from sage.matrix.constructor import matrix

from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra


class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanSubalgebra):
    def __init__(self, elt, orthonormalize_basis):
        self._superalgebra = elt.parent()
        category = self._superalgebra.category().Associative()
        V = self._superalgebra.vector_space()
        field = self._superalgebra.base_ring()

        # This list is guaranteed to contain all independent powers,
        # because it's the maximal set of powers that could possibly
        # be independent (by a dimension argument).
        powers = [ elt**k for k in range(V.dimension()) ]
        power_vectors = [ p.to_vector() for p in powers ]
        P = matrix(field, power_vectors)

        if orthonormalize_basis == False:
            # In this case, we just need to figure out which elements
            # of the "powers" list are redundant... First compute the
            # vector subspace spanned by the powers of the given
            # element.

            # Figure out which powers form a linearly-independent set.
            ind_rows = P.pivot_rows()

            # Pick those out of the list of all powers.
            superalgebra_basis = tuple(map(powers.__getitem__, ind_rows))

            # If our superalgebra is a subalgebra of something else, then
            # these vectors won't have the right coordinates for
            # V.span_of_basis() unless we use V.from_vector() on them.
            basis_vectors = map(power_vectors.__getitem__, ind_rows)
        else:
            # If we're going to orthonormalize the basis anyway, we
            # might as well just do Gram-Schmidt on the whole list of
            # powers. The redundant ones will get zero'd out. If this
            # looks like a roundabout way to orthonormalize, it is.
            # But converting everything from algebra elements to vectors
            # to matrices and then back again turns out to be about
            # as fast as reimplementing our own Gram-Schmidt that
            # works in an EJA.
            G,_ = P.gram_schmidt(orthonormal=True)
            basis_vectors = [ g for g in G.rows() if not g.is_zero() ]
            superalgebra_basis = [ self._superalgebra.from_vector(b)
                                   for b in basis_vectors ]

        W = V.span_of_basis( V.from_vector(v) for v in basis_vectors )

        fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self)
        fdeja.__init__(self._superalgebra,
                       superalgebra_basis,
                       category=category,
                       check_axioms=False)

        # The rank is the highest possible degree of a minimal
        # polynomial, and is bounded above by the dimension. We know
        # in this case that there's an element whose minimal
        # polynomial has the same degree as the space's dimension
        # (remember how we constructed the space?), so that must be
        # its rank too.
        self.rank.set_cache(W.dimension())


    def _a_regular_element(self):
        """
        Override the superalgebra method to return the one
        regular element that is sure to exist in this
        subalgebra, namely the element that generated it.

        SETUP::

            sage: from mjo.eja.eja_algebra import random_eja

        TESTS::

            sage: set_random_seed()
            sage: J = random_eja().random_element().subalgebra_generated_by()
            sage: J._a_regular_element().is_regular()
            True

        """
        if self.dimension() == 0:
            return self.zero()
        else:
            return self.monomial(1)


    def one(self):
        """
        Return the multiplicative identity element of this algebra.

        The superclass method computes the identity element, which is
        beyond overkill in this case: the superalgebra identity
        restricted to this algebra is its identity. Note that we can't
        count on the first basis element being the identity -- it migth
        have been scaled if we orthonormalized the basis.

        SETUP::

            sage: from mjo.eja.eja_algebra import (HadamardEJA,
            ....:                                  random_eja)

        EXAMPLES::

            sage: J = HadamardEJA(5)
            sage: J.one()
            e0 + e1 + e2 + e3 + e4
            sage: x = sum(J.gens())
            sage: A = x.subalgebra_generated_by()
            sage: A.one()
            f0
            sage: A.one().superalgebra_element()
            e0 + e1 + e2 + e3 + e4

        TESTS:

        The identity element acts like the identity over the rationals::

            sage: set_random_seed()
            sage: x = random_eja(field=QQ).random_element()
            sage: A = x.subalgebra_generated_by()
            sage: x = A.random_element()
            sage: A.one()*x == x and x*A.one() == x
            True

        The identity element acts like the identity over the algebraic
        reals with an orthonormal basis::

            sage: set_random_seed()
            sage: x = random_eja().random_element()
            sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
            sage: x = A.random_element()
            sage: A.one()*x == x and x*A.one() == x
            True

        The matrix of the unit element's operator is the identity over
        the rationals::

            sage: set_random_seed()
            sage: x = random_eja(field=QQ).random_element()
            sage: A = x.subalgebra_generated_by()
            sage: actual = A.one().operator().matrix()
            sage: expected = matrix.identity(A.base_ring(), A.dimension())
            sage: actual == expected
            True

        The matrix of the unit element's operator is the identity over
        the algebraic reals with an orthonormal basis::

            sage: set_random_seed()
            sage: x = random_eja().random_element()
            sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
            sage: actual = A.one().operator().matrix()
            sage: expected = matrix.identity(A.base_ring(), A.dimension())
            sage: actual == expected
            True

        """
        if self.dimension() == 0:
            return self.zero()
        else:
            sa_one = self.superalgebra().one().to_vector()
            # The extra hackery is because foo.to_vector() might not
            # live in foo.parent().vector_space()!
            coords = sum( a*b for (a,b)
                          in zip(sa_one,
                                 self.superalgebra().vector_space().basis()) )
            return self.from_vector(self.vector_space().coordinate_vector(coords))