quadratic form

QuadraticForm

asked 2017-06-30 01:17:32 +0100

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Write a function in Sage that accepts as input a symmetrical bilinear (not trivial) form B [caracterized by the associated matrix respect to the canonical base in R^n] and gives in output a vector subspace W ⊆ R^n such that: - Dim W is maximal - the restriction B|wxw has maximum rank

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This looks like homework.

If you want some help, you should ask more precise questions related to your research in solving those exercises, especially where you are locked.

tmonteil ( 2017-06-30 10:35:27 +0100 )edit

i'm a sage beginner, i know what to do but i don't know how to implement it.. the idea is that i want to give in input a symmetrical nontrivial matrix A associated to the bilinear form B respect to the canonical base in R^n..then i want to obtain a subspace W⊆ R^n of maximal dimension such that the restriction B|wxw has max rank, in other words the matrix associated to this restriction has maximum rank and this condition occurs if kerW=[0].So i have to : 1. Give in input a matrix, 2. Declare the canonical basis of R^n and R 3. If B is the bilinear form and ei elements of the canonical base of R^n, i say that B(ei,ej)= A(i,j) with A(i,j) the i,j element of A 4. Obtain a subspace W ...(more)

ciao ( 2017-06-30 13:22:34 +0100 )edit

Please give us a symmetric matrix A to work with. Relevant code to initialize it can be found after typing ?matrix . Sample code for a matrix of my choice (maximal rank):

sage: A = matrix( QQ, 3, 3, [8,1,1, 1,8,1, 1,1,8 ] )
sage: A
[8 1 1]
[1 8 1]
[1 1 8]
sage: A.kernel()
Vector space of degree 3 and dimension 0 over Rational Field
Basis matrix:
[]

dan_fulea ( 2017-06-30 17:30:04 +0100 )edit

def function(Q):

#(Sylvester theorem to obtain a diagonalizing base)

n=Q.dim()

D=Q.rational_diagonal_form(return_matrix=True)[0]

M=Q.rational_diagonal_form(return_matrix=True)[1]

base=[M.column(i) for i in range(0,n)]

BASEpos=[]

BASEneg=[]

BASEnull=[]

for i in range(0,n):

    if D[i,i]>0: BASEpos=BASEpos+[1/sqrt(D[i,i])*base[i]]

    if D[i,i]<0: BASEneg=BASEneg+[1/sqrt(-D[i,i])*base[i]]

    if D[i,i]==0: BASEnull=BASEnull+[base[i]]

Base1= BASEpos+BASEneg      #(base without radical's vector to generate W)       

Base2= BASEpos+BASEneg+BASEnull

W=span(SR,B1)

return W

When i give as input a quadratic form Q such as Q=QuadraticForm(SR,4,[0,0,0,1,0,-1,0,0,0,0]) W=function(Q) the program give me a lot of errors

ciao ( 2017-07-01 03:03:26 +0100 )edit

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answered 2017-07-02 04:33:15 +0100

dan_fulea
5537 ●5 ●43 ●95

The following works (over QQ instead of SR):

def f( Q ):
    D, M = Q.rational_diagonal_form( return_matrix=True )
    base = M.columns()

    BASEpos  = []
    BASEneg  = []
    BASEnull = []

    for j in range( 0, Q.dim() ):
        if   D[j,j] > 0 :    BASEpos  += [ 1/sqrt(  D[j,j] ) * base[j] ]
        elif D[j,j] < 0 :    BASEneg  += [ 1/sqrt( -D[j,j] ) * base[j] ]
        elif D[j,j] ==0 :    BASEnull += [                     base[j] ]

    return span( QQ, BASEpos + BASEneg )

Q = QuadraticForm( QQ, 4, [ 0,0,0,1,  0,-1,0,  0,0,  0 ] )
f( Q )

Results:

Vector space of degree 4 and dimension 4 over Rational Field
Basis matrix:
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]

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