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 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)
