Find (representatives of) all the similarity classes of matrices over which satisfy . Do the same for matrices which satisfy .
We begin by factoring into irreducibles. Note that since in . Moreover, , and is irreducible over since it has no roots. Thus we have the irreducible factorization .
Suppose now that is a matrix over satisfying ; then the minimal polynomial of divides and has degree at most 3 (since it divides the characteristic polynomial, which has degree 3). The divisors of having degree at most 3 are , , , and . Note that the third polynomial, , cannot be the minimal polynomial of a matrix since no power of is divisible by a degree 3 polynomial. (See Prop. 20 in D&F.) For each remaining divisor of , once fixed as the minimal polynomial of , the remaining invariant factors are determined. The possible lists of invariant factors for are thus as follows.
- , ,
The corresponding matrices in rational canonical form are then as follows.
(Note that these matrices are also representatives of the conjugacy classes of the representation of by permutation matrices. Does this mean anything?)
Next we factor . We have , , and . We claim that is irreducible over . Certainly has no roots, and thus no linear factors. Suppose now that has a quadratic factor. Multiplying out and comparing coefficients, we have , , , and . Now by the fourth equation, that by the second. Now . But then , a contradiction. So has no quadratic factors, and thus is irreducible over . So factors into irreducibles as .
Suppose now that is a matrix satisfying . Then the minimal polynomial divides and has degree at most 4. The divisors of with this property are , , , , and . The possible lists of invariant factors for are then as follows.
- , , ,
- , ,
The corresponding matrices in rational canonical form are thus as follows.
Every matrix over such that is similar to exactly one matrix in this list.