TITLE: What are some meaningful connections between the minimal polynomial and other concepts in linear algebra?
QUESTION [16 upvotes]: I’ve found that the most effective way for me to deeply grasp mathematical concepts is to connect them to as many other concepts as I can. Unfortunately, I’m seeing neither the importance nor the relevance of the minimal polynomial at all. Are there any significant connections between the minimal polynomial and linear algebra? Particularly regarding the relationship between the minimal polynomial of a linear operator on a vector space and other properties of that operator?
I think the problem is partially due to my textbook's emphasis on crunching out as many theorems about a concept as possible, rather than explaining it on a deep level and demonstrating its importance. But part of it is definitely also that this concept is not clicking well for me. Thanks for any help.

REPLY [8 votes]: Let $V$ be a finite dimensional vector space over a field $F$.
If a polynomial $f \in F[X]$ is the product $g_1 \cdots g_m$ of coprime polynomials, then
$$
\ker f(T) = \ker g_1(T) \oplus \cdots \oplus \ker g_m(T)
$$
When $f(T)=0$, we have $\ker f(T)=V$ and a decomposition of $V$ into invariant subspaces. This an important tool for understanding $T$.
It's natural to consider the simplest $f$ such that $f(T)=0$. This is the minimal polynomial.
By factoring $f$ into irreducible polynomials, we get the primary decomposition theorem.
When $f$ splits into linear factors (which happens for instance when $F$ is algebraically closed), we get a decomposition into generalized eigenspaces, which leads to the Jordan canonical form.
Thus, there is a close connection between polynomials and linear transformations. Somewhat surprisingly, this connection depends on the arithmetic properties of the field $F$, which for the most part of linear algebra does not really play a part, until you reach the decomposition theorems.