TITLE: Minimal polynomial of restriction to invariant subspace divides minimal polynomial
QUESTION [13 upvotes]: I'm trying to prove this:
$T : V \to V$ linear transformation. $W$ subspace of $V$. If $W$ is $T$-invariant then the minimal polynomial for the restriction operator $T|_W$ divides the minimal polynomial for $T$.

REPLY [13 votes]: For any linear operator $S$ of a space $W$, the set of polynomials $P$ such that $P[S]=0$ (on $W$) is the set (ideal in $K[X]$) of multiples of the minimal polynomial $\mu_S$ of $S$. Apply this in the question to the restriction $S=T|_W$, with $P=\mu_T$, the minimal polynomial of$~T$ on all of$~V$ (since $P[T]=0$ on all of$~V$, its restriction $P[S]$ is certainly $0$ on$~W$). This gives that $P=\mu_T$ is a multiple of$~\mu_S$.