Hello,

I want to prove,

Let the minimal polynomial of $T$ on a finite dimensional vectore space $V$ be p^2 where $p$ is irreducible. Is it true that $V$ contains a proper $T$ invariant subspace?

(I'm using theorems out of Hoffman and Kunze. $\S$ 6.1)

So the minimal polynomial divides the characteristic polynomial $\det (\lambda - T) = c(\lambda)$ so in particular $c(p) = 0$ and therefore, $p$ is an eigenvalue of $T$.

I can let $E_p$ be a projection onto the subspace $p v$ where $v$ is the eigenvector associated with $p$. We have $E_p w = pw`$ for all $w \in V$.

To show that $T$ is invariant on $p v$ I can check that $T$ commutes with $E_p$.

I tried writing:

$$(E_p T) y = E_p (T y) = y` = p (\alpha v) = p E_p y = T (E_p y) = (TE_p)y.$$

Here $y`$ is the projection of $Ty$ onto $ v$. Since $y`$ is colinear to $v$ I should be able to find a scalar such that $p(\alpha v ) = y`$.

I'm not super confidant with this: is there any advice?