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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](http://math.stackexchange.com/questions/177232/linear-algebra-minimal-polynomial-problem) and repost?2012-08-01
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    I corrected some writting, it didnot get answer for the day2012-08-01
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    To correct writing, editing seems to work well. There is no need to delete and repost. [This thread on meta](http://meta.math.stackexchange.com/questions/3795/should-i-delete-and-repost-my-question-to-regenerate-interest) might be relevant.2012-08-01
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    What is $p_\alpha$? $W$ does not have to equal $W_c$.2012-08-01
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    $p_\alpha$ is the T-annihilator associated to $\alpha$, which divides $p_T$. can you offer me some hints?2012-08-01
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    I have revised my answer, since I realised that "characteristic subspace" here probably just means "eigenspace". Although probably somewhat late, confirmation by OP of this interpretation would be nice.2015-03-16

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