On Dec 30, 1:50 pm, Rob Beezer <goo...@beezer.cotse.net> wrote: > I'd like to improve the current state of the linear algebra code over > CDF (and by extension, over RDF). The purpose would be to make Sage > more usable for teaching various topics involving matrices with > complex entries and orthogonal vectors (thus introducing square > roots). You can actually do quite a bit of this over the rationals, > where (to my surprise) the square roots get approximated by rationals, > and the accumulated errors are quite small. But it makes more sense > to me to actually work over CDF and liberally employ the .zero_at() > method when showing students how to interpret the results. > > In any event, the goal would be to work up to things like QR > decompositions, unitary matrices, orthogonal projections, etc. A lot > of this is present, but not always documented well. I have not done a > comprehensive survey yet, but for example, everything about QR > decomposition talks about (and doctests) real matrices, even though > the calls to NumPy are doing the right thing with complex entries and > returning Q as unitary (not just orthogonal). But there needs to be > some prerequiste gaps filled for students to study these objects, the > algorithms, etc (such as there is not yet a function to take the > conjugate of a vector). > > The inner product especially has me stumped. The following does not > seem to be what we want at all: > > sage: v = vector(CDF, [I, I]) > sage: v.inner_product(v) > -2.0 > > I'd like to have the "usual" version with a conjugate of one of the > vectors prior to the dot product (the sesquilinear form). I can't see > that setting a custom inner product matrix can make this happen, but > maybe I'm missing something. Should the inner product be over-ridden > for vector spaces over the complex numbers? Or, I hate to ask, should > there be something new, like .complex_inner_product(), > or .sesquilinear_inner_product()? (Just kidding about that last > one.) Or maybe this inner product is lurking somewhere and I'm not > noticing it.
I'd call this one a Hermitian inner product. Well, I do not see anything wrong with sesquilinear either (especially if this is done consistently, i.e. for any given field automorphism). Dima > > I know Jason Grout has done a lot of work to integrate this with > NumPy. I'm trolling for anything else folks can send me that might > help: advice, informative Trac tickets, potential pitfalls, secret > desires. Thanks! > > 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
