"""
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_minimal_polynomial(x):
    """
    Return the minimal polynomial of ``x`` in its parent EJA
    """
    return x._x.matrix().minimal_polynomial()


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)


def eja_ln(dimension, field=QQ):
    """
    Return the Jordan algebra corresponding to the Lorenzt "ice cream"
    cone of the given ``dimension``.

    EXAMPLES:

    This multiplication table can be verified by hand::

        sage: J = eja_ln(4)
        sage: e0,e1,e2,e3 = J.gens()
        sage: e0*e0
        e0
        sage: e0*e1
        e1
        sage: e0*e2
        e2
        sage: e0*e3
        e3
        sage: e1*e2
        0
        sage: e1*e3
        0
        sage: e2*e3
        0

    In one dimension, this is the reals under multiplication::

      sage: J1 = eja_ln(1)
      sage: J2 = eja_rn(1)
      sage: J1 == J2
      True

    """
    Qs = []
    id_matrix = identity_matrix(field,dimension)
    for i in xrange(dimension):
        ei = id_matrix.column(i)
        Qi = zero_matrix(field,dimension)
        Qi.set_row(0, ei)
        Qi.set_column(0, ei)
        Qi += diagonal_matrix(dimension, [ei[0]]*dimension)
        # The addition of the diagonal matrix adds an extra ei[0] in the
        # upper-left corner of the matrix.
        Qi[0,0] = Qi[0,0] * ~field(2)
        Qs.append(Qi)

    A = FiniteDimensionalAlgebra(QQ,Qs,assume_associative=True)
    return JordanAlgebra(A)