minimal polynomial

Definition

Let $M$ be a finitely generated torsion $k[x]$-module for a field $k$. Note that $M$ is equivalent to a vector space with an accompanying endomorphism as explained here.

The minimal polynomial of $M$ is the monic polynomial $q_M\in k[x]$ which generates the kernel of the ring homomorphism

$$k[x]⟶{\text{End}}_{\text{ab}}(M)$$

Cyclic modules and companion matrices

Let $M$ be a cyclic with generator $m\in M$. Then $M$ is spanned (over $k$) by $\{m,Xm,X^{2}m,\dots \}$. We can write the minimal polynomial as

$$q_M(x)=x^d+a_{d-1}{x}^{d-1}+\cdots +a_{1}x+a_0,a_i\in k.$$

Using this we can see that $\{m,Xm,X^{2}m,\dots ,X^{d-1}m\}$ is a $k$-basis for $M$. This can be seen to be linearly independent using the minimal polynomial and is spanning using the division algorithm.

Using the ordered basis $\{v_i=X^{i}m|0\le i\le d-1\}$, we have

$$X{v}_i=v_{i+1},0\le i\le d-2$$

and for the top level degree, it “wraps around” and relates back to the minimal polynomial

$$X{v}_{d-1}=-a_0{v}_0-a_1{v}_1-\cdots -a_{d-1}{v}_{d-1}$$

This gives that

$$C_{q_M}=\left[\begin{array}{cccccc}0& \cdots & & \cdots & 0& -a_0\\ 1& \ddots & & & ⋮& ⋮\\ 0& \ddots & & \ddots & ⋮& ⋮\\ ⋮& \ddots & & \ddots & 0& -a_{d-2}\\ 0& \cdots & & 0& 1& -a_{d-1}\end{array}\right]$$

$C_{q_M}$ is called the companion matrix of $q_M$.