 Solution

Take $V={\mathbb{R}}^3$, $\alpha =\mathrm{diag}(0,1,0)$ and $W=\mathrm{span}(e_1,e_2)$. Then $W$ is a proper invariant subspace, and the restriction has matrix $\mathrm{diag}(0,1)$. In each case $X(X-1)$ annihilates the operator, while any annihilator must vanish at both zero and one. ConsequentlyThis same example works over any scalar field.