This post arose from an attempt to solve a question in this past Waterloo pure mathematics PhD comprehensive exam.

Let be an algebraically closed field. Let denote the set of all matrices with entries in . Let

be a Jordan block. Let be the standard basis vectors, i.e. the -th component of is . Note the action of on the basis vectors: for each (where we take ).

Suppose commutes with . Then

i.e. if is the -th column of , then for each . Thus

i.e. if , then , a polynomial in . Further, since , it follows that is a polynomial in . So we deduce:

**Fact 1. ** commutes with iff is a polynomial in .

**Fact 2. ** commutes with iff is a polynomial in **.**

If we denote

for , then we’ve just shown

Now let have minimal and characteristic polynomial . This means the Jordan normal form of is . So there exists an invertible matrix such that . Thus

Thus

**Fact 3. ** commutes with iff is a polynomial in .