minimal polynomial

Definition

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

The minimal polynomial of is the monic polynomial which generates the kernel of the ring homomorphism

Cyclic modules and companion matrices

Let be a cyclic with generator . Then is spanned (over ) by . We can write the minimal polynomial as

Using this we can see that is a -basis for . This can be seen to be linearly independent using the minimal polynomial and is spanning using the division algorithm.

Using the ordered basis , we have

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

This gives that

is called the companion matrix of .