Hi, Backwards compatibility or not, I consider this either a bad design or a bug:
sage: M = Matrix([[1,-1],[1,0]]) sage: f = M.minimal_polynomial() sage: f.roots() [] sage: M.eigenvalues() [0.50000000000000000? - 0.866025403784439?*I, 0.50000000000000000? + 0.866025403784439?*I] First of all, why return roots in some arbitrary field? I don't agree that this is the most natural result for first year calculus or linear algebra. Many will only consider real solutions to be valid. Certainly an algebraist will expect solutions over the given base ring. There is also the lack of consistency with sqrt(3), which returns a symbolic ring element - why not the same for eigenvalues? Over a finite field, one should ask if Sage is consistent: sage: FF = FiniteField(Integer(5)) sage: M = Matrix(FF,[1,-1],[1,0]]) sage: M.eigenvalues() I get a 'Not implemented error' -- but this should be implemented and should give something consistent. In a general system, I can't see anything reasonable that one can do in general other than returning the roots of the characteristic polynomial in the base ring. I find the syntax: M.eigenvalues(ring=None, extend=False) perfectly well-defined. Since extend=True will fail for most rings, since a natural algebraically closed field containing the base ring is unlikely to exist or be implemented in Sage, so I don't think that default can be extend=True. So I must vote -1 to default extend=True. On the other hand, I think M.eigenvalues(CC) gives a simple syntax for linear algebra students, and also lets them do M.eigenvalues(RR) in order to distinguish between real vector spaces, complex ones, or different behaviors of these spaces. I'm not sure all teachers and students would prefer CC over RR -- in my experience in Sydney complex numbers only existed for "advanced" first year students, and for students in the "ordinary" track the real numbers were enough (one could argue that such relative mathematics should not exist but there are still different teaching needs). When studying or teaching real solutions for a problem, one needs a simple syntax for equation solving over RR or CC. I would hope that Sage could keep a generic syntax for linear algebra which doesn't require knowing that one behavior will result for matrices over the integers, rationals, or reals, a different one for linear operators, and a completely different set of behaviors when the base ring is a finite field, number field, power series ring, or function field. I would argue that one should always ask first whether a given function makes sense in a general context, and not implement something which conflicts with the function in the general context. --David On Jun 14, 7:14 pm, Rob Beezer <goo...@beezer.cotse.net> wrote: > Miguel, > > Having read #8974 carefully, I could see the default for endomorphisms > going either way. My main concern is that matrices follow practice > and default to providing eigenvalues (and eigenvectors) outside the > base field. Endomorphisms could default to behave identically to > matrices, or they could respect the notion of being functions and only > return "proper" elements of the domain and range, as you have argued > in regards to your own teaching. But with options available for > matrices and endomorphisms, individual opinions/tastes can be > accomodated. > > Thanks for bringing the discussion to sage-devel. > > Rob -- To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
