def isom(A,B):
# First check that A is a symmetric matrix.
if not matrix(A).is_symmetric():
return False
# Then check A against the viable database candidates.
else:
n=len(A[0])
m=len(B[0])
Avec=[]
Bvec=[]
for i in range(n):
for j in range(i,n):
if i==j:
Avec+=[A[i][j]]
else:
Avec+=[2*A[i][j]]
for i in range(m):
for j in range(i,m):
if i==j:
Bvec+=[B[i][j]]
else:
Bvec+=[2*B[i][j]]
Aquad=QuadraticForm(ZZ,len(A[0]),Avec)
# check positive definite
if Aquad.is_positive_definite():
Bquad=QuadraticForm(ZZ,len(B[0]),Bvec)
return Aquad.is_globally_equivalent_to(Bquad)
else:
return False
def isom(A, B):
# First check that A is a symmetric matrix.
if not matrix(A).is_symmetric():
return False
# Then check A against the viable database candidates.
else:
n = len(A[0])
m = len(B[0])
Avec = []
Bvec = []
for i in range(n):
for j in range(i, n):
if i == j:
Avec += [A[i][j]]
else:
Avec += [2 * A[i][j]]
for i in range(m):
for j in range(i, m):
if i == j:
Bvec += [B[i][j]]
else:
Bvec += [2 * B[i][j]]
Aquad = QuadraticForm(ZZ, len(A[0]), Avec)
# check positive definite
if Aquad.is_positive_definite():
Bquad = QuadraticForm(ZZ, len(B[0]), Bvec)
return Aquad.is_globally_equivalent_to(Bquad)
else:
return False
def assert_jordan_blcs(self, p, mat):
p = ZZ(p)
if p == 2:
blcs = jordan_blocks_2(mat)
else:
blcs = jordan_blocks_odd(mat, p)
q = QuadraticForm(ZZ, ZZ(2) * mat)
q1 = _blocks_to_quad_form(blcs.blocks, p)
if p == 2:
self.assertTrue((q.det() / q1.det()) % 8 == 1)
else:
self.assertTrue(kronecker_symbol(q.det() / q1.det(), p) == 1)
self.assertEqual(
q.hasse_invariant__OMeara(p), q1.hasse_invariant__OMeara(p))
def assert_jordan_blocks_method(self, p, mat):
p = ZZ(p)
if p == 2:
blcs = jordan_blocks_2(mat)
else:
blcs = jordan_blocks_odd(mat, p)
q = QuadraticForm(ZZ, 2 * mat)
if p == 2:
should1 = (q.Gram_det() / blcs.Gram_det()) % 8
else:
should1 = kronecker_symbol((q.Gram_det() / blcs.Gram_det()), p)
self.assertEqual(should1, 1)
self.assertEqual(
q.hasse_invariant__OMeara(p), blcs.hasse_invariant__OMeara())
self.assertEqual(q.dim(), blcs.dim())
self.assertEqual(q.content().valuation(p), blcs.content_order())
def QuadraticForm_from_quadric(Q):
"""
On input a homogeneous polynomial of degree 2 return the Quadratic Form corresponding to it.
"""
R = Q.parent()
assert all([sum(e)==2 for e in Q.exponents()])
M = copy(zero_matrix(R.ngens(),R.ngens()))
for i in range(R.ngens()):
for j in range(R.ngens()):
if i==j:
M[i,j]=2*Q.coefficient(R.gen(i)*R.gen(j))
else:
M[i,j]=Q.coefficient(R.gen(i)*R.gen(j))
return QuadraticForm(M)