Define minimal polynomial for any square matrix over any field

[darijgr]

This implements a function minpoly_lin on the Matrix class that computes the minimal polynomial of a square matrix over any field, notwithstanding the possible lack of polynomial factorization algorithms.

It is slow, which is why I am not overwriting the existing minpoly method. More should probably be done to expose this method as an algorithm for minpoly.

Fixes #39696.

📝 Checklist

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⌛ Dependencies

[user202729]

We could probably use https://flintlib.org/doc/gr_mat.html#c.gr_mat_minpoly_field instead. (but the first step is to convert arbitrary matrix to flint)

[maxale]

This looks a bit more straightforward:

        pow = self**0
        Mpows = matrix(F,n**2,1,pow.list())
        for k in range(1,n+1):
            pow *= self
            v = vector(pow.list())
            try:
                cs = Mpows.solve_right(v)
            except ValueError:
                Mpows = Mpows.augment(v)
                continue
            ....

[darijgr]

user202729

Does flint know how to deal with arbitrary Sage fields? I thought this is not how packages worked...

[user202729]

You can probably implement a generic wrapper with https://flintlib.org/doc/gr_implementing.html

[github-actions]

Documentation preview for this PR (built with commit a758537; changes) is ready! 🎉
This preview will update shortly after each push to this PR.

[darijgr]

You can probably implement a generic wrapper with https://flintlib.org/doc/gr_implementing.html

Can I really outsource linear algebra over any field to an external package? How does it compute in that field?

Sorry, I'm a complete noob in these tehcnical aspects...

[user202729]

Probably if you write the correct wrappers, I don't know. I haven't looked carefully either.