Construct (representatives of) the similarity classes of matrices over having characteristic polynomial .
First, note that factors over as .
Recall that the characteristic polynomial of a matrix is the product of its invariant factors, that the minimal polynomial is the divisibility-largest invariant factor, and that the characteristic polynomial divides a power of the minimal polynomial (that is, since is a UFD, each factor of the characteristic polynomial appears in the factorization of the minimal polynomial).
If is a matrix having characteristic polynomial , then the minimal polynomial of is one of the following.
In each case, the remaining invariant factors are determined. The possible lists of invariant factors are thus as follows.
- ,
- ,
- ,
The corresponding matrices are thus as follows.
Every matrix over with characteristic polynomial is similar to exactly one matrix in this list.