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"""
return cls(n, field, **kwargs)
-class HadamardEJA(RationalBasisEuclideanJordanAlgebra,
+class HadamardEJA(RationalBasisEuclideanJordanAlgebraNg,
ConcreteEuclideanJordanAlgebra):
"""
Return the Euclidean Jordan Algebra corresponding to the set
"""
def __init__(self, n, field=AA, **kwargs):
V = VectorSpace(field, n)
- mult_table = [ [ V.gen(i)*(i == j) for j in range(n) ]
- for i in range(n) ]
+ basis = V.basis()
- # Inner products are real numbers and not algebra
- # elements, so once we turn the algebra element
- # into a vector in inner_product(), we never go
- # back. As a result -- contrary to what we do with
- # self._multiplication_table -- we store the inner
- # product table as a plain old matrix and not as
- # an algebra operator.
- ip_table = matrix.identity(field,n)
- self._inner_product_matrix = ip_table
+ def jordan_product(x,y):
+ return V([ xi*yi for (xi,yi) in zip(x,y) ])
+ def inner_product(x,y):
+ return x.inner_product(y)
super(HadamardEJA, self).__init__(field,
- mult_table,
- check_axioms=False,
+ basis,
+ jordan_product,
+ inner_product,
**kwargs)
self.rank.set_cache(n)