eja: add an implementation warning for characteristic_polynomial().

[sage.d.git] / mjo / eja / euclidean_jordan_algebra.py

"""
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.categories.magmatic_algebras import MagmaticAlgebras
from sage.structure.element import is_Matrix
from sage.structure.category_object import normalize_names

from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra import FiniteDimensionalAlgebra
from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_element import FiniteDimensionalAlgebraElement

class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
    @staticmethod
    def __classcall_private__(cls,
                              field,
                              mult_table,
                              names='e',
                              assume_associative=False,
                              category=None,
                              rank=None,
                              natural_basis=None):
        n = len(mult_table)
        mult_table = [b.base_extend(field) for b in mult_table]
        for b in mult_table:
            b.set_immutable()
            if not (is_Matrix(b) and b.dimensions() == (n, n)):
                raise ValueError("input is not a multiplication table")
        mult_table = tuple(mult_table)

        cat = MagmaticAlgebras(field).FiniteDimensional().WithBasis()
        cat.or_subcategory(category)
        if assume_associative:
            cat = cat.Associative()

        names = normalize_names(n, names)

        fda = super(FiniteDimensionalEuclideanJordanAlgebra, cls)
        return fda.__classcall__(cls,
                                 field,
                                 mult_table,
                                 assume_associative=assume_associative,
                                 names=names,
                                 category=cat,
                                 rank=rank,
                                 natural_basis=natural_basis)


    def __init__(self,
                 field,
                 mult_table,
                 names='e',
                 assume_associative=False,
                 category=None,
                 rank=None,
                 natural_basis=None):
        """
        EXAMPLES:

        By definition, Jordan multiplication commutes::

            sage: set_random_seed()
            sage: J = random_eja()
            sage: x = J.random_element()
            sage: y = J.random_element()
            sage: x*y == y*x
            True

        """
        self._rank = rank
        self._natural_basis = natural_basis
        self._multiplication_table = mult_table
        fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
        fda.__init__(field,
                     mult_table,
                     names=names,
                     category=category)


    def _repr_(self):
        """
        Return a string representation of ``self``.
        """
        fmt = "Euclidean Jordan algebra of degree {} over {}"
        return fmt.format(self.degree(), self.base_ring())



    @cached_method
    def _charpoly_coeff(self, i):
        """
        Return the coefficient polynomial "a_{i}" of this algebra's
        general characteristic polynomial.

        Having this be a separate cached method lets us compute and
        store the trace/determinant (a_{r-1} and a_{0} respectively)
        separate from the entire characteristic polynomial.
        """
        (A_of_x, x) = self._charpoly_matrix()
        R = A_of_x.base_ring()
        A_cols = A_of_x.columns()
        A_cols[i] = (x**self.rank()).vector()
        numerator = column_matrix(A_of_x.base_ring(), A_cols).det()
        denominator = A_of_x.det()

        # We're relying on the theory here to ensure that each a_i is
        # indeed back in R, and the added negative signs are to make
        # the whole charpoly expression sum to zero.
        return R(-numerator/denominator)


    @cached_method
    def _charpoly_matrix(self):
        """
        Compute the matrix whose entries A_ij are polynomials in
        X1,...,XN. This same matrix is used in more than one method and
        it's not so fast to construct.
        """
        r = self.rank()
        n = self.dimension()

        # Construct a new algebra over a multivariate polynomial ring...
        names = ['X' + str(i) for i in range(1,n+1)]
        R = PolynomialRing(self.base_ring(), names)
        J = FiniteDimensionalEuclideanJordanAlgebra(R,
                                                    self._multiplication_table,
                                                    rank=r)

        idmat = identity_matrix(J.base_ring(), n)

        x = J(vector(R, R.gens()))
        l1 = [column_matrix((x**k).vector()) for k in range(r)]
        l2 = [idmat.column(k-1).column() for k in range(r+1, n+1)]
        A_of_x = block_matrix(R, 1, n, (l1 + l2))
        return (A_of_x, x)


    @cached_method
    def characteristic_polynomial(self):
        """

        .. WARNING::

            This implementation doesn't guarantee that the polynomial
            denominator in the coefficients is not identically zero, so
            theoretically it could crash. The way that this is handled
            in e.g. Faraut and Koranyi is to use a basis that guarantees
            the denominator is non-zero. But, doing so requires knowledge
            of at least one regular element, and we don't even know how
            to do that. The trade-off is that, if we use the standard basis,
            the resulting polynomial will accept the "usual" coordinates. In
            other words, we don't have to do a change of basis before e.g.
            computing the trace or determinant.

        EXAMPLES:

        The characteristic polynomial in the spin algebra is given in
        Alizadeh, Example 11.11::

            sage: J = JordanSpinEJA(3)
            sage: p = J.characteristic_polynomial(); p
            X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
            sage: xvec = J.one().vector()
            sage: p(*xvec)
            t^2 - 2*t + 1

        """
        r = self.rank()
        n = self.dimension()

        # The list of coefficient polynomials a_1, a_2, ..., a_n.
        a = [ self._charpoly_coeff(i) for i in range(n) ]

        # We go to a bit of trouble here to reorder the
        # indeterminates, so that it's easier to evaluate the
        # characteristic polynomial at x's coordinates and get back
        # something in terms of t, which is what we want.
        R = a[0].parent()
        S = PolynomialRing(self.base_ring(),'t')
        t = S.gen(0)
        S = PolynomialRing(S, R.variable_names())
        t = S(t)

        # Note: all entries past the rth should be zero. The
        # coefficient of the highest power (x^r) is 1, but it doesn't
        # appear in the solution vector which contains coefficients
        # for the other powers (to make them sum to x^r).
        if (r < n):
            a[r] = 1 # corresponds to x^r
        else:
            # When the rank is equal to the dimension, trying to
            # assign a[r] goes out-of-bounds.
            a.append(1) # corresponds to x^r

        return sum( a[k]*(t**k) for k in range(len(a)) )


    def inner_product(self, x, y):
        """
        The inner product associated with this Euclidean Jordan algebra.

        Defaults to the trace inner product, but can be overridden by
        subclasses if they are sure that the necessary properties are
        satisfied.

        EXAMPLES:

        The inner product must satisfy its axiom for this algebra to truly
        be a Euclidean Jordan Algebra::

            sage: set_random_seed()
            sage: J = random_eja()
            sage: x = J.random_element()
            sage: y = J.random_element()
            sage: z = J.random_element()
            sage: (x*y).inner_product(z) == y.inner_product(x*z)
            True

        """
        if (not x in self) or (not y in self):
            raise TypeError("arguments must live in this algebra")
        return x.trace_inner_product(y)


    def natural_basis(self):
        """
        Return a more-natural representation of this algebra's basis.

        Every finite-dimensional Euclidean Jordan Algebra is a direct
        sum of five simple algebras, four of which comprise Hermitian
        matrices. This method returns the original "natural" basis
        for our underlying vector space. (Typically, the natural basis
        is used to construct the multiplication table in the first place.)

        Note that this will always return a matrix. The standard basis
        in `R^n` will be returned as `n`-by-`1` column matrices.

        EXAMPLES::

            sage: J = RealSymmetricEJA(2)
            sage: J.basis()
            Family (e0, e1, e2)
            sage: J.natural_basis()
            (
            [1 0]  [0 1]  [0 0]
            [0 0], [1 0], [0 1]
            )

        ::

            sage: J = JordanSpinEJA(2)
            sage: J.basis()
            Family (e0, e1)
            sage: J.natural_basis()
            (
            [1]  [0]
            [0], [1]
            )

        """
        if self._natural_basis is None:
            return tuple( b.vector().column() for b in self.basis() )
        else:
            return self._natural_basis


    def rank(self):
        """
        Return the rank of this EJA.
        """
        if self._rank is None:
            raise ValueError("no rank specified at genesis")
        else:
            return self._rank


    class Element(FiniteDimensionalAlgebraElement):
        """
        An element of a Euclidean Jordan algebra.
        """

        def __init__(self, A, elt=None):
            """
            EXAMPLES:

            The identity in `S^n` is converted to the identity in the EJA::

                sage: J = RealSymmetricEJA(3)
                sage: I = identity_matrix(QQ,3)
                sage: J(I) == J.one()
                True

            This skew-symmetric matrix can't be represented in the EJA::

                sage: J = RealSymmetricEJA(3)
                sage: A = matrix(QQ,3, lambda i,j: i-j)
                sage: J(A)
                Traceback (most recent call last):
                ...
                ArithmeticError: vector is not in free module

            """
            # Goal: if we're given a matrix, and if it lives in our
            # parent algebra's "natural ambient space," convert it
            # into an algebra element.
            #
            # The catch is, we make a recursive call after converting
            # the given matrix into a vector that lives in the algebra.
            # This we need to try the parent class initializer first,
            # to avoid recursing forever if we're given something that
            # already fits into the algebra, but also happens to live
            # in the parent's "natural ambient space" (this happens with
            # vectors in R^n).
            try:
                FiniteDimensionalAlgebraElement.__init__(self, A, elt)
            except ValueError:
                natural_basis = A.natural_basis()
                if elt in natural_basis[0].matrix_space():
                    # Thanks for nothing! Matrix spaces aren't vector
                    # spaces in Sage, so we have to figure out its
                    # natural-basis coordinates ourselves.
                    V = VectorSpace(elt.base_ring(), elt.nrows()**2)
                    W = V.span( _mat2vec(s) for s in natural_basis )
                    coords =  W.coordinates(_mat2vec(elt))
                    FiniteDimensionalAlgebraElement.__init__(self, A, coords)

        def __pow__(self, n):
            """
            Return ``self`` raised to the power ``n``.

            Jordan algebras are always power-associative; see for
            example Faraut and Koranyi, Proposition II.1.2 (ii).

            .. WARNING:

                We have to override this because our superclass uses row vectors
                instead of column vectors! We, on the other hand, assume column
                vectors everywhere.

            EXAMPLES::

                sage: set_random_seed()
                sage: x = random_eja().random_element()
                sage: x.operator_matrix()*x.vector() == (x^2).vector()
                True

            A few examples of power-associativity::

                sage: set_random_seed()
                sage: x = random_eja().random_element()
                sage: x*(x*x)*(x*x) == x^5
                True
                sage: (x*x)*(x*x*x) == x^5
                True

            We also know that powers operator-commute (Koecher, Chapter
            III, Corollary 1)::

                sage: set_random_seed()
                sage: x = random_eja().random_element()
                sage: m = ZZ.random_element(0,10)
                sage: n = ZZ.random_element(0,10)
                sage: Lxm = (x^m).operator_matrix()
                sage: Lxn = (x^n).operator_matrix()
                sage: Lxm*Lxn == Lxn*Lxm
                True

            """
            A = self.parent()
            if n == 0:
                return A.one()
            elif n == 1:
                return self
            else:
                return A( (self.operator_matrix()**(n-1))*self.vector() )


        def apply_univariate_polynomial(self, p):
            """
            Apply the univariate polynomial ``p`` to this element.

            A priori, SageMath won't allow us to apply a univariate
            polynomial to an element of an EJA, because we don't know
            that EJAs are rings (they are usually not associative). Of
            course, we know that EJAs are power-associative, so the
            operation is ultimately kosher. This function sidesteps
            the CAS to get the answer we want and expect.

            EXAMPLES::

                sage: R = PolynomialRing(QQ, 't')
                sage: t = R.gen(0)
                sage: p = t^4 - t^3 + 5*t - 2
                sage: J = RealCartesianProductEJA(5)
                sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
                True

            TESTS:

            We should always get back an element of the algebra::

                sage: set_random_seed()
                sage: p = PolynomialRing(QQ, 't').random_element()
                sage: J = random_eja()
                sage: x = J.random_element()
                sage: x.apply_univariate_polynomial(p) in J
                True

            """
            if len(p.variables()) > 1:
                raise ValueError("not a univariate polynomial")
            P = self.parent()
            R = P.base_ring()
            # Convert the coeficcients to the parent's base ring,
            # because a priori they might live in an (unnecessarily)
            # larger ring for which P.sum() would fail below.
            cs = [ R(c) for c in p.coefficients(sparse=False) ]
            return P.sum( cs[k]*(self**k) for k in range(len(cs)) )


        def characteristic_polynomial(self):
            """
            Return the characteristic polynomial of this element.

            EXAMPLES:

            The rank of `R^3` is three, and the minimal polynomial of
            the identity element is `(t-1)` from which it follows that
            the characteristic polynomial should be `(t-1)^3`::

                sage: J = RealCartesianProductEJA(3)
                sage: J.one().characteristic_polynomial()
                t^3 - 3*t^2 + 3*t - 1

            Likewise, the characteristic of the zero element in the
            rank-three algebra `R^{n}` should be `t^{3}`::

                sage: J = RealCartesianProductEJA(3)
                sage: J.zero().characteristic_polynomial()
                t^3

            The characteristic polynomial of an element should evaluate
            to zero on that element::

                sage: set_random_seed()
                sage: x = RealCartesianProductEJA(3).random_element()
                sage: p = x.characteristic_polynomial()
                sage: x.apply_univariate_polynomial(p)
                0

            """
            p = self.parent().characteristic_polynomial()
            return p(*self.vector())


        def inner_product(self, other):
            """
            Return the parent algebra's inner product of myself and ``other``.

            EXAMPLES:

            The inner product in the Jordan spin algebra is the usual
            inner product on `R^n` (this example only works because the
            basis for the Jordan algebra is the standard basis in `R^n`)::

                sage: J = JordanSpinEJA(3)
                sage: x = vector(QQ,[1,2,3])
                sage: y = vector(QQ,[4,5,6])
                sage: x.inner_product(y)
                32
                sage: J(x).inner_product(J(y))
                32

            The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
            multiplication is the usual matrix multiplication in `S^n`,
            so the inner product of the identity matrix with itself
            should be the `n`::

                sage: J = RealSymmetricEJA(3)
                sage: J.one().inner_product(J.one())
                3

            Likewise, the inner product on `C^n` is `<X,Y> =
            Re(trace(X*Y))`, where we must necessarily take the real
            part because the product of Hermitian matrices may not be
            Hermitian::

                sage: J = ComplexHermitianEJA(3)
                sage: J.one().inner_product(J.one())
                3

            Ditto for the quaternions::

                sage: J = QuaternionHermitianEJA(3)
                sage: J.one().inner_product(J.one())
                3

            TESTS:

            Ensure that we can always compute an inner product, and that
            it gives us back a real number::

                sage: set_random_seed()
                sage: J = random_eja()
                sage: x = J.random_element()
                sage: y = J.random_element()
                sage: x.inner_product(y) in RR
                True

            """
            P = self.parent()
            if not other in P:
                raise TypeError("'other' must live in the same algebra")

            return P.inner_product(self, other)


        def operator_commutes_with(self, other):
            """
            Return whether or not this element operator-commutes
            with ``other``.

            EXAMPLES:

            The definition of a Jordan algebra says that any element
            operator-commutes with its square::

                sage: set_random_seed()
                sage: x = random_eja().random_element()
                sage: x.operator_commutes_with(x^2)
                True

            TESTS:

            Test Lemma 1 from Chapter III of Koecher::

                sage: set_random_seed()
                sage: J = random_eja()
                sage: u = J.random_element()
                sage: v = J.random_element()
                sage: lhs = u.operator_commutes_with(u*v)
                sage: rhs = v.operator_commutes_with(u^2)
                sage: lhs == rhs
                True

            """
            if not other in self.parent():
                raise TypeError("'other' must live in the same algebra")

            A = self.operator_matrix()
            B = other.operator_matrix()
            return (A*B == B*A)


        def det(self):
            """
            Return my determinant, the product of my eigenvalues.

            EXAMPLES::

                sage: J = JordanSpinEJA(2)
                sage: e0,e1 = J.gens()
                sage: x = sum( J.gens() )
                sage: x.det()
                0

            ::

                sage: J = JordanSpinEJA(3)
                sage: e0,e1,e2 = J.gens()
                sage: x = sum( J.gens() )
                sage: x.det()
                -1

            TESTS:

            An element is invertible if and only if its determinant is
            non-zero::

                sage: set_random_seed()
                sage: x = random_eja().random_element()
                sage: x.is_invertible() == (x.det() != 0)
                True

            """
            P = self.parent()
            r = P.rank()
            p = P._charpoly_coeff(0)
            # The _charpoly_coeff function already adds the factor of
            # -1 to ensure that _charpoly_coeff(0) is really what
            # appears in front of t^{0} in the charpoly. However,
            # we want (-1)^r times THAT for the determinant.
            return ((-1)**r)*p(*self.vector())


        def inverse(self):
            """
            Return the Jordan-multiplicative inverse of this element.

            We can't use the superclass method because it relies on the
            algebra being associative.

            EXAMPLES:

            The inverse in the spin factor algebra is given in Alizadeh's
            Example 11.11::

                sage: set_random_seed()
                sage: n = ZZ.random_element(1,10)
                sage: J = JordanSpinEJA(n)
                sage: x = J.random_element()
                sage: while x.is_zero():
                ....:     x = J.random_element()
                sage: x_vec = x.vector()
                sage: x0 = x_vec[0]
                sage: x_bar = x_vec[1:]
                sage: coeff = 1/(x0^2 - x_bar.inner_product(x_bar))
                sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
                sage: x_inverse = coeff*inv_vec
                sage: x.inverse() == J(x_inverse)
                True

            TESTS:

            The identity element is its own inverse::

                sage: set_random_seed()
                sage: J = random_eja()
                sage: J.one().inverse() == J.one()
                True

            If an element has an inverse, it acts like one::

                sage: set_random_seed()
                sage: J = random_eja()
                sage: x = J.random_element()
                sage: (not x.is_invertible()) or (x.inverse()*x == J.one())
                True

            """
            if self.parent().is_associative():
                elt = FiniteDimensionalAlgebraElement(self.parent(), self)
                return elt.inverse()

            # TODO: we can do better once the call to is_invertible()
            # doesn't crash on irregular elements.
            #if not self.is_invertible():
            #    raise ValueError('element is not invertible')

            # We do this a little different than the usual recursive
            # call to a finite-dimensional algebra element, because we
            # wind up with an inverse that lives in the subalgebra and
            # we need information about the parent to convert it back.
            V = self.span_of_powers()
            assoc_subalg = self.subalgebra_generated_by()
            # Mis-design warning: the basis used for span_of_powers()
            # and subalgebra_generated_by() must be the same, and in
            # the same order!
            elt = assoc_subalg(V.coordinates(self.vector()))

            # This will be in the subalgebra's coordinates...
            fda_elt = FiniteDimensionalAlgebraElement(assoc_subalg, elt)
            subalg_inverse = fda_elt.inverse()

            # So we have to convert back...
            basis = [ self.parent(v) for v in V.basis() ]
            pairs = zip(subalg_inverse.vector(), basis)
            return self.parent().linear_combination(pairs)


        def is_invertible(self):
            """
            Return whether or not this element is invertible.

            We can't use the superclass method because it relies on
            the algebra being associative.

            ALGORITHM:

            The usual way to do this is to check if the determinant is
            zero, but we need the characteristic polynomial for the
            determinant. The minimal polynomial is a lot easier to get,
            so we use Corollary 2 in Chapter V of Koecher to check
            whether or not the paren't algebra's zero element is a root
            of this element's minimal polynomial.

            TESTS:

            The identity element is always invertible::

                sage: set_random_seed()
                sage: J = random_eja()
                sage: J.one().is_invertible()
                True

            The zero element is never invertible::

                sage: set_random_seed()
                sage: J = random_eja()
                sage: J.zero().is_invertible()
                False

            """
            zero = self.parent().zero()
            p = self.minimal_polynomial()
            return not (p(zero) == zero)


        def is_nilpotent(self):
            """
            Return whether or not some power of this element is zero.

            The superclass method won't work unless we're in an
            associative algebra, and we aren't. However, we generate
            an assocoative subalgebra and we're nilpotent there if and
            only if we're nilpotent here (probably).

            TESTS:

            The identity element is never nilpotent::

                sage: set_random_seed()
                sage: random_eja().one().is_nilpotent()
                False

            The additive identity is always nilpotent::

                sage: set_random_seed()
                sage: random_eja().zero().is_nilpotent()
                True

            """
            # The element we're going to call "is_nilpotent()" on.
            # Either myself, interpreted as an element of a finite-
            # dimensional algebra, or an element of an associative
            # subalgebra.
            elt = None

            if self.parent().is_associative():
                elt = FiniteDimensionalAlgebraElement(self.parent(), self)
            else:
                V = self.span_of_powers()
                assoc_subalg = self.subalgebra_generated_by()
                # Mis-design warning: the basis used for span_of_powers()
                # and subalgebra_generated_by() must be the same, and in
                # the same order!
                elt = assoc_subalg(V.coordinates(self.vector()))

            # Recursive call, but should work since elt lives in an
            # associative algebra.
            return elt.is_nilpotent()


        def is_regular(self):
            """
            Return whether or not this is a regular element.

            EXAMPLES:

            The identity element always has degree one, but any element
            linearly-independent from it is regular::

                sage: J = JordanSpinEJA(5)
                sage: J.one().is_regular()
                False
                sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
                sage: for x in J.gens():
                ....:     (J.one() + x).is_regular()
                False
                True
                True
                True
                True

            """
            return self.degree() == self.parent().rank()


        def degree(self):
            """
            Compute the degree of this element the straightforward way
            according to the definition; by appending powers to a list
            and figuring out its dimension (that is, whether or not
            they're linearly dependent).

            EXAMPLES::

                sage: J = JordanSpinEJA(4)
                sage: J.one().degree()
                1
                sage: e0,e1,e2,e3 = J.gens()
                sage: (e0 - e1).degree()
                2

            In the spin factor algebra (of rank two), all elements that
            aren't multiples of the identity are regular::

                sage: set_random_seed()
                sage: n = ZZ.random_element(1,10)
                sage: J = JordanSpinEJA(n)
                sage: x = J.random_element()
                sage: x == x.coefficient(0)*J.one() or x.degree() == 2
                True

            """
            return self.span_of_powers().dimension()


        def minimal_polynomial(self):
            """
            Return the minimal polynomial of this element,
            as a function of the variable `t`.

            ALGORITHM:

            We restrict ourselves to the associative subalgebra
            generated by this element, and then return the minimal
            polynomial of this element's operator matrix (in that
            subalgebra). This works by Baes Proposition 2.3.16.

            TESTS:

            The minimal polynomial of the identity and zero elements are
            always the same::

                sage: set_random_seed()
                sage: J = random_eja()
                sage: J.one().minimal_polynomial()
                t - 1
                sage: J.zero().minimal_polynomial()
                t

            The degree of an element is (by one definition) the degree
            of its minimal polynomial::

                sage: set_random_seed()
                sage: x = random_eja().random_element()
                sage: x.degree() == x.minimal_polynomial().degree()
                True

            The minimal polynomial and the characteristic polynomial coincide
            and are known (see Alizadeh, Example 11.11) for all elements of
            the spin factor algebra that aren't scalar multiples of the
            identity::

                sage: set_random_seed()
                sage: n = ZZ.random_element(2,10)
                sage: J = JordanSpinEJA(n)
                sage: y = J.random_element()
                sage: while y == y.coefficient(0)*J.one():
                ....:     y = J.random_element()
                sage: y0 = y.vector()[0]
                sage: y_bar = y.vector()[1:]
                sage: actual = y.minimal_polynomial()
                sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
                sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
                sage: bool(actual == expected)
                True

            The minimal polynomial should always kill its element::

                sage: set_random_seed()
                sage: x = random_eja().random_element()
                sage: p = x.minimal_polynomial()
                sage: x.apply_univariate_polynomial(p)
                0

            """
            V = self.span_of_powers()
            assoc_subalg = self.subalgebra_generated_by()
            # Mis-design warning: the basis used for span_of_powers()
            # and subalgebra_generated_by() must be the same, and in
            # the same order!
            elt = assoc_subalg(V.coordinates(self.vector()))

            # We get back a symbolic polynomial in 'x' but want a real
            # polynomial in 't'.
            p_of_x = elt.operator_matrix().minimal_polynomial()
            return p_of_x.change_variable_name('t')


        def natural_representation(self):
            """
            Return a more-natural representation of this element.

            Every finite-dimensional Euclidean Jordan Algebra is a
            direct sum of five simple algebras, four of which comprise
            Hermitian matrices. This method returns the original
            "natural" representation of this element as a Hermitian
            matrix, if it has one. If not, you get the usual representation.

            EXAMPLES::

                sage: J = ComplexHermitianEJA(3)
                sage: J.one()
                e0 + e5 + e8
                sage: J.one().natural_representation()
                [1 0 0 0 0 0]
                [0 1 0 0 0 0]
                [0 0 1 0 0 0]
                [0 0 0 1 0 0]
                [0 0 0 0 1 0]
                [0 0 0 0 0 1]

            ::

                sage: J = QuaternionHermitianEJA(3)
                sage: J.one()
                e0 + e9 + e14
                sage: J.one().natural_representation()
                [1 0 0 0 0 0 0 0 0 0 0 0]
                [0 1 0 0 0 0 0 0 0 0 0 0]
                [0 0 1 0 0 0 0 0 0 0 0 0]
                [0 0 0 1 0 0 0 0 0 0 0 0]
                [0 0 0 0 1 0 0 0 0 0 0 0]
                [0 0 0 0 0 1 0 0 0 0 0 0]
                [0 0 0 0 0 0 1 0 0 0 0 0]
                [0 0 0 0 0 0 0 1 0 0 0 0]
                [0 0 0 0 0 0 0 0 1 0 0 0]
                [0 0 0 0 0 0 0 0 0 1 0 0]
                [0 0 0 0 0 0 0 0 0 0 1 0]
                [0 0 0 0 0 0 0 0 0 0 0 1]

            """
            B = self.parent().natural_basis()
            W = B[0].matrix_space()
            return W.linear_combination(zip(self.vector(), B))


        def operator_matrix(self):
            """
            Return the matrix that represents left- (or right-)
            multiplication by this element in the parent algebra.

            We have to override this because the superclass method
            returns a matrix that acts on row vectors (that is, on
            the right).

            EXAMPLES:

            Test the first polarization identity from my notes, Koecher Chapter
            III, or from Baes (2.3)::

                sage: set_random_seed()
                sage: J = random_eja()
                sage: x = J.random_element()
                sage: y = J.random_element()
                sage: Lx = x.operator_matrix()
                sage: Ly = y.operator_matrix()
                sage: Lxx = (x*x).operator_matrix()
                sage: Lxy = (x*y).operator_matrix()
                sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
                True

            Test the second polarization identity from my notes or from
            Baes (2.4)::

                sage: set_random_seed()
                sage: J = random_eja()
                sage: x = J.random_element()
                sage: y = J.random_element()
                sage: z = J.random_element()
                sage: Lx = x.operator_matrix()
                sage: Ly = y.operator_matrix()
                sage: Lz = z.operator_matrix()
                sage: Lzy = (z*y).operator_matrix()
                sage: Lxy = (x*y).operator_matrix()
                sage: Lxz = (x*z).operator_matrix()
                sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
                True

            Test the third polarization identity from my notes or from
            Baes (2.5)::

                sage: set_random_seed()
                sage: J = random_eja()
                sage: u = J.random_element()
                sage: y = J.random_element()
                sage: z = J.random_element()
                sage: Lu = u.operator_matrix()
                sage: Ly = y.operator_matrix()
                sage: Lz = z.operator_matrix()
                sage: Lzy = (z*y).operator_matrix()
                sage: Luy = (u*y).operator_matrix()
                sage: Luz = (u*z).operator_matrix()
                sage: Luyz = (u*(y*z)).operator_matrix()
                sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
                sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
                sage: bool(lhs == rhs)
                True

            """
            fda_elt = FiniteDimensionalAlgebraElement(self.parent(), self)
            return fda_elt.matrix().transpose()


        def quadratic_representation(self, other=None):
            """
            Return the quadratic representation of this element.

            EXAMPLES:

            The explicit form in the spin factor algebra is given by
            Alizadeh's Example 11.12::

                sage: set_random_seed()
                sage: n = ZZ.random_element(1,10)
                sage: J = JordanSpinEJA(n)
                sage: x = J.random_element()
                sage: x_vec = x.vector()
                sage: x0 = x_vec[0]
                sage: x_bar = x_vec[1:]
                sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
                sage: B = 2*x0*x_bar.row()
                sage: C = 2*x0*x_bar.column()
                sage: D = identity_matrix(QQ, n-1)
                sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
                sage: D = D + 2*x_bar.tensor_product(x_bar)
                sage: Q = block_matrix(2,2,[A,B,C,D])
                sage: Q == x.quadratic_representation()
                True

            Test all of the properties from Theorem 11.2 in Alizadeh::

                sage: set_random_seed()
                sage: J = random_eja()
                sage: x = J.random_element()
                sage: y = J.random_element()

            Property 1:

                sage: actual = x.quadratic_representation(y)
                sage: expected = ( (x+y).quadratic_representation()
                ....:              -x.quadratic_representation()
                ....:              -y.quadratic_representation() ) / 2
                sage: actual == expected
                True

            Property 2:

                sage: alpha = QQ.random_element()
                sage: actual = (alpha*x).quadratic_representation()
                sage: expected = (alpha^2)*x.quadratic_representation()
                sage: actual == expected
                True

            Property 5:

                sage: Qy = y.quadratic_representation()
                sage: actual = J(Qy*x.vector()).quadratic_representation()
                sage: expected = Qy*x.quadratic_representation()*Qy
                sage: actual == expected
                True

            Property 6:

                sage: k = ZZ.random_element(1,10)
                sage: actual = (x^k).quadratic_representation()
                sage: expected = (x.quadratic_representation())^k
                sage: actual == expected
                True

            """
            if other is None:
                other=self
            elif not other in self.parent():
                raise TypeError("'other' must live in the same algebra")

            L = self.operator_matrix()
            M = other.operator_matrix()
            return ( L*M + M*L - (self*other).operator_matrix() )


        def span_of_powers(self):
            """
            Return the vector space spanned by successive powers of
            this element.
            """
            # The dimension of the subalgebra can't be greater than
            # the big algebra, so just put everything into a list
            # and let span() get rid of the excess.
            #
            # We do the extra ambient_vector_space() in case we're messing
            # with polynomials and the direct parent is a module.
            V = self.vector().parent().ambient_vector_space()
            return V.span( (self**d).vector() for d in xrange(V.dimension()) )


        def subalgebra_generated_by(self):
            """
            Return the associative subalgebra of the parent EJA generated
            by this element.

            TESTS::

                sage: set_random_seed()
                sage: x = random_eja().random_element()
                sage: x.subalgebra_generated_by().is_associative()
                True

            Squaring in the subalgebra should be the same thing as
            squaring in the superalgebra::

                sage: set_random_seed()
                sage: x = random_eja().random_element()
                sage: u = x.subalgebra_generated_by().random_element()
                sage: u.operator_matrix()*u.vector() == (u**2).vector()
                True

            """
            # First get the subspace spanned by the powers of myself...
            V = self.span_of_powers()
            F = self.base_ring()

            # Now figure out the entries of the right-multiplication
            # matrix for the successive basis elements b0, b1,... of
            # that subspace.
            mats = []
            for b_right in V.basis():
                eja_b_right = self.parent()(b_right)
                b_right_rows = []
                # The first row of the right-multiplication matrix by
                # b1 is what we get if we apply that matrix to b1. The
                # second row of the right multiplication matrix by b1
                # is what we get when we apply that matrix to b2...
                #
                # IMPORTANT: this assumes that all vectors are COLUMN
                # vectors, unlike our superclass (which uses row vectors).
                for b_left in V.basis():
                    eja_b_left = self.parent()(b_left)
                    # Multiply in the original EJA, but then get the
                    # coordinates from the subalgebra in terms of its
                    # basis.
                    this_row = V.coordinates((eja_b_left*eja_b_right).vector())
                    b_right_rows.append(this_row)
                b_right_matrix = matrix(F, b_right_rows)
                mats.append(b_right_matrix)

            # It's an algebra of polynomials in one element, and EJAs
            # are power-associative.
            #
            # TODO: choose generator names intelligently.
            return FiniteDimensionalEuclideanJordanAlgebra(F, mats, assume_associative=True, names='f')


        def subalgebra_idempotent(self):
            """
            Find an idempotent in the associative subalgebra I generate
            using Proposition 2.3.5 in Baes.

            TESTS::

                sage: set_random_seed()
                sage: J = RealCartesianProductEJA(5)
                sage: c = J.random_element().subalgebra_idempotent()
                sage: c^2 == c
                True
                sage: J = JordanSpinEJA(5)
                sage: c = J.random_element().subalgebra_idempotent()
                sage: c^2 == c
                True

            """
            if self.is_nilpotent():
                raise ValueError("this only works with non-nilpotent elements!")

            V = self.span_of_powers()
            J = self.subalgebra_generated_by()
            # Mis-design warning: the basis used for span_of_powers()
            # and subalgebra_generated_by() must be the same, and in
            # the same order!
            u = J(V.coordinates(self.vector()))

            # The image of the matrix of left-u^m-multiplication
            # will be minimal for some natural number s...
            s = 0
            minimal_dim = V.dimension()
            for i in xrange(1, V.dimension()):
                this_dim = (u**i).operator_matrix().image().dimension()
                if this_dim < minimal_dim:
                    minimal_dim = this_dim
                    s = i

            # Now minimal_matrix should correspond to the smallest
            # non-zero subspace in Baes's (or really, Koecher's)
            # proposition.
            #
            # However, we need to restrict the matrix to work on the
            # subspace... or do we? Can't we just solve, knowing that
            # A(c) = u^(s+1) should have a solution in the big space,
            # too?
            #
            # Beware, solve_right() means that we're using COLUMN vectors.
            # Our FiniteDimensionalAlgebraElement superclass uses rows.
            u_next = u**(s+1)
            A = u_next.operator_matrix()
            c_coordinates = A.solve_right(u_next.vector())

            # Now c_coordinates is the idempotent we want, but it's in
            # the coordinate system of the subalgebra.
            #
            # We need the basis for J, but as elements of the parent algebra.
            #
            basis = [self.parent(v) for v in V.basis()]
            return self.parent().linear_combination(zip(c_coordinates, basis))


        def trace(self):
            """
            Return my trace, the sum of my eigenvalues.

            EXAMPLES::

                sage: J = JordanSpinEJA(3)
                sage: x = sum(J.gens())
                sage: x.trace()
                2

            ::

                sage: J = RealCartesianProductEJA(5)
                sage: J.one().trace()
                5

            TESTS:

            The trace of an element is a real number::

                sage: set_random_seed()
                sage: J = random_eja()
                sage: J.random_element().trace() in J.base_ring()
                True

            """
            P = self.parent()
            r = P.rank()
            p = P._charpoly_coeff(r-1)
            # The _charpoly_coeff function already adds the factor of
            # -1 to ensure that _charpoly_coeff(r-1) is really what
            # appears in front of t^{r-1} in the charpoly. However,
            # we want the negative of THAT for the trace.
            return -p(*self.vector())


        def trace_inner_product(self, other):
            """
            Return the trace inner product of myself and ``other``.

            TESTS:

            The trace inner product is commutative::

                sage: set_random_seed()
                sage: J = random_eja()
                sage: x = J.random_element(); y = J.random_element()
                sage: x.trace_inner_product(y) == y.trace_inner_product(x)
                True

            The trace inner product is bilinear::

                sage: set_random_seed()
                sage: J = random_eja()
                sage: x = J.random_element()
                sage: y = J.random_element()
                sage: z = J.random_element()
                sage: a = QQ.random_element();
                sage: actual = (a*(x+z)).trace_inner_product(y)
                sage: expected = ( a*x.trace_inner_product(y) +
                ....:              a*z.trace_inner_product(y) )
                sage: actual == expected
                True
                sage: actual = x.trace_inner_product(a*(y+z))
                sage: expected = ( a*x.trace_inner_product(y) +
                ....:              a*x.trace_inner_product(z) )
                sage: actual == expected
                True

            The trace inner product satisfies the compatibility
            condition in the definition of a Euclidean Jordan algebra::

                sage: set_random_seed()
                sage: J = random_eja()
                sage: x = J.random_element()
                sage: y = J.random_element()
                sage: z = J.random_element()
                sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
                True

            """
            if not other in self.parent():
                raise TypeError("'other' must live in the same algebra")

            return (self*other).trace()


class RealCartesianProductEJA(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.

    EXAMPLES:

    This multiplication table can be verified by hand::

        sage: J = RealCartesianProductEJA(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

    """
    @staticmethod
    def __classcall_private__(cls, n, field=QQ):
        # 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, n, n, lambda k,j: 1*(k == j == i))
               for i in xrange(n) ]

        fdeja = super(RealCartesianProductEJA, cls)
        return fdeja.__classcall_private__(cls, field, Qs, rank=n)

    def inner_product(self, x, y):
        return _usual_ip(x,y)


def random_eja():
    """
    Return a "random" finite-dimensional Euclidean Jordan Algebra.

    ALGORITHM:

    For now, we choose a random natural number ``n`` (greater than zero)
    and then give you back one of the following:

      * The cartesian product of the rational numbers ``n`` times; this is
        ``QQ^n`` with the Hadamard product.

      * The Jordan spin algebra on ``QQ^n``.

      * The ``n``-by-``n`` rational symmetric matrices with the symmetric
        product.

      * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
        in the space of ``2n``-by-``2n`` real symmetric matrices.

      * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
        in the space of ``4n``-by-``4n`` real symmetric matrices.

    Later this might be extended to return Cartesian products of the
    EJAs above.

    TESTS::

        sage: random_eja()
        Euclidean Jordan algebra of degree...

    """

    # The max_n component lets us choose different upper bounds on the
    # value "n" that gets passed to the constructor. This is needed
    # because e.g. R^{10} is reasonable to test, while the Hermitian
    # 10-by-10 quaternion matrices are not.
    (constructor, max_n) = choice([(RealCartesianProductEJA, 6),
                                   (JordanSpinEJA, 6),
                                   (RealSymmetricEJA, 5),
                                   (ComplexHermitianEJA, 4),
                                   (QuaternionHermitianEJA, 3)])
    n = ZZ.random_element(1, max_n)
    return constructor(n, field=QQ)



def _real_symmetric_basis(n, field=QQ):
    """
    Return a basis for the space of real symmetric n-by-n matrices.
    """
    # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
    # coordinates.
    S = []
    for i in xrange(n):
        for j in xrange(i+1):
            Eij = matrix(field, n, lambda k,l: k==i and l==j)
            if i == j:
                Sij = Eij
            else:
                # Beware, orthogonal but not normalized!
                Sij = Eij + Eij.transpose()
            S.append(Sij)
    return tuple(S)


def _complex_hermitian_basis(n, field=QQ):
    """
    Returns a basis for the space of complex Hermitian n-by-n matrices.

    TESTS::

        sage: set_random_seed()
        sage: n = ZZ.random_element(1,5)
        sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
        True

    """
    F = QuadraticField(-1, 'I')
    I = F.gen()

    # This is like the symmetric case, but we need to be careful:
    #
    #   * We want conjugate-symmetry, not just symmetry.
    #   * The diagonal will (as a result) be real.
    #
    S = []
    for i in xrange(n):
        for j in xrange(i+1):
            Eij = matrix(field, n, lambda k,l: k==i and l==j)
            if i == j:
                Sij = _embed_complex_matrix(Eij)
                S.append(Sij)
            else:
                # Beware, orthogonal but not normalized! The second one
                # has a minus because it's conjugated.
                Sij_real = _embed_complex_matrix(Eij + Eij.transpose())
                S.append(Sij_real)
                Sij_imag = _embed_complex_matrix(I*Eij - I*Eij.transpose())
                S.append(Sij_imag)
    return tuple(S)


def _quaternion_hermitian_basis(n, field=QQ):
    """
    Returns a basis for the space of quaternion Hermitian n-by-n matrices.

    TESTS::

        sage: set_random_seed()
        sage: n = ZZ.random_element(1,5)
        sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) )
        True

    """
    Q = QuaternionAlgebra(QQ,-1,-1)
    I,J,K = Q.gens()

    # This is like the symmetric case, but we need to be careful:
    #
    #   * We want conjugate-symmetry, not just symmetry.
    #   * The diagonal will (as a result) be real.
    #
    S = []
    for i in xrange(n):
        for j in xrange(i+1):
            Eij = matrix(Q, n, lambda k,l: k==i and l==j)
            if i == j:
                Sij = _embed_quaternion_matrix(Eij)
                S.append(Sij)
            else:
                # Beware, orthogonal but not normalized! The second,
                # third, and fourth ones have a minus because they're
                # conjugated.
                Sij_real = _embed_quaternion_matrix(Eij + Eij.transpose())
                S.append(Sij_real)
                Sij_I = _embed_quaternion_matrix(I*Eij - I*Eij.transpose())
                S.append(Sij_I)
                Sij_J = _embed_quaternion_matrix(J*Eij - J*Eij.transpose())
                S.append(Sij_J)
                Sij_K = _embed_quaternion_matrix(K*Eij - K*Eij.transpose())
                S.append(Sij_K)
    return tuple(S)


def _mat2vec(m):
        return vector(m.base_ring(), m.list())

def _vec2mat(v):
        return matrix(v.base_ring(), sqrt(v.degree()), v.list())

def _multiplication_table_from_matrix_basis(basis):
    """
    At least three of the five simple Euclidean Jordan algebras have the
    symmetric multiplication (A,B) |-> (AB + BA)/2, where the
    multiplication on the right is matrix multiplication. Given a basis
    for the underlying matrix space, this function returns a
    multiplication table (obtained by looping through the basis
    elements) for an algebra of those matrices. A reordered copy
    of the basis is also returned to work around the fact that
    the ``span()`` in this function will change the order of the basis
    from what we think it is, to... something else.
    """
    # In S^2, for example, we nominally have four coordinates even
    # though the space is of dimension three only. The vector space V
    # is supposed to hold the entire long vector, and the subspace W
    # of V will be spanned by the vectors that arise from symmetric
    # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
    field = basis[0].base_ring()
    dimension = basis[0].nrows()

    V = VectorSpace(field, dimension**2)
    W = V.span( _mat2vec(s) for s in basis )

    # Taking the span above reorders our basis (thanks, jerk!) so we
    # need to put our "matrix basis" in the same order as the
    # (reordered) vector basis.
    S = tuple( _vec2mat(b) for b in W.basis() )

    Qs = []
    for s in S:
        # Brute force the multiplication-by-s matrix by looping
        # through all elements of the basis and doing the computation
        # to find out what the corresponding row should be. BEWARE:
        # these multiplication tables won't be symmetric! It therefore
        # becomes REALLY IMPORTANT that the underlying algebra
        # constructor uses ROW vectors and not COLUMN vectors. That's
        # why we're computing rows here and not columns.
        Q_rows = []
        for t in S:
            this_row = _mat2vec((s*t + t*s)/2)
            Q_rows.append(W.coordinates(this_row))
        Q = matrix(field, W.dimension(), Q_rows)
        Qs.append(Q)

    return (Qs, S)


def _embed_complex_matrix(M):
    """
    Embed the n-by-n complex matrix ``M`` into the space of real
    matrices of size 2n-by-2n via the map the sends each entry `z = a +
    bi` to the block matrix ``[[a,b],[-b,a]]``.

    EXAMPLES::

        sage: F = QuadraticField(-1,'i')
        sage: x1 = F(4 - 2*i)
        sage: x2 = F(1 + 2*i)
        sage: x3 = F(-i)
        sage: x4 = F(6)
        sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
        sage: _embed_complex_matrix(M)
        [ 4 -2| 1  2]
        [ 2  4|-2  1]
        [-----+-----]
        [ 0 -1| 6  0]
        [ 1  0| 0  6]

    TESTS:

    Embedding is a homomorphism (isomorphism, in fact)::

        sage: set_random_seed()
        sage: n = ZZ.random_element(5)
        sage: F = QuadraticField(-1, 'i')
        sage: X = random_matrix(F, n)
        sage: Y = random_matrix(F, n)
        sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
        sage: expected = _embed_complex_matrix(X*Y)
        sage: actual == expected
        True

    """
    n = M.nrows()
    if M.ncols() != n:
        raise ValueError("the matrix 'M' must be square")
    field = M.base_ring()
    blocks = []
    for z in M.list():
        a = z.real()
        b = z.imag()
        blocks.append(matrix(field, 2, [[a,b],[-b,a]]))

    # We can drop the imaginaries here.
    return block_matrix(field.base_ring(), n, blocks)


def _unembed_complex_matrix(M):
    """
    The inverse of _embed_complex_matrix().

    EXAMPLES::

        sage: A = matrix(QQ,[ [ 1,  2,   3,  4],
        ....:                 [-2,  1,  -4,  3],
        ....:                 [ 9,  10, 11, 12],
        ....:                 [-10, 9, -12, 11] ])
        sage: _unembed_complex_matrix(A)
        [  2*i + 1   4*i + 3]
        [ 10*i + 9 12*i + 11]

    TESTS:

    Unembedding is the inverse of embedding::

        sage: set_random_seed()
        sage: F = QuadraticField(-1, 'i')
        sage: M = random_matrix(F, 3)
        sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
        True

    """
    n = ZZ(M.nrows())
    if M.ncols() != n:
        raise ValueError("the matrix 'M' must be square")
    if not n.mod(2).is_zero():
        raise ValueError("the matrix 'M' must be a complex embedding")

    F = QuadraticField(-1, 'i')
    i = F.gen()

    # Go top-left to bottom-right (reading order), converting every
    # 2-by-2 block we see to a single complex element.
    elements = []
    for k in xrange(n/2):
        for j in xrange(n/2):
            submat = M[2*k:2*k+2,2*j:2*j+2]
            if submat[0,0] != submat[1,1]:
                raise ValueError('bad on-diagonal submatrix')
            if submat[0,1] != -submat[1,0]:
                raise ValueError('bad off-diagonal submatrix')
            z = submat[0,0] + submat[0,1]*i
            elements.append(z)

    return matrix(F, n/2, elements)


def _embed_quaternion_matrix(M):
    """
    Embed the n-by-n quaternion matrix ``M`` into the space of real
    matrices of size 4n-by-4n by first sending each quaternion entry
    `z = a + bi + cj + dk` to the block-complex matrix
    ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
    a real matrix.

    EXAMPLES::

        sage: Q = QuaternionAlgebra(QQ,-1,-1)
        sage: i,j,k = Q.gens()
        sage: x = 1 + 2*i + 3*j + 4*k
        sage: M = matrix(Q, 1, [[x]])
        sage: _embed_quaternion_matrix(M)
        [ 1  2  3  4]
        [-2  1 -4  3]
        [-3  4  1 -2]
        [-4 -3  2  1]

    Embedding is a homomorphism (isomorphism, in fact)::

        sage: set_random_seed()
        sage: n = ZZ.random_element(5)
        sage: Q = QuaternionAlgebra(QQ,-1,-1)
        sage: X = random_matrix(Q, n)
        sage: Y = random_matrix(Q, n)
        sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y)
        sage: expected = _embed_quaternion_matrix(X*Y)
        sage: actual == expected
        True

    """
    quaternions = M.base_ring()
    n = M.nrows()
    if M.ncols() != n:
        raise ValueError("the matrix 'M' must be square")

    F = QuadraticField(-1, 'i')
    i = F.gen()

    blocks = []
    for z in M.list():
        t = z.coefficient_tuple()
        a = t[0]
        b = t[1]
        c = t[2]
        d = t[3]
        cplx_matrix = matrix(F, 2, [[ a + b*i, c + d*i],
                                    [-c + d*i, a - b*i]])
        blocks.append(_embed_complex_matrix(cplx_matrix))

    # We should have real entries by now, so use the realest field
    # we've got for the return value.
    return block_matrix(quaternions.base_ring(), n, blocks)


def _unembed_quaternion_matrix(M):
    """
    The inverse of _embed_quaternion_matrix().

    EXAMPLES::

        sage: M = matrix(QQ, [[ 1,  2,  3,  4],
        ....:                 [-2,  1, -4,  3],
        ....:                 [-3,  4,  1, -2],
        ....:                 [-4, -3,  2,  1]])
        sage: _unembed_quaternion_matrix(M)
        [1 + 2*i + 3*j + 4*k]

    TESTS:

    Unembedding is the inverse of embedding::

        sage: set_random_seed()
        sage: Q = QuaternionAlgebra(QQ, -1, -1)
        sage: M = random_matrix(Q, 3)
        sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
        True

    """
    n = ZZ(M.nrows())
    if M.ncols() != n:
        raise ValueError("the matrix 'M' must be square")
    if not n.mod(4).is_zero():
        raise ValueError("the matrix 'M' must be a complex embedding")

    Q = QuaternionAlgebra(QQ,-1,-1)
    i,j,k = Q.gens()

    # Go top-left to bottom-right (reading order), converting every
    # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
    # quaternion block.
    elements = []
    for l in xrange(n/4):
        for m in xrange(n/4):
            submat = _unembed_complex_matrix(M[4*l:4*l+4,4*m:4*m+4])
            if submat[0,0] != submat[1,1].conjugate():
                raise ValueError('bad on-diagonal submatrix')
            if submat[0,1] != -submat[1,0].conjugate():
                raise ValueError('bad off-diagonal submatrix')
            z  = submat[0,0].real() + submat[0,0].imag()*i
            z += submat[0,1].real()*j + submat[0,1].imag()*k
            elements.append(z)

    return matrix(Q, n/4, elements)


# The usual inner product on R^n.
def _usual_ip(x,y):
    return x.vector().inner_product(y.vector())

# The inner product used for the real symmetric simple EJA.
# We keep it as a separate function because e.g. the complex
# algebra uses the same inner product, except divided by 2.
def _matrix_ip(X,Y):
    X_mat = X.natural_representation()
    Y_mat = Y.natural_representation()
    return (X_mat*Y_mat).trace()


class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra):
    """
    The rank-n simple EJA consisting of real symmetric n-by-n
    matrices, the usual symmetric Jordan product, and the trace inner
    product. It has dimension `(n^2 + n)/2` over the reals.

    EXAMPLES::

        sage: J = RealSymmetricEJA(2)
        sage: e0, e1, e2 = J.gens()
        sage: e0*e0
        e0
        sage: e1*e1
        e0 + e2
        sage: e2*e2
        e2

    TESTS:

    The degree of this algebra is `(n^2 + n) / 2`::

        sage: set_random_seed()
        sage: n = ZZ.random_element(1,5)
        sage: J = RealSymmetricEJA(n)
        sage: J.degree() == (n^2 + n)/2
        True

    The Jordan multiplication is what we think it is::

        sage: set_random_seed()
        sage: n = ZZ.random_element(1,5)
        sage: J = RealSymmetricEJA(n)
        sage: x = J.random_element()
        sage: y = J.random_element()
        sage: actual = (x*y).natural_representation()
        sage: X = x.natural_representation()
        sage: Y = y.natural_representation()
        sage: expected = (X*Y + Y*X)/2
        sage: actual == expected
        True
        sage: J(expected) == x*y
        True

    """
    @staticmethod
    def __classcall_private__(cls, n, field=QQ):
        S = _real_symmetric_basis(n, field=field)
        (Qs, T) = _multiplication_table_from_matrix_basis(S)

        fdeja = super(RealSymmetricEJA, cls)
        return fdeja.__classcall_private__(cls,
                                           field,
                                           Qs,
                                           rank=n,
                                           natural_basis=T)

    def inner_product(self, x, y):
        return _matrix_ip(x,y)


class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):
    """
    The rank-n simple EJA consisting of complex Hermitian n-by-n
    matrices over the real numbers, the usual symmetric Jordan product,
    and the real-part-of-trace inner product. It has dimension `n^2` over
    the reals.

    TESTS:

    The degree of this algebra is `n^2`::

        sage: set_random_seed()
        sage: n = ZZ.random_element(1,5)
        sage: J = ComplexHermitianEJA(n)
        sage: J.degree() == n^2
        True

    The Jordan multiplication is what we think it is::

        sage: set_random_seed()
        sage: n = ZZ.random_element(1,5)
        sage: J = ComplexHermitianEJA(n)
        sage: x = J.random_element()
        sage: y = J.random_element()
        sage: actual = (x*y).natural_representation()
        sage: X = x.natural_representation()
        sage: Y = y.natural_representation()
        sage: expected = (X*Y + Y*X)/2
        sage: actual == expected
        True
        sage: J(expected) == x*y
        True

    """
    @staticmethod
    def __classcall_private__(cls, n, field=QQ):
        S = _complex_hermitian_basis(n)
        (Qs, T) = _multiplication_table_from_matrix_basis(S)

        fdeja = super(ComplexHermitianEJA, cls)
        return fdeja.__classcall_private__(cls,
                                           field,
                                           Qs,
                                           rank=n,
                                           natural_basis=T)

    def inner_product(self, x, y):
        # Since a+bi on the diagonal is represented as
        #
        #   a + bi  = [  a  b  ]
        #             [ -b  a  ],
        #
        # we'll double-count the "a" entries if we take the trace of
        # the embedding.
        return _matrix_ip(x,y)/2


class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):
    """
    The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
    matrices, the usual symmetric Jordan product, and the
    real-part-of-trace inner product. It has dimension `2n^2 - n` over
    the reals.

    TESTS:

    The degree of this algebra is `n^2`::

        sage: set_random_seed()
        sage: n = ZZ.random_element(1,5)
        sage: J = QuaternionHermitianEJA(n)
        sage: J.degree() == 2*(n^2) - n
        True

    The Jordan multiplication is what we think it is::

        sage: set_random_seed()
        sage: n = ZZ.random_element(1,5)
        sage: J = QuaternionHermitianEJA(n)
        sage: x = J.random_element()
        sage: y = J.random_element()
        sage: actual = (x*y).natural_representation()
        sage: X = x.natural_representation()
        sage: Y = y.natural_representation()
        sage: expected = (X*Y + Y*X)/2
        sage: actual == expected
        True
        sage: J(expected) == x*y
        True

    """
    @staticmethod
    def __classcall_private__(cls, n, field=QQ):
        S = _quaternion_hermitian_basis(n)
        (Qs, T) = _multiplication_table_from_matrix_basis(S)

        fdeja = super(QuaternionHermitianEJA, cls)
        return fdeja.__classcall_private__(cls,
                                           field,
                                           Qs,
                                           rank=n,
                                           natural_basis=T)

    def inner_product(self, x, y):
        # Since a+bi+cj+dk on the diagonal is represented as
        #
        #   a + bi +cj + dk = [  a  b  c  d]
        #                     [ -b  a -d  c]
        #                     [ -c  d  a -b]
        #                     [ -d -c  b  a],
        #
        # we'll quadruple-count the "a" entries if we take the trace of
        # the embedding.
        return _matrix_ip(x,y)/4


class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra):
    """
    The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
    with the usual inner product and jordan product ``x*y =
    (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
    the reals.

    EXAMPLES:

    This multiplication table can be verified by hand::

        sage: J = JordanSpinEJA(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

    """
    @staticmethod
    def __classcall_private__(cls, n, field=QQ):
        Qs = []
        id_matrix = identity_matrix(field, n)
        for i in xrange(n):
            ei = id_matrix.column(i)
            Qi = zero_matrix(field, n)
            Qi.set_row(0, ei)
            Qi.set_column(0, ei)
            Qi += diagonal_matrix(n, [ei[0]]*n)
            # 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)

        # The rank of the spin algebra is two, unless we're in a
        # one-dimensional ambient space (because the rank is bounded by
        # the ambient dimension).
        fdeja = super(JordanSpinEJA, cls)
        return fdeja.__classcall_private__(cls, field, Qs, rank=min(n,2))

    def inner_product(self, x, y):
        return _usual_ip(x,y)