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Possible Duplicate:
Property of an operator in a finite-dimensional vector space $V$ over $R$

How do we prove the stabilization property:there exists $N$ such that for all $n \geq N$, we have $K_n=K_N$, $I_n= I_N$?

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    You haven't defined $K_n$ and $I_n$. Your question is not clear...2012-12-14

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Hint for (a): Note that $K_n \subseteq K_{n+1}$ and $I_{n+1} \subseteq I_n$. What can you say about the dimensions if they are not equal?

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    If $L^N x = 0$, what is $L^N (L x)$? If $y = L^N x$ for some $x$, what can you say about $Ly$?2012-12-13
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    If the characteristic polynomial of $L^N$ is $p(x) = x^m q(x)$ where $q(0) \ne 0$, then for any vector $y$, what can you say about $q(L^N) y$?2012-12-13
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    Where did you ues the fact if p∈R[x] is the characteristic polynomial of L , then p(L)=0 )? @Robert Israel2012-12-13