Let $M$ be a matrix over $\mathbb Q$ with an irreducible minimal polynomial $m_M$.
Suppose we have a polynomial $P \in \mathbb Z[x]$ such $P(M)(e_i) = \underline{0}$ for some $i$, i.e. the matrix $P(M)$ has a zero column.
(If this is too restrictive, I am also interested in the case when $P(M) $ annihilates some non-zero vector, i.e. $P(M)(v)=\underline{0}$ for some non-zero vector $v$. )
Question: Is it always the case that $m_M$ divides $P$?
I feel there should be other polynomials that annihilate some none zero vectors, but I couldn't find any examples of such matrices. Any hints to construct an example would be really appreciated. Thank you.
Yes, it is always the case that $m_M$ divides $P$.
Here is one argument that this should be the case. Suppose that $P(M)(v) = 0$ for some non-zero vector $v$. Note that $\ker(P(M))$ is a non-trivial invariant subspace of $M$. Thus, the restriction $R = M|_{\ker(P(M))}$ is well defined. On the one hand, we quickly see that $P(R) = 0$, which means that $m_R \mid P$. On the other hand, the minimal polynomial of $M$ is irreducible, so we have $m_R = m_M$. So, $m_M \mid P$.