On Tue, Nov 17, 2009 at 09:23:40PM +0100, Francois Maltey wrote:
> This calculus is maple-right but is user-discourteous.
>
> M = matrix ([[a,b],[c,d]])
> 0*M = 0 with maple, and all other systems answer matrix([[0,0],[0,0]])
>
> Even if you explain that for maple syntax, it's normal to get 0*A = 0
> because the right way is "evalm(0*M)", I repeat : "everyone thinks that
> 0*M = matrix 0, not number 0". This 0*XYZ=0 rule isn't fine.
> Only(?) an object language (as python) can treat this "multi-sens" of zero.
I'm not sure I understand this paragraph. Mathematically, 0*M is
always the zero matrix and never the number 0. So it seems to me that
maple screws up if it returns the number 0. In Sage:
sage: var("a b c d")
(a, b, c, d)
sage: M = matrix([[a,b],[c,d]])
sage: 0*M
[0 0]
[0 0]
> I have an other question : how can you easily verify this theorem in
> sage ?
>
> M = matrix([[a,b],[c,d]]) # or an nxn matrix with any parameters...
> P = det (M - x*matrix ONE) # Call Cayley-Hamilton therem in France
> eval (P with x=M) answers matrix([[0,0],[0,0]]).
Here is one way, I don't know if there's an easier one:
sage: var("a b c d")
(a, b, c, d)
sage: M = matrix([[a,b],[c,d]])
sage: I = M.parent()(1)
sage: P = lambda x: det(M - x*I)
sage: P(M)
0
>
> The two first methods I don't find in sage was :
>
> 1/ Sum as sum(q^k, k=0..N)=(1-q^(n+1))/(1-q) and sum(x^k/k!, k=0..+oo)=e^x
This is in the works presently, wrapping Maxima's symbolic summation.
It's ticket 3587, and might make it into sage-4.3:
http://trac.sagemath.org/sage_trac/ticket/3587
> 2/ kernel over matrix is right, but I don't find the maple intbasis and
> sumbasis which build a basis for F cap G and F+G where the F and G
> subspaces are described by a list of vectors.
>
You have to work with the objects in Sage (which in my opinion is good
because students are forced to think about subspaces rather than lists
of vectors):
sage: V = VectorSpace(QQ, 5)
sage: F = V.subspace([(0,1,2,3,4), (1,2,3,4,6), (0,1,0,1,0)])
sage: G = V.subspace([(1, 1, 0, 0, 0), (0, 1, 1, 2, 2), (5, 4, 3, 2, 1)])
sage: I = F.intersection(G)
sage: I.basis()
[
(1, 0, -1, -2, -2),
(0, 1, 1, 2, 2)
]
sage: S = F + G
sage: S.basis()
[
(1, 0, 0, -1, 0),
(0, 1, 0, 1, 0),
(0, 0, 1, 1, 0),
(0, 0, 0, 0, 1)
]
Best,
Alex
--
Alex Ghitza -- Lecturer in Mathematics -- The University of Melbourne
-- Australia -- http://www.ms.unimelb.edu.au/~aghitza/
--
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