Find all the possible rational canonical forms of matrices having characteristic polynomial .

Let be a field. Our solution depends on how the polynomial factors over .

Suppose is irreducible in . (For instance, in or .) If is a matrix with characteristic polynomial , then the minimal polynomial of must divide and be divisible by and by . There are four such divisors, and with the minimal polynomial chosen, in each case the remaining invariant factors are determined. These are as follows.

- ,
- ,
- ,

The corresponding rational canonical forms are as follows.

Every matrix over whose characteristic polynomial is is similar to exactly one matrix on this list.

Now suppose factors over in . (For instance, in or .) Since has degree 2, there is only one way this can happen- namely, contains both roots of . If these roots are called and , then comparing coefficients in , we see that . Now the irreducible factorization of over is . Now if is a matrix with characteristic polynomial , then the minimal polynomial of must divide and is divisible by . There are eight such divisors of , and for each choice of the minimal polynomial, the remaining invariant factors are determined. The possible lists of invariant factors for are as follows.

- ,
- ,
- ,
- ,
- ,
- ,
- ,

The corresponding rational canonical form matrices are as follows.

Every matrix over having characteristic polynomial is similar to exactly one matrix in this list.

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