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Let $T$ be a linear transformation on $V$ over field $F$. Let $c\in F$ be a characteristic value of $T$, and $W_c$ the characteristic subspace of $T$ associated with $c$. Suppose a proper $T$-invariant subspace $W \supset W_c$, and there is a vector $\alpha\in V\setminus W$, such that $(T-cI)\alpha \in W$. Prove that the minimal polynomial $p_T$ of $T$ is in the form of $(x-c)^2q$ for nonzero polynomial $q\in F[x]$

My progress:

My first thought is to use $p_\alpha\mid p_T$, and I need to show $(T-cI)^2\alpha=0$. for a certain vector $\alpha$ in $V\setminus W$, $(T-cI)\alpha$ is not zero because otherwise $\alpha$ is in $W_c$ and thus is in $W$ (contradiction).

So if I can show $W=W_c$, then it's done. But I fail to show $W=W_c$. Or am I in the wrong way?

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Out of curiosity, why delete and repost? –  Jonas Meyer Aug 1 '12 at 1:50
    
I corrected some writting, it didnot get answer for the day –  David Aug 1 '12 at 3:30
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To correct writing, editing seems to work well. There is no need to delete and repost. This thread on meta might be relevant. –  Jonas Meyer Aug 1 '12 at 3:34
    
What is $p_\alpha$? $W$ does not have to equal $W_c$. –  Jonas Meyer Aug 1 '12 at 3:39
    
$p_\alpha$ is the T-annihilator associated to $\alpha$, which divides $p_T$. can you offer me some hints? –  David Aug 1 '12 at 17:02
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1 Answer

You should start realising how little you are asked to do: it is given that $c$ is a characteristic value, so a root of $p_T$, and the (monic) factor $c$ can absorb anything; therefore all that is asked is to show that $c$ is a multiple root of $p_T$. If it were a simple root, then $W_c$ would be the (ordinary) eigenspace for$~c$ of$~T$, which means $(T-cI)v=0$ for all $v\in W_c$. Now looking at the question you will see that it does not exclude that possibility, and therefore the question is wrong: $T-cI$ and $W=\{0\}$ provides a counterexample (one has $p_T=x-c$ here). Probably you mistyped the condition $(T-cI)v\in W$ which should have been $(T-cI)v\notin W$, in which case one can exclude the possibility that $c$ is a simple root, since $(T-cI)v=0$ will be in any subspace whatsoever.

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