If $T$ is an endomorphism of a finite-dimensional vector space $V$ over a finite field, then how can I show that there exists a positive integer $r$ such that $T^r$ is a projection operator?
Some iterate of a linear operator over $\mathbf F_q$ is a projection
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linear-algebra
finite-fields
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0Please do not ask questions in the imperative, you make it sound like a homework question. And if it is indeed a homework question, please first read our FAQ on that: http://meta.math.stackexchange.com/q/1803/1543 – 2011-08-13
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0Do you know what is a Jordan normal form? – 2011-08-13
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0@Sarah I tried to fix up the question a little bit (the problems were not mathematical). Hope that's alright! – 2011-08-13
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1@Prometheus, if I remember correctly, you can't use the Jordan normal form for matrices over finite fields. – 2011-08-13
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2@Sarah: what have you tried so far? What tools did you consider? – 2011-08-13
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2@Calle Are you certain? The proofs I've seen of the JNF only rely on the structure theorem for modules over a PID, and $k[x]$ is a PID for any field $k$. You need the characteristic polynomial to split, but this only requires moving up to a larger finite field. – 2011-08-13