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An endomorphism $f: V \rightarrow V$ of an $F$-vector space is called nilpotent iff there exists $ \delta \in \mathbb N$ such that $f^\delta=0$. Suppose that $f : V\rightarrow V$ is a nilpotent endomorphism of a finite dimensional vector space. Show that the vector space $V$ has an ordered basis $A$ such that the representing matrix $_A [f]_A$ of $f$ with respect to the basis has the form of an upper triangular matrix with only $0$s along the diagonal.

I've read proofs in the opposite order, but I'm not going to fall for that logical flaw. So I've been trying to expand out the multiplication but that seems long winded, and I know that once I've proven it's upper triangular that I can use the Cayley-Hamilton theorem to show that the diagonal is zero. Any hints?

Cameron Buie
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Tessa Danger Bamkin
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1 Answers1

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Let $V_0=V$, $V_1=f(V)$, $V_2=f(V_1)$ and so on.

  • Show that there is a $c\geq0$ such that $$V=V_0\supsetneq V_1\supsetneq V_2\cdots\supsetneq V_{c}\supsetneq V_{c+1}=0$$
  • For each $i\in\{0,\dots,c\}$ pick a basis $B_i$ of a complement of $V_{i+1}$ in $V_i$.
  • See what $f$ does to the elements of the set $B_0\cup\cdots\cup B_c$.
Mariano Suárez-Álvarez
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  • not really sure of the meaning of this to be honest. I mean I see that you're saying there is a vector which is in the vector space and another vector is the function applied to the original vector then the same in recursive form. Then there is some constank for which v is something which i dont understand. Not really sure how to go about this still. Been mulling it over but no improvment. – Tessa Danger Bamkin Oct 08 '13 at 20:40
  • To be honest, I cannot imagine how it is you even understand the statement of Engel's theorem, in that case. The constrcution I sketched is explained in every textbook on linear algebra which deals with Jordan's canonical form. If what I wrote is that opaque to you, I can't think of much more than suggest you read one such textbook. – Mariano Suárez-Álvarez Oct 09 '13 at 00:58
  • we've not covered jordan;s canonical form. I dont completely understand the statement of Engel's theorem but from your hint and the formatting of the question I can tell that they are very closely intertwijned. – Tessa Danger Bamkin Oct 09 '13 at 07:35
  • Your questio is precisely the special case of Engel's theorem for 1-dimensional algebras. – Mariano Suárez-Álvarez Oct 09 '13 at 07:41