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 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?