Singular

Thank you for Singular.
I want to do what seems a simple job: to find the minimal polynomial of a square matrix.
The matrix entries are polynomials in the ring Q[x], so the minimal polynomial will also be in this ring.
The algorithmic complications seem to be
- although all the entities involved are in commutative rings, the calculations of powers of the original matrix are made in the ring of all such matrices over Q[x], which is not commutative;
- the tools of Singular seem designed (for good reason) to hide the information I want. They tell the user whether an element is in an ideal, but not (obviously) how the element can be expressed in terms of the ideal's generators.
Any guidance welcome.

1. don't you need to adjoin another commutative variable to your ring (e.g. Q[x] -> Q[x][T])?
2. matrix multiplication is generally non-commutative, no?
3. one uses 'lift' for "how the element can be expressed in terms of the ideal's generators"

4. i'd suggest you to start with characteristic polynomial (of M) as follows:
a. compute D = det(M) in our original ring,
b. switch to enriched ring (e.g. new variable T), and imap D and M there
c. compute det(D * E * T - M) (where E is the identity matrix)
it can be used for testing and checking your results...


ps: Happy New Year

Thanks, malek.
This makes things much clearer.
Alan.