for j in range(n) ]
# take advantage of symmetry
for i in range(n):
- for j in range(n):
+ for j in range(i+1):
elt = self.from_vector(mult_table[i][j])
self._multiplication_table[i][j] = elt
self._multiplication_table[j][i] = elt
Element = FiniteDimensionalEuclideanJordanAlgebraElement
+class RationalBasisEuclideanJordanAlgebraNg(FiniteDimensionalEuclideanJordanAlgebra):
+ def __init__(self,
+ field,
+ basis,
+ jordan_product,
+ inner_product,
+ orthonormalize=True,
+ prefix='e',
+ category=None,
+ check_field=True,
+ check_axioms=True):
+
+ n = len(basis)
+ vector_basis = basis
+
+ from sage.structure.element import is_Matrix
+ basis_is_matrices = False
+
+ degree = 0
+ if n > 0:
+ if is_Matrix(basis[0]):
+ basis_is_matrices = True
+ vector_basis = tuple( map(_mat2vec,basis) )
+ degree = basis[0].nrows()**2
+ else:
+ degree = basis[0].degree()
+
+ V = VectorSpace(field, degree)
+
+ # Compute this from "Q" (obtained from Gram-Schmidt) below as
+ # R = Q.solve_right(A), where the rows of "Q" are the
+ # orthonormalized vector_basis and and the rows of "A" are the
+ # original vector_basis.
+ self._deorthonormalization_matrix = None
+
+ if orthonormalize:
+ from mjo.eja.eja_utils import gram_schmidt
+ vector_basis = gram_schmidt(vector_basis, inner_product)
+ W = V.span_of_basis( vector_basis )
+ if basis_is_matrices:
+ from mjo.eja.eja_utils import _vec2mat
+ basis = tuple( map(_vec2mat,vector_basis) )
+
+ W = V.span_of_basis( vector_basis )
+
+ mult_table = [ [0 for i in range(n)] for j in range(n) ]
+ ip_table = [ [0 for i in range(n)] for j in range(n) ]
+
+ for i in range(n):
+ for j in range(i+1):
+ # do another mat2vec because the multiplication
+ # table is in terms of vectors
+ elt = _mat2vec(jordan_product(basis[i],basis[j]))
+ elt = W.coordinate_vector(elt)
+ mult_table[i][j] = elt
+ mult_table[j][i] = elt
+ ip = inner_product(basis[i],basis[j])
+ ip_table[i][j] = ip
+ ip_table[j][i] = ip
+
+ self._inner_product_matrix = matrix(field,ip_table)
+
+ if basis_is_matrices:
+ for m in basis:
+ m.set_immutable()
+ else:
+ basis = tuple( x.column() for x in basis )
+
+ super().__init__(field,
+ mult_table,
+ prefix,
+ category,
+ basis, # matrix basis
+ check_field,
+ check_axioms)
class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra):
r"""