[Math] What are some meaningful connections between the minimal polynomial and other concepts in linear algebra

Question

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.

Answer 1

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 $E$.