What are some meaningful connections between the minimal polynomial and other concepts in linear algebra?

This answer is devoted to the connection between commutative groups and linear algebra. It especially focuses on the link between exponent and minimal polynomial of an endomorphism, as the original poster rightfully find useful to connect notions between them.

Definition. Let $R$ be a commutative ring with unity, then a $R$-module is a triple $(M,+,\cdot)$, where $(M,+)$ is a commutative group and $\cdot\colon R\times M\rightarrow M$ is such that the following properties hold: $$\begin{align}r\cdot(x+y)&=r\cdot x+r\cdot y\\(r+s)\cdot x&=r\cdot x+s\cdot x\\(rs)\cdot x&=r\cdot (s\cdot x)\\1\cdot x&=x\end{align}$$

Remark. Modules over a field are exactly vector spaces over this field so that the notion of module is a generalisation of vector spaces to rings.

Observation. The $\mathbb{Z}$-modules are exactly commutative groups.

Proof. The direct implication follows from the definition.

Let $(G,+)$ be a commutative group and let define $\cdot\colon\mathbb{Z}\times G\rightarrow G$ by: $$0\cdot g:=0_G,n\cdot g:=(n-1)\cdot g+g,(-n)\cdot g:=n\cdot(-g).$$ Then, $(G,+,\cdot)$ is a $\mathbb{Z}$-module. $\Box$

Observation. Let $k$ be a field, then $k[T]$-module are exactly the $k$-vector spaces endowed with an endomorphism.

Proof. Let $(M,+,\cdot)$ be a $k[T]$-module, then notice that by restriction, $(M,+,\cdot_{\vert k\times M})$ is a $k$-vector space. Furthermore, $\varphi\colon M\rightarrow M$ defined by: $$\varphi(m):=T\cdot m$$ is an endomorphism of $M$.

Conversely, let $(E,+,\cdot)$ be a vector space and $\varphi\in\textrm{End}(E)$, then let define $\star\colon k[T]\times E\rightarrow E$ by: $$f\star x:=f(\varphi)(x).$$ Then, $(E,+,\star)$ is a $k[T]$-vector space. $\Box$

Definition. Let $M$ be a $R$-module, then $M$ is finitely generated if and only if there exists finetely many elements $x_1,\ldots,x_n$ of $M$ such that: $$M=\bigoplus_{k=1}^nRx_k:=\left\{\sum_{k=1}^nr_kx_k;r_k\in R\right\}.$$

Remark. Respectively, this extends the notion of finitely generated commutative groups and finite-dimensional vector spaces.

Theorem. Let $R$ be a principal ideal domain and $M$ be a finitely generated $R$-module, then there exists $d_1\vert\cdots\vert d_n$ elements of $R\setminus R^\times$ such that: $$M=\bigoplus_{k=1}^nM/(d_k).$$ Furthermore, the $d_k$ are unique up to multiplication by a unit of $R$.

Proof. See the corresponding chapter in Basic Algebra I by N. Jacobson. $\Box$

Remark. This theorem gives the structure of finitely generated commutative group as a direct sum of cyclic groups and the Frobenius decomposition.

Finely, here is the connection I claimed:

In the case of $R=\mathbb{Z}$, the least common multiple of the $d_k$ in the theorem leads to the notion of the exponent of a commutative group and for $R=k[T]$ to the minimal polynomial of an endomorphism.

In a sense, every theorem on the exponent of a commutative group can be transposed in the linear algebra world through the minimal polynomial. Here are a few examples, in particular, you will see how the minimal polynomial offers informations on the linear transformation:

Proposition. Let $G$ be a finite abelian group, then $G$ is cyclic if and only if its order equals its exponent.

Proposition. Let $E$ be a $n$-dimensional vector space and $\varphi\in\textrm{End}(E)$, then there exists $x\in E$ such that $$\left\{x,\varphi(x),\cdots,\varphi^{n-1}(x)\right\}$$ is a basis of $E$ if and only if the minimal polynomial of $\varphi$ equals its characteristic polynomial.

Remark. In the case $R=\mathbb{Z}$, the product of the $d_k$ in the theorem is equal to the order of the group and for $R=k[T]$ to the characteristic polynomial.

Proposition. Let $G$ be a group of prime order, then $G$ is a simple group.

Proposition. Let $E$ be a $n$-dimensional vector space and $\varphi\in\textrm{End}(E)$, if the minimal polynomial of $\varphi$ is irreducible, then $\{0\}$ and $E$ are the only $\varphi$-invariant subvector spaces of $E$.

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.

Here is one. The following two conditions are equivalent:

(I) the characteristic polynomial of square ($n$ by $n$) matrix $A$ and the minimal polynomial are the same

(II) the only matrices that commute with $A$ are of the form $$ a_0 I + a_1 A + a_2 A^2 + \cdots + a_{n-1} A^{n-1} $$

The set of matrices in (II) make up a vector space of dimension $n.$ The set of all matrices (such as commute with $I$) is of dimension $n^2,$ much bigger