This post arose from an attempt to solve a question in this past Waterloo pure mathematics PhD comprehensive exam.
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
Fact 3. commutes with iff is a polynomial in .