-class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra):
- """
- Return the Euclidean Jordan Algebra corresponding to the set
- `R^n` under the Hadamard product.
-
- Note: this is nothing more than the Cartesian product of ``n``
- copies of the spin algebra. Once Cartesian product algebras
- are implemented, this can go.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import HadamardEJA
-
- EXAMPLES:
-
- This multiplication table can be verified by hand::
-
- sage: J = HadamardEJA(3)
- sage: e0,e1,e2 = J.gens()
- sage: e0*e0
- e0
- sage: e0*e1
- 0
- sage: e0*e2
- 0
- sage: e1*e1
- e1
- sage: e1*e2
- 0
- sage: e2*e2
- e2
-
- TESTS:
-
- We can change the generator prefix::
-
- sage: HadamardEJA(3, prefix='r').gens()
- (r0, r1, r2)
-
- """
- 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) ]
-
- fdeja = super(HadamardEJA, self)
- fdeja.__init__(field, mult_table, **kwargs)
- self.rank.set_cache(n)
-
- def inner_product(self, x, y):
- """
- Faster to reimplement than to use natural representations.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import HadamardEJA
-
- TESTS:
-
- Ensure that this is the usual inner product for the algebras
- over `R^n`::
-
- sage: set_random_seed()
- sage: J = HadamardEJA.random_instance()
- sage: x,y = J.random_elements(2)
- sage: X = x.natural_representation()
- sage: Y = y.natural_representation()
- sage: x.inner_product(y) == J.natural_inner_product(X,Y)
- True
-
- """
- return x.to_vector().inner_product(y.to_vector())
-