Question. I am working on Hoffman and Kunze, page 219, question 4c, they ask: Let $T$ be a linear operator on $V$. Suppose $V = W_1 \oplus \cdots \oplus W_k$ where each $W_i$ is invariant under $T$. Let $T_i$ be the induced restriction operator on $W_i$. Prove that the minimal polynomial for $T$ is the least common multiple of the minimal polynomials for $T_1, \dots, T_k$.
Solution. To solve this, I let $m_i$ be the minimal polynomial of $T_i$ and $m$ the minimal polynomial of $T$. My strategy is to prove that:
$p$ is an annihilating polynomial of $T$ if and only if $p$ is a common multiple of the $m_i$. (1)
Then, assuming (1) is true, $m$ divides every annihilating polynomial $p$, so $m$ is the least common multiple of the $m_i$. I think I have proved (1), and I won't bother with the details for now because my question is: Is (1) a true statement? Or have I "proved" something that isn't true at all?
Follow Up. I'm asking, in part, because a recent question on a qualifying exam at my school asked almost exactly this, except they say that $V = W_1 + \cdots + W_k$. So, assuming (1) is true, is (1) also true under the weaker assumption that $V = W_1 + \cdots + W_k$?
Thanks in advance!
You must of course in all cases assume the subspaces $W_i$ to be invariant under$~T$, so that restriction of $T$ to $W_i$ makes sense as a linear operator in $W_i$, and has a minimal polynomial in the first place. Now clearly any polynomial annihilating $T$ will annihilate any restriction of$~T$ as well. On the other hand, a polynomial $P$ annihilating $T|_{W_i}$ means that $W_i\subseteq\ker(P[T])$, and if this holds for every$~i$, then $\ker(P[T])$ contains the sum of the subspaces$~W_i$. But if that sum equals the whole space$~V$, then this means that $P[T]=0$ and $P$ is an annihilating polynomial of$~T$. So indeed, if $V=W_1+\cdots+W_k$ then $P$ is an annihilating polynomial of$~T$ if and only if it is a multiple of the minimal polynomial for every restriction of $T$ to a $W_i$, and the minimal degree (monic) such polynomial is by definition the $\operatorname{lcm}$ of those minimal polynomials. No direct sum required.