"""
Euclidean Jordan Algebras. These are formally-real Jordan Algebras;
specifically those where u^2 + v^2 = 0 implies that u = v = 0. They
are used in optimization, and have some additional nice methods beyond
what can be supported in a general Jordan Algebra.
"""

from sage.all import *

def eja_rn(dimension, field=QQ):
    """
    Return the Euclidean Jordan Algebra corresponding to the set
    `R^n` under the Hadamard product.

    EXAMPLES:

    This multiplication table can be verified by hand::

        sage: J = eja_rn(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

    """
    # The FiniteDimensionalAlgebra constructor takes a list of
    # matrices, the ith representing right multiplication by the ith
    # basis element in the vector space. So if e_1 = (1,0,0), then
    # right (Hadamard) multiplication of x by e_1 picks out the first
    # component of x; and likewise for the ith basis element e_i.
    Qs = [ matrix(field, dimension, dimension, lambda k,j: 1*(k == j == i))
           for i in xrange(dimension) ]
    A = FiniteDimensionalAlgebra(QQ,Qs,assume_associative=True)
    return JordanAlgebra(A)