minimal polynomials of non-invariant subspaces

Question

I know that given an endomorphism $f$ over a vectorial space $V$, if I have an $f$-invariant subspace $W$, then I know the minimum polynomial (let's say mp from now on) of $f$ restricted to $W$ divides the mp of $f$ over the whole space $V$. I was wondering, what could I say about the mp of non-invariant subspaces? Does it divides the mp of $f$ over the whole $V$? I think so, but I have found the former result quite a lot, while nobody talks about the mp of non-invariant subspaces and its correlation with the mp of $f$ over $V$. Thanks for the help.

Answer 1

Given an endomorphism $f$ of a $K$-vector space $V$, the space can be viewed as $K[X]$-module, with $X$ acting as $f$. Submodules of this module are precisely $f$-invariant subspaces. Every subset $S$ of a module spans a submodule $K[X]S$, in the same way as a subset of a vector space spans a subspace. This has the property that if $P\in K[X]$ acts as $0$ on each element of $S$, then it acts as $0$ on $K[X]S$ (and of course conversely, since $K[X]S$ contains $S$). So the ideal of such polynomials is the same for $S$ as for $K[X]S$, and the generator of this ideal is the minimal polynomial of the restriction of $f$ to the $f$-invariant subspace $K[X]S$.

In other words, you get nothing new by considering non-invariant subspaces, their "minimal polynomial" is just the same as that associated to the invariant subspace they span.