X^k$ for some $k\in\mathbf N$, nilpotent matrices always have a Jordan normal form.

This gives you point (b). Point (a) is a consequence of the fact that in characterestic $0$ the coefficients of the characteristic polynomial of $\phi$ can be expressed in terms of the values $\operatorname{tr}(\phi^k)$ for $0<k\leq n)$ via Newton's identities; if all of those traces are $0$ then the characteristic polynomial is $X^n$ and $\phi$ nilpotent. Apart from its characteristic, this does not depend on the ground field.

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In fact, looking at the proof of the Newton identities, we see that b) holds whenever $\mathrm{char}(K)>n$ –  M Turgeon Sep 6 '12 at 12:26
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