I faced that kind of decissions when i implemented the eigen-stuff for endomorphisms (see ticket 8974). My opinion was to stick to the base field, and only look for extensions when directly requested. David Loefler argued that, for consistency reasons, it would be preferable to continue with the same philosophy. The compromise solution was to introduce the option extend=False to say sage to work only on the base field.
I don't remember now if it solved all the issues you present here, but i do remember that it helped to solve some cases over finite fields where the option extend=True (default) fails. On 21 jun, 22:59, Rob Beezer <[email protected]> wrote: > I think I have one more big push left in me as I try to tidy up linear > algebra in Sage to make it even more useful for students studying the > subject for the first time. Eigen-stuff is on my radar. Some > behaviors that I find problematic, most vexing first. > > 1. Eigenspaces > > sage: A = matrix(QQ, [[0,-1,0,0],[1,0,0,0],[0,0,0,-1],[0,0,1,0]]) > sage: A.eigenspaces_right() > [ > (a0, Vector space of degree 4 and dimension 2 over Number Field in a0 > with defining polynomial x^2 + 1 > User basis matrix: > [ 1 -a0 0 0] > [ 0 0 1 -a0]) > ] > > It is real impressive that we can do computations in QQbar and get > Galois conjugates and all, but there is no possible way to explain > this to a student who is fresh out of calculus. I would like to > retain getting eigenspaces as "true" vector spaces, since I think this > is a real strength of Sage. But I'd rather have the textual versions > of the QQbar elements, like the output for eigenvectors: > > sage: A.eigenvectors_right() > [(-1*I, [(1, 1*I, 0, 0), (0, 0, 1, 1*I)], 2), (1*I, [(1, -1*I, 0, 0), > (0, 0, 1, -1*I)], 2)] > > Question: is it important to only get a limited number of non- > isomorphic eigenspaces back? Most of that information seems available > to me from the factored characteristic polynomial, easily obtainable > with the fcp() method. > > sage: A.fcp() > (x^2 + 1)^2 > > Thoughts or reactions to returning an eigenspace per eigenvalue, with > QQbar elements as entries of basis vectors, etc? > > 2. Eigenspaces for matrices over RDF/CDF > > These eigenspaces come back as dimension 1, always. The method was > adjusted to match the format of returned values for exact matrices > (and I was a party to that). With more experience with the floating > point matrix class, I've come to the conclusion this is a really bad > idea, bordering on criminal. So, I'd propose raising a NotImplemented > exception for an eigenspace request, since all the information > presently returned for eigenspaces can be found with the existing > eigenmatrix commands. Reactions? > > 3. Algebraically Closed Fields > > We return eigenvalues outside the base field, by default, only for the > rationals. Not sure I'm suggesting anything here, it just seems odd > that we do this in the one case, just because we can? > > 4. Sorting Eigenvalues > > Easy and natural to sort eigenvalues over RDF and QQ. Messier in > other cases. Docstrings say things like: > > The eigenspaces are returned sorted by the corresponding > characteristic polynomials, where polynomials are sorted in > dictionary order starting with constant terms. > > But I do not quite see that implemented unless it comes from deep in > the factorization of the characteristic polynomial somehow. In any > event, trying to get a consistent approach to the sort order looks > like a big job with the existing code. Maybe somebody has some > thoughts about this. > > Thanks in advance for thoughts, affirmative or dissident. > > Rob -- To post to this group, send an email to [email protected] To unsubscribe from this group, send an email to [email protected] For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
