Define map with vector argument(s)

asked 2017-06-22 00:14:46 +0100

515 ●9 ●20 ●37

updated 2017-06-22 01:25:27 +0100

(Is it possible to use LaTeX code here in order to get a nice output?)

Let's say I want to put the map phi: Q^3 x Q^3 -> Q defined by (x,y) -> x1y1 - x2y2 + x3*y3 into Sage. How do I do this? I want to have later for example:

sage: x=vector((1,2,3)); y=vector((2,3,1))
sage: phi(x,y)

Then the output should be the value phi(x,y).

Is it only possible via the def command (withx[1] etc.)? Or how can you specify the domain of a map? How would you do this?

edit retag flag offensive close merge delete

add a comment

2 Answers

Sort by » oldest newest most voted

answered 2017-06-22 11:05:57 +0100

eric_g
8424 ●10 ●62 ●166 https://luth.obspm.fr/...

updated 2017-06-22 11:53:14 +0100

Assuming phi is bilinear, one possibility is

sage: Q3 = FiniteRankFreeModule(QQ, 3, 'Q^3', start_index=1)
sage: e = Q3.basis('e') # Q^3 canonical basis
sage: phi = Q3.tensor((0,2))
sage: phi[1,1], phi[2,2], phi[3,3] = 1, -1, 1
sage: phi
Type-(0,2) tensor on the 3-dimensional vector space Q^3 over the Rational Field
sage: phi.display()
e^1*e^1 - e^2*e^2 + e^3*e^3
sage: x = Q3((1,2,3)); x
Element of the 3-dimensional vector space Q^3 over the Rational Field
sage: y = Q3((2,3,1))
sage: phi(x,y)
-1
sage: phi(x,y) == x[1]*y[1] - x[2]*y[2] + x[3]*y[3]
True

edit flag offensive delete link

add a comment

answered 2017-06-22 11:36:17 +0100

dan_fulea
5537 ●5 ●43 ●95

updated 2017-06-22 11:49:54 +0100

The inner_product is the implemented method in the case of the phi associated to the $(1,1,1)$ diagonal matrix, for instance:

sage: x=vector((1,2,3)); y=vector((2,3,11))
sage: x.inner_product(y)
41

(That 11 instead of the one in the postshould make the result bigger and avoid "coincidences".)

An explicit check:

sage: sum( [x[k]*y[k] for k in range(len(x)) ] )
41

(The range(len(x)) could have been simply (0,1,2) in this simple case.)

In the more general case the matrix giving the bilinear form can be inserted, we have in the same spirit:

sage: PHI = diagonal_matrix( 3, [1,-1,1] )
sage: x * PHI * y.column()
(29)
sage: sum( [ (-1)^k*x[k]*y[k] for k in range(len(x)) ] )
29
sage: ( x*PHI ).inner_product(y) 
29

Above, y.column() is the column vector related to y.

sage: y.column()
[ 2]
[ 3]
[11]

edit flag offensive delete link

add a comment

Your Answer

Please start posting anonymously - your entry will be published after you log in or create a new account.

Add Answer

Define map with vector argument(s)

2 Answers

Your Answer

Question Tools

Stats

Define map with vector argument(s) edit

2 Answers

Your Answer

Question Tools

Stats

Define map with vector argument(s)