eja: rename orthogonal_idempotents() to a_jordan_frame().

[sage.d.git] / mjo / eja / eja_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 itertools import repeat

from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra
from sage.categories.magmatic_algebras import MagmaticAlgebras
from sage.combinat.free_module import CombinatorialFreeModule
from sage.matrix.constructor import matrix
from sage.matrix.matrix_space import MatrixSpace
from sage.misc.cachefunc import cached_method
from sage.misc.lazy_import import lazy_import
from sage.misc.prandom import choice
from sage.misc.table import table
from sage.modules.free_module import FreeModule, VectorSpace
from sage.rings.all import (ZZ, QQ, RR, RLF, CLF,
                            PolynomialRing,
                            QuadraticField)
from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
lazy_import('mjo.eja.eja_subalgebra',
            'FiniteDimensionalEuclideanJordanSubalgebra')
from mjo.eja.eja_utils import _mat2vec

class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):

    def _coerce_map_from_base_ring(self):
        """
        Disable the map from the base ring into the algebra.

        Performing a nonsense conversion like this automatically
        is counterpedagogical. The fallback is to try the usual
        element constructor, which should also fail.

        SETUP::

            sage: from mjo.eja.eja_algebra import random_eja

        TESTS::

            sage: set_random_seed()
            sage: J = random_eja()
            sage: J(1)
            Traceback (most recent call last):
            ...
            ValueError: not a naturally-represented algebra element

        """
        return None

    def __init__(self,
                 field,
                 mult_table,
                 rank,
                 prefix='e',
                 category=None,
                 natural_basis=None,
                 check=True):
        """
        SETUP::

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

        EXAMPLES:

        By definition, Jordan multiplication commutes::

            sage: set_random_seed()
            sage: J = random_eja()
            sage: x,y = J.random_elements(2)
            sage: x*y == y*x
            True

        TESTS:

        The ``field`` we're given must be real::

            sage: JordanSpinEJA(2,QQbar)
            Traceback (most recent call last):
            ...
            ValueError: field is not real

        """
        if check:
            if not field.is_subring(RR):
                # Note: this does return true for the real algebraic
                # field, and any quadratic field where we've specified
                # a real embedding.
                raise ValueError('field is not real')

        self._rank = rank
        self._natural_basis = natural_basis

        if category is None:
            category = MagmaticAlgebras(field).FiniteDimensional()
            category = category.WithBasis().Unital()

        fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
        fda.__init__(field,
                     range(len(mult_table)),
                     prefix=prefix,
                     category=category)
        self.print_options(bracket='')

        # The multiplication table we're given is necessarily in terms
        # of vectors, because we don't have an algebra yet for
        # anything to be an element of. However, it's faster in the
        # long run to have the multiplication table be in terms of
        # algebra elements. We do this after calling the superclass
        # constructor so that from_vector() knows what to do.
        self._multiplication_table = [
            list(map(lambda x: self.from_vector(x), ls))
            for ls in mult_table
        ]


    def _element_constructor_(self, elt):
        """
        Construct an element of this algebra from its natural
        representation.

        This gets called only after the parent element _call_ method
        fails to find a coercion for the argument.

        SETUP::

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

        EXAMPLES:

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

            sage: J = RealSymmetricEJA(3)
            sage: I = matrix.identity(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

        TESTS:

        Ensure that we can convert any element of the two non-matrix
        simple algebras (whose natural representations are their usual
        vector representations) back and forth faithfully::

            sage: set_random_seed()
            sage: J = HadamardEJA.random_instance()
            sage: x = J.random_element()
            sage: J(x.to_vector().column()) == x
            True
            sage: J = JordanSpinEJA.random_instance()
            sage: x = J.random_element()
            sage: J(x.to_vector().column()) == x
            True

        """
        msg = "not a naturally-represented algebra element"
        if elt == 0:
            # The superclass implementation of random_element()
            # needs to be able to coerce "0" into the algebra.
            return self.zero()
        elif elt in self.base_ring():
            # Ensure that no base ring -> algebra coercion is performed
            # by this method. There's some stupidity in sage that would
            # otherwise propagate to this method; for example, sage thinks
            # that the integer 3 belongs to the space of 2-by-2 matrices.
            raise ValueError(msg)

        natural_basis = self.natural_basis()
        basis_space = natural_basis[0].matrix_space()
        if elt not in basis_space:
            raise ValueError(msg)

        # Thanks for nothing! Matrix spaces aren't vector spaces in
        # Sage, so we have to figure out its natural-basis coordinates
        # ourselves. We use the basis space's ring instead of the
        # element's ring because the basis space might be an algebraic
        # closure whereas the base ring of the 3-by-3 identity matrix
        # could be QQ instead of QQbar.
        V = VectorSpace(basis_space.base_ring(), elt.nrows()*elt.ncols())
        W = V.span_of_basis( _mat2vec(s) for s in natural_basis )
        coords =  W.coordinate_vector(_mat2vec(elt))
        return self.from_vector(coords)


    def _repr_(self):
        """
        Return a string representation of ``self``.

        SETUP::

            sage: from mjo.eja.eja_algebra import JordanSpinEJA

        TESTS:

        Ensure that it says what we think it says::

            sage: JordanSpinEJA(2, field=QQ)
            Euclidean Jordan algebra of dimension 2 over Rational Field
            sage: JordanSpinEJA(3, field=RDF)
            Euclidean Jordan algebra of dimension 3 over Real Double Field

        """
        fmt = "Euclidean Jordan algebra of dimension {} over {}"
        return fmt.format(self.dimension(), self.base_ring())

    def product_on_basis(self, i, j):
        return self._multiplication_table[i][j]

    def _a_regular_element(self):
        """
        Guess a regular element. Needed to compute the basis for our
        characteristic polynomial coefficients.

        SETUP::

            sage: from mjo.eja.eja_algebra import random_eja

        TESTS:

        Ensure that this hacky method succeeds for every algebra that we
        know how to construct::

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

        """
        gs = self.gens()
        z = self.sum( (i+1)*gs[i] for i in range(len(gs)) )
        if not z.is_regular():
            raise ValueError("don't know a regular element")
        return z


    @cached_method
    def _charpoly_basis_space(self):
        """
        Return the vector space spanned by the basis used in our
        characteristic polynomial coefficients. This is used not only to
        compute those coefficients, but also any time we need to
        evaluate the coefficients (like when we compute the trace or
        determinant).
        """
        z = self._a_regular_element()
        # Don't use the parent vector space directly here in case this
        # happens to be a subalgebra. In that case, we would be e.g.
        # two-dimensional but span_of_basis() would expect three
        # coordinates.
        V = VectorSpace(self.base_ring(), self.vector_space().dimension())
        basis = [ (z**k).to_vector() for k in range(self.rank()) ]
        V1 = V.span_of_basis( basis )
        b =  (V1.basis() + V1.complement().basis())
        return V.span_of_basis(b)



    @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, xr, detA) = self._charpoly_matrix_system()
        R = A_of_x.base_ring()

        if i == self.rank():
            return R.one()
        if i > self.rank():
            # Guaranteed by theory
            return R.zero()

        # Danger: the in-place modification is done for performance
        # reasons (reconstructing a matrix with huge polynomial
        # entries is slow), but I don't know how cached_method works,
        # so it's highly possible that we're modifying some global
        # list variable by reference, here. In other words, you
        # probably shouldn't call this method twice on the same
        # algebra, at the same time, in two threads
        Ai_orig = A_of_x.column(i)
        A_of_x.set_column(i,xr)
        numerator = A_of_x.det()
        A_of_x.set_column(i,Ai_orig)

        # 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/detA)


    @cached_method
    def _charpoly_matrix_system(self):
        """
        Compute the matrix whose entries A_ij are polynomials in
        X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector
        corresponding to `x^r` and the determinent of the matrix A =
        [A_ij]. In other words, all of the fixed (cachable) data needed
        to compute the coefficients of the characteristic polynomial.
        """
        r = self.rank()
        n = self.dimension()

        # Turn my vector space into a module so that "vectors" can
        # have multivatiate polynomial entries.
        names = tuple('X' + str(i) for i in range(1,n+1))
        R = PolynomialRing(self.base_ring(), names)

        # Using change_ring() on the parent's vector space doesn't work
        # here because, in a subalgebra, that vector space has a basis
        # and change_ring() tries to bring the basis along with it. And
        # that doesn't work unless the new ring is a PID, which it usually
        # won't be.
        V = FreeModule(R,n)

        # Now let x = (X1,X2,...,Xn) be the vector whose entries are
        # indeterminates...
        x = V(names)

        # And figure out the "left multiplication by x" matrix in
        # that setting.
        lmbx_cols = []
        monomial_matrices = [ self.monomial(i).operator().matrix()
                              for i in range(n) ] # don't recompute these!
        for k in range(n):
            ek = self.monomial(k).to_vector()
            lmbx_cols.append(
              sum( x[i]*(monomial_matrices[i]*ek)
                   for i in range(n) ) )
        Lx = matrix.column(R, lmbx_cols)

        # Now we can compute powers of x "symbolically"
        x_powers = [self.one().to_vector(), x]
        for d in range(2, r+1):
            x_powers.append( Lx*(x_powers[-1]) )

        idmat = matrix.identity(R, n)

        W = self._charpoly_basis_space()
        W = W.change_ring(R.fraction_field())

        # Starting with the standard coordinates x = (X1,X2,...,Xn)
        # and then converting the entries to W-coordinates allows us
        # to pass in the standard coordinates to the charpoly and get
        # back the right answer. Specifically, with x = (X1,X2,...,Xn),
        # we have
        #
        #   W.coordinates(x^2) eval'd at (standard z-coords)
        #     =
        #   W-coords of (z^2)
        #     =
        #   W-coords of (standard coords of x^2 eval'd at std-coords of z)
        #
        # We want the middle equivalent thing in our matrix, but use
        # the first equivalent thing instead so that we can pass in
        # standard coordinates.
        x_powers = [ W.coordinate_vector(xp) for xp in x_powers ]
        l2 = [idmat.column(k-1) for k in range(r+1, n+1)]
        A_of_x = matrix.column(R, n, (x_powers[:r] + l2))
        return (A_of_x, x, x_powers[r], A_of_x.det())


    @cached_method
    def characteristic_polynomial(self):
        """
        Return a characteristic polynomial that works for all elements
        of this algebra.

        The resulting polynomial has `n+1` variables, where `n` is the
        dimension of this algebra. The first `n` variables correspond to
        the coordinates of an algebra element: when evaluated at the
        coordinates of an algebra element with respect to a certain
        basis, the result is a univariate polynomial (in the one
        remaining variable ``t``), namely the characteristic polynomial
        of that element.

        SETUP::

            sage: from mjo.eja.eja_algebra import JordanSpinEJA, TrivialEJA

        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().to_vector()
            sage: p(*xvec)
            t^2 - 2*t + 1

        By definition, the characteristic polynomial is a monic
        degree-zero polynomial in a rank-zero algebra. Note that
        Cayley-Hamilton is indeed satisfied since the polynomial
        ``1`` evaluates to the identity element of the algebra on
        any argument::

            sage: J = TrivialEJA()
            sage: J.characteristic_polynomial()
            1

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

        # The list of coefficient polynomials a_0, a_1, a_2, ..., a_n.
        a = [ self._charpoly_coeff(i) for i in range(r+1) ]

        # 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)

        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.

        SETUP::

            sage: from mjo.eja.eja_algebra import random_eja

        EXAMPLES:

        Our inner product is "associative," which means the following for
        a symmetric bilinear form::

            sage: set_random_seed()
            sage: J = random_eja()
            sage: x,y,z = J.random_elements(3)
            sage: (x*y).inner_product(z) == y.inner_product(x*z)
            True

        """
        X = x.natural_representation()
        Y = y.natural_representation()
        return self.natural_inner_product(X,Y)


    def is_trivial(self):
        """
        Return whether or not this algebra is trivial.

        A trivial algebra contains only the zero element.

        SETUP::

            sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
            ....:                                  TrivialEJA)

        EXAMPLES::

            sage: J = ComplexHermitianEJA(3)
            sage: J.is_trivial()
            False

        ::

            sage: J = TrivialEJA()
            sage: J.is_trivial()
            True

        """
        return self.dimension() == 0


    def multiplication_table(self):
        """
        Return a visual representation of this algebra's multiplication
        table (on basis elements).

        SETUP::

            sage: from mjo.eja.eja_algebra import JordanSpinEJA

        EXAMPLES::

            sage: J = JordanSpinEJA(4)
            sage: J.multiplication_table()
            +----++----+----+----+----+
            | *  || e0 | e1 | e2 | e3 |
            +====++====+====+====+====+
            | e0 || e0 | e1 | e2 | e3 |
            +----++----+----+----+----+
            | e1 || e1 | e0 | 0  | 0  |
            +----++----+----+----+----+
            | e2 || e2 | 0  | e0 | 0  |
            +----++----+----+----+----+
            | e3 || e3 | 0  | 0  | e0 |
            +----++----+----+----+----+

        """
        M = list(self._multiplication_table) # copy
        for i in range(len(M)):
            # M had better be "square"
            M[i] = [self.monomial(i)] + M[i]
        M = [["*"] + list(self.gens())] + M
        return table(M, header_row=True, header_column=True, frame=True)


    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.

        SETUP::

            sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
            ....:                                  RealSymmetricEJA)

        EXAMPLES::

            sage: J = RealSymmetricEJA(2)
            sage: J.basis()
            Finite family {0: e0, 1: e1, 2: e2}
            sage: J.natural_basis()
            (
            [1 0]  [        0 1/2*sqrt2]  [0 0]
            [0 0], [1/2*sqrt2         0], [0 1]
            )

        ::

            sage: J = JordanSpinEJA(2)
            sage: J.basis()
            Finite family {0: e0, 1: e1}
            sage: J.natural_basis()
            (
            [1]  [0]
            [0], [1]
            )

        """
        if self._natural_basis is None:
            M = self.natural_basis_space()
            return tuple( M(b.to_vector()) for b in self.basis() )
        else:
            return self._natural_basis


    def natural_basis_space(self):
        """
        Return the matrix space in which this algebra's natural basis
        elements live.
        """
        if self._natural_basis is None or len(self._natural_basis) == 0:
            return MatrixSpace(self.base_ring(), self.dimension(), 1)
        else:
            return self._natural_basis[0].matrix_space()


    @staticmethod
    def natural_inner_product(X,Y):
        """
        Compute the inner product of two naturally-represented elements.

        For example in the real symmetric matrix EJA, this will compute
        the trace inner-product of two n-by-n symmetric matrices. The
        default should work for the real cartesian product EJA, the
        Jordan spin EJA, and the real symmetric matrices. The others
        will have to be overridden.
        """
        return (X.conjugate_transpose()*Y).trace()


    @cached_method
    def one(self):
        """
        Return the unit element of this algebra.

        SETUP::

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

        EXAMPLES::

            sage: J = HadamardEJA(5)
            sage: J.one()
            e0 + e1 + e2 + e3 + e4

        TESTS:

        The identity element acts like the identity::

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

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

            sage: set_random_seed()
            sage: J = random_eja()
            sage: actual = J.one().operator().matrix()
            sage: expected = matrix.identity(J.base_ring(), J.dimension())
            sage: actual == expected
            True

        """
        # We can brute-force compute the matrices of the operators
        # that correspond to the basis elements of this algebra.
        # If some linear combination of those basis elements is the
        # algebra identity, then the same linear combination of
        # their matrices has to be the identity matrix.
        #
        # Of course, matrices aren't vectors in sage, so we have to
        # appeal to the "long vectors" isometry.
        oper_vecs = [ _mat2vec(g.operator().matrix()) for g in self.gens() ]

        # Now we use basis linear algebra to find the coefficients,
        # of the matrices-as-vectors-linear-combination, which should
        # work for the original algebra basis too.
        A = matrix.column(self.base_ring(), oper_vecs)

        # We used the isometry on the left-hand side already, but we
        # still need to do it for the right-hand side. Recall that we
        # wanted something that summed to the identity matrix.
        b = _mat2vec( matrix.identity(self.base_ring(), self.dimension()) )

        # Now if there's an identity element in the algebra, this should work.
        coeffs = A.solve_right(b)
        return self.linear_combination(zip(self.gens(), coeffs))


    def peirce_decomposition(self, c):
        """
        The Peirce decomposition of this algebra relative to the
        idempotent ``c``.

        In the future, this can be extended to a complete system of
        orthogonal idempotents.

        INPUT:

          - ``c`` -- an idempotent of this algebra.

        OUTPUT:

        A triple (J0, J5, J1) containing two subalgebras and one subspace
        of this algebra,

          - ``J0`` -- the algebra on the eigenspace of ``c.operator()``
            corresponding to the eigenvalue zero.

          - ``J5`` -- the eigenspace (NOT a subalgebra) of ``c.operator()``
            corresponding to the eigenvalue one-half.

          - ``J1`` -- the algebra on the eigenspace of ``c.operator()``
            corresponding to the eigenvalue one.

        These are the only possible eigenspaces for that operator, and this
        algebra is a direct sum of them. The spaces ``J0`` and ``J1`` are
        orthogonal, and are subalgebras of this algebra with the appropriate
        restrictions.

        SETUP::

            sage: from mjo.eja.eja_algebra import random_eja, RealSymmetricEJA

        EXAMPLES:

        The canonical example comes from the symmetric matrices, which
        decompose into diagonal and off-diagonal parts::

            sage: J = RealSymmetricEJA(3)
            sage: C = matrix(QQ, [ [1,0,0],
            ....:                  [0,1,0],
            ....:                  [0,0,0] ])
            sage: c = J(C)
            sage: J0,J5,J1 = J.peirce_decomposition(c)
            sage: J0
            Euclidean Jordan algebra of dimension 1...
            sage: J5
            Vector space of degree 6 and dimension 2...
            sage: J1
            Euclidean Jordan algebra of dimension 3...

        TESTS:

        Every algebra decomposes trivially with respect to its identity
        element::

            sage: set_random_seed()
            sage: J = random_eja()
            sage: J0,J5,J1 = J.peirce_decomposition(J.one())
            sage: J0.dimension() == 0 and J5.dimension() == 0
            True
            sage: J1.superalgebra() == J and J1.dimension() == J.dimension()
            True

        The identity elements in the two subalgebras are the
        projections onto their respective subspaces of the
        superalgebra's identity element::

            sage: set_random_seed()
            sage: J = random_eja()
            sage: x = J.random_element()
            sage: if not J.is_trivial():
            ....:     while x.is_nilpotent():
            ....:         x = J.random_element()
            sage: c = x.subalgebra_idempotent()
            sage: J0,J5,J1 = J.peirce_decomposition(c)
            sage: J1(c) == J1.one()
            True
            sage: J0(J.one() - c) == J0.one()
            True

        """
        if not c.is_idempotent():
            raise ValueError("element is not idempotent: %s" % c)

        # Default these to what they should be if they turn out to be
        # trivial, because eigenspaces_left() won't return eigenvalues
        # corresponding to trivial spaces (e.g. it returns only the
        # eigenspace corresponding to lambda=1 if you take the
        # decomposition relative to the identity element).
        trivial = FiniteDimensionalEuclideanJordanSubalgebra(self, ())
        J0 = trivial                          # eigenvalue zero
        J5 = VectorSpace(self.base_ring(), 0) # eigenvalue one-half
        J1 = trivial                          # eigenvalue one

        for (eigval, eigspace) in c.operator().matrix().left_eigenspaces():
            if eigval == ~(self.base_ring()(2)):
                J5 = eigspace
            else:
                gens = tuple( self.from_vector(b) for b in eigspace.basis() )
                subalg = FiniteDimensionalEuclideanJordanSubalgebra(self, gens)
                if eigval == 0:
                    J0 = subalg
                elif eigval == 1:
                    J1 = subalg
                else:
                    raise ValueError("unexpected eigenvalue: %s" % eigval)

        return (J0, J5, J1)


    def a_jordan_frame(self):
        r"""
        Generate a Jordan frame for this algebra.

        This implementation is based on the so-called "central
        orthogonal idempotents" implemented for (semisimple) centers
        of SageMath ``FiniteDimensionalAlgebrasWithBasis``. Since all
        Euclidean Jordan algebas are commutative (and thus equal to
        their own centers) and semisimple, the method should work more
        or less as implemented, if it ever worked in the first place.
        (I don't know the justification for the original implementation.
        yet).

        How it works: we loop through the algebras generators, looking
        for their eigenspaces. If there's more than one eigenspace,
        and if they result in more than one subalgebra, then we split
        those subalgebras recursively until we get to subalgebras of
        dimension one (whose idempotent is the unit element). Why does
        some generator have to produce at least two subalgebras? I
        dunno. But it seems to work.

        Beware that Koecher defines the "center" of a Jordan algebra to
        be something else, because the usual definition is stupid in a
        (necessarily commutative) Jordan algebra.

        SETUP::

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

        EXAMPLES:

        A Jordan frame for the trivial algebra has to be empty
        (zero-length) since its rank is zero. More to the point, there
        are no non-zero idempotents in the trivial EJA. This does not
        cause any problems so long as we adopt the convention that the
        empty sum is zero, since then the sole element of the trivial
        EJA has an (empty) spectral decomposition::

            sage: J = TrivialEJA()
            sage: J.a_jordan_frame()
            ()

        A one-dimensional algebra has rank one (equal to its dimension),
        and only one primitive idempotent, namely the algebra's unit
        element::

            sage: J = JordanSpinEJA(1)
            sage: J.a_jordan_frame()
            (e0,)

        TESTS::

            sage: J = random_eja()
            sage: c = J.a_jordan_frame()
            sage: all( x^2 == x for x in c )
            True
            sage: r = len(c)
            sage: all( c[i]*c[j] == c[i]*(i==j) for i in range(r)
            ....:                               for j in range(r) )
            True

        """
        if self.dimension() == 0:
            return ()
        if self.dimension() == 1:
            return (self.one(),)

        for g in self.gens():
            eigenpairs = g.operator().matrix().right_eigenspaces()
            if len(eigenpairs) >= 2:
                subalgebras = []
                for eigval, eigspace in eigenpairs:
                    # Make sub-EJAs from the matrix eigenspaces...
                    sb = tuple( self.from_vector(b) for b in eigspace.basis() )
                    try:
                        # This will fail if e.g. the eigenspace basis
                        # contains two elements and their product
                        # isn't a linear combination of the two of
                        # them (i.e. the generated EJA isn't actually
                        # two dimensional).
                        s = FiniteDimensionalEuclideanJordanSubalgebra(self, sb)
                        subalgebras.append(s)
                    except:
                        pass
                if len(subalgebras) >= 2:
                    # apply this method recursively.
                    return tuple( c.superalgebra_element()
                                  for subalgebra in subalgebras
                                  for c in subalgebra.a_jordan_frame() )

        # If we got here, the algebra didn't decompose, at least not when we looked at
        # the eigenspaces corresponding only to basis elements of the algebra. The
        # implementation I stole says that this should work because of Schur's Lemma,
        # so I personally blame Schur's Lemma if it does not.
        raise Exception("Schur's Lemma didn't work!")


    def random_elements(self, count):
        """
        Return ``count`` random elements as a tuple.

        SETUP::

            sage: from mjo.eja.eja_algebra import JordanSpinEJA

        EXAMPLES::

            sage: J = JordanSpinEJA(3)
            sage: x,y,z = J.random_elements(3)
            sage: all( [ x in J, y in J, z in J ])
            True
            sage: len( J.random_elements(10) ) == 10
            True

        """
        return  tuple( self.random_element() for idx in range(count) )


    def rank(self):
        """
        Return the rank of this EJA.

        ALGORITHM:

        The author knows of no algorithm to compute the rank of an EJA
        where only the multiplication table is known. In lieu of one, we
        require the rank to be specified when the algebra is created,
        and simply pass along that number here.

        SETUP::

            sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
            ....:                                  RealSymmetricEJA,
            ....:                                  ComplexHermitianEJA,
            ....:                                  QuaternionHermitianEJA,
            ....:                                  random_eja)

        EXAMPLES:

        The rank of the Jordan spin algebra is always two::

            sage: JordanSpinEJA(2).rank()
            2
            sage: JordanSpinEJA(3).rank()
            2
            sage: JordanSpinEJA(4).rank()
            2

        The rank of the `n`-by-`n` Hermitian real, complex, or
        quaternion matrices is `n`::

            sage: RealSymmetricEJA(4).rank()
            4
            sage: ComplexHermitianEJA(3).rank()
            3
            sage: QuaternionHermitianEJA(2).rank()
            2

        TESTS:

        Ensure that every EJA that we know how to construct has a
        positive integer rank, unless the algebra is trivial in
        which case its rank will be zero::

            sage: set_random_seed()
            sage: J = random_eja()
            sage: r = J.rank()
            sage: r in ZZ
            True
            sage: r > 0 or (r == 0 and J.is_trivial())
            True

        """
        return self._rank


    def vector_space(self):
        """
        Return the vector space that underlies this algebra.

        SETUP::

            sage: from mjo.eja.eja_algebra import RealSymmetricEJA

        EXAMPLES::

            sage: J = RealSymmetricEJA(2)
            sage: J.vector_space()
            Vector space of dimension 3 over...

        """
        return self.zero().to_vector().parent().ambient_vector_space()


    Element = FiniteDimensionalEuclideanJordanAlgebraElement


class KnownRankEJA(object):
    """
    A class for algebras that we actually know we can construct.  The
    main issue is that, for most of our methods to make sense, we need
    to know the rank of our algebra. Thus we can't simply generate a
    "random" algebra, or even check that a given basis and product
    satisfy the axioms; because even if everything looks OK, we wouldn't
    know the rank we need to actuallty build the thing.

    Not really a subclass of FDEJA because doing that causes method
    resolution errors, e.g.

      TypeError: Error when calling the metaclass bases
      Cannot create a consistent method resolution
      order (MRO) for bases FiniteDimensionalEuclideanJordanAlgebra,
      KnownRankEJA

    """
    @staticmethod
    def _max_test_case_size():
        """
        Return an integer "size" that is an upper bound on the size of
        this algebra when it is used in a random test
        case. Unfortunately, the term "size" is quite vague -- when
        dealing with `R^n` under either the Hadamard or Jordan spin
        product, the "size" refers to the dimension `n`. When dealing
        with a matrix algebra (real symmetric or complex/quaternion
        Hermitian), it refers to the size of the matrix, which is
        far less than the dimension of the underlying vector space.

        We default to five in this class, which is safe in `R^n`. The
        matrix algebra subclasses (or any class where the "size" is
        interpreted to be far less than the dimension) should override
        with a smaller number.
        """
        return 5

    @classmethod
    def random_instance(cls, field=QQ, **kwargs):
        """
        Return a random instance of this type of algebra.

        Beware, this will crash for "most instances" because the
        constructor below looks wrong.
        """
        if cls is TrivialEJA:
            # The TrivialEJA class doesn't take an "n" argument because
            # there's only one.
            return cls(field)

        n = ZZ.random_element(cls._max_test_case_size()) + 1
        return cls(n, field, **kwargs)


class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA):
    """
    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=QQ, **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)
        return fdeja.__init__(field, mult_table, rank=n, **kwargs)

    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())


def random_eja(field=QQ, nontrivial=False):
    """
    Return a "random" finite-dimensional Euclidean Jordan Algebra.

    SETUP::

        sage: from mjo.eja.eja_algebra import random_eja

    TESTS::

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

    """
    eja_classes = KnownRankEJA.__subclasses__()
    if nontrivial:
        eja_classes.remove(TrivialEJA)
    classname = choice(eja_classes)
    return classname.random_instance(field=field)






class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra):
    @staticmethod
    def _max_test_case_size():
        # Play it safe, since this will be squared and the underlying
        # field can have dimension 4 (quaternions) too.
        return 2

    def __init__(self, field, basis, rank, normalize_basis=True, **kwargs):
        """
        Compared to the superclass constructor, we take a basis instead of
        a multiplication table because the latter can be computed in terms
        of the former when the product is known (like it is here).
        """
        # Used in this class's fast _charpoly_coeff() override.
        self._basis_normalizers = None

        # We're going to loop through this a few times, so now's a good
        # time to ensure that it isn't a generator expression.
        basis = tuple(basis)

        if rank > 1 and normalize_basis:
            # We'll need sqrt(2) to normalize the basis, and this
            # winds up in the multiplication table, so the whole
            # algebra needs to be over the field extension.
            R = PolynomialRing(field, 'z')
            z = R.gen()
            p = z**2 - 2
            if p.is_irreducible():
                field = field.extension(p, 'sqrt2', embedding=RLF(2).sqrt())
                basis = tuple( s.change_ring(field) for s in basis )
            self._basis_normalizers = tuple(
                ~(self.natural_inner_product(s,s).sqrt()) for s in basis )
            basis = tuple(s*c for (s,c) in zip(basis,self._basis_normalizers))

        Qs = self.multiplication_table_from_matrix_basis(basis)

        fdeja = super(MatrixEuclideanJordanAlgebra, self)
        return fdeja.__init__(field,
                              Qs,
                              rank=rank,
                              natural_basis=basis,
                              **kwargs)


    @cached_method
    def _charpoly_coeff(self, i):
        """
        Override the parent method with something that tries to compute
        over a faster (non-extension) field.
        """
        if self._basis_normalizers is None:
            # We didn't normalize, so assume that the basis we started
            # with had entries in a nice field.
            return super(MatrixEuclideanJordanAlgebra, self)._charpoly_coeff(i)
        else:
            basis = ( (b/n) for (b,n) in zip(self.natural_basis(),
                                             self._basis_normalizers) )

            # Do this over the rationals and convert back at the end.
            J = MatrixEuclideanJordanAlgebra(QQ,
                                             basis,
                                             self.rank(),
                                             normalize_basis=False)
            (_,x,_,_) = J._charpoly_matrix_system()
            p = J._charpoly_coeff(i)
            # p might be missing some vars, have to substitute "optionally"
            pairs = zip(x.base_ring().gens(), self._basis_normalizers)
            substitutions = { v: v*c for (v,c) in pairs }
            result = p.subs(substitutions)

            # The result of "subs" can be either a coefficient-ring
            # element or a polynomial. Gotta handle both cases.
            if result in QQ:
                return self.base_ring()(result)
            else:
                return result.change_ring(self.base_ring())


    @staticmethod
    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.
        """
        # 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_of_basis( _mat2vec(s) for s in basis )
        n = len(basis)
        mult_table = [[W.zero() for j in range(n)] for i in range(n)]
        for i in range(n):
            for j in range(n):
                mat_entry = (basis[i]*basis[j] + basis[j]*basis[i])/2
                mult_table[i][j] = W.coordinate_vector(_mat2vec(mat_entry))

        return mult_table


    @staticmethod
    def real_embed(M):
        """
        Embed the matrix ``M`` into a space of real matrices.

        The matrix ``M`` can have entries in any field at the moment:
        the real numbers, complex numbers, or quaternions. And although
        they are not a field, we can probably support octonions at some
        point, too. This function returns a real matrix that "acts like"
        the original with respect to matrix multiplication; i.e.

          real_embed(M*N) = real_embed(M)*real_embed(N)

        """
        raise NotImplementedError


    @staticmethod
    def real_unembed(M):
        """
        The inverse of :meth:`real_embed`.
        """
        raise NotImplementedError


    @classmethod
    def natural_inner_product(cls,X,Y):
        Xu = cls.real_unembed(X)
        Yu = cls.real_unembed(Y)
        tr = (Xu*Yu).trace()

        if tr in RLF:
            # It's real already.
            return tr

        # Otherwise, try the thing that works for complex numbers; and
        # if that doesn't work, the thing that works for quaternions.
        try:
            return tr.vector()[0] # real part, imag part is index 1
        except AttributeError:
            # A quaternions doesn't have a vector() method, but does
            # have coefficient_tuple() method that returns the
            # coefficients of 1, i, j, and k -- in that order.
            return tr.coefficient_tuple()[0]


class RealMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra):
    @staticmethod
    def real_embed(M):
        """
        The identity function, for embedding real matrices into real
        matrices.
        """
        return M

    @staticmethod
    def real_unembed(M):
        """
        The identity function, for unembedding real matrices from real
        matrices.
        """
        return M


class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA):
    """
    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.

    SETUP::

        sage: from mjo.eja.eja_algebra import RealSymmetricEJA

    EXAMPLES::

        sage: J = RealSymmetricEJA(2)
        sage: e0, e1, e2 = J.gens()
        sage: e0*e0
        e0
        sage: e1*e1
        1/2*e0 + 1/2*e2
        sage: e2*e2
        e2

    In theory, our "field" can be any subfield of the reals::

        sage: RealSymmetricEJA(2, AA)
        Euclidean Jordan algebra of dimension 3 over Algebraic Real Field
        sage: RealSymmetricEJA(2, RR)
        Euclidean Jordan algebra of dimension 3 over Real Field with
        53 bits of precision

    TESTS:

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

        sage: set_random_seed()
        sage: n_max = RealSymmetricEJA._max_test_case_size()
        sage: n = ZZ.random_element(1, n_max)
        sage: J = RealSymmetricEJA(n)
        sage: J.dimension() == (n^2 + n)/2
        True

    The Jordan multiplication is what we think it is::

        sage: set_random_seed()
        sage: J = RealSymmetricEJA.random_instance()
        sage: x,y = J.random_elements(2)
        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

    We can change the generator prefix::

        sage: RealSymmetricEJA(3, prefix='q').gens()
        (q0, q1, q2, q3, q4, q5)

    Our natural basis is normalized with respect to the natural inner
    product unless we specify otherwise::

        sage: set_random_seed()
        sage: J = RealSymmetricEJA.random_instance()
        sage: all( b.norm() == 1 for b in J.gens() )
        True

    Since our natural basis is normalized with respect to the natural
    inner product, and since we know that this algebra is an EJA, any
    left-multiplication operator's matrix will be symmetric because
    natural->EJA basis representation is an isometry and within the EJA
    the operator is self-adjoint by the Jordan axiom::

        sage: set_random_seed()
        sage: x = RealSymmetricEJA.random_instance().random_element()
        sage: x.operator().matrix().is_symmetric()
        True

    """
    @classmethod
    def _denormalized_basis(cls, n, field):
        """
        Return a basis for the space of real symmetric n-by-n matrices.

        SETUP::

            sage: from mjo.eja.eja_algebra import RealSymmetricEJA

        TESTS::

            sage: set_random_seed()
            sage: n = ZZ.random_element(1,5)
            sage: B = RealSymmetricEJA._denormalized_basis(n,QQ)
            sage: all( M.is_symmetric() for M in  B)
            True

        """
        # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
        # coordinates.
        S = []
        for i in range(n):
            for j in range(i+1):
                Eij = matrix(field, n, lambda k,l: k==i and l==j)
                if i == j:
                    Sij = Eij
                else:
                    Sij = Eij + Eij.transpose()
                S.append(Sij)
        return S


    @staticmethod
    def _max_test_case_size():
        return 4 # Dimension 10


    def __init__(self, n, field=QQ, **kwargs):
        basis = self._denormalized_basis(n, field)
        super(RealSymmetricEJA, self).__init__(field, basis, n, **kwargs)


class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra):
    @staticmethod
    def real_embed(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]]``.

        SETUP::

            sage: from mjo.eja.eja_algebra import \
            ....:   ComplexMatrixEuclideanJordanAlgebra

        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: ComplexMatrixEuclideanJordanAlgebra.real_embed(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_max = ComplexMatrixEuclideanJordanAlgebra._max_test_case_size()
            sage: n = ZZ.random_element(n_max)
            sage: F = QuadraticField(-1, 'i')
            sage: X = random_matrix(F, n)
            sage: Y = random_matrix(F, n)
            sage: Xe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X)
            sage: Ye = ComplexMatrixEuclideanJordanAlgebra.real_embed(Y)
            sage: XYe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X*Y)
            sage: Xe*Ye == XYe
            True

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

        # We don't need any adjoined elements...
        field = M.base_ring().base_ring()

        blocks = []
        for z in M.list():
            a = z.list()[0] # real part, I guess
            b = z.list()[1] # imag part, I guess
            blocks.append(matrix(field, 2, [[a,b],[-b,a]]))

        return matrix.block(field, n, blocks)


    @staticmethod
    def real_unembed(M):
        """
        The inverse of _embed_complex_matrix().

        SETUP::

            sage: from mjo.eja.eja_algebra import \
            ....:   ComplexMatrixEuclideanJordanAlgebra

        EXAMPLES::

            sage: A = matrix(QQ,[ [ 1,  2,   3,  4],
            ....:                 [-2,  1,  -4,  3],
            ....:                 [ 9,  10, 11, 12],
            ....:                 [-10, 9, -12, 11] ])
            sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(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: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
            sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == 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")

        # If "M" was normalized, its base ring might have roots
        # adjoined and they can stick around after unembedding.
        field = M.base_ring()
        R = PolynomialRing(field, 'z')
        z = R.gen()
        F = field.extension(z**2 + 1, 'i', embedding=CLF(-1).sqrt())
        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 range(n/2):
            for j in range(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)


    @classmethod
    def natural_inner_product(cls,X,Y):
        """
        Compute a natural inner product in this algebra directly from
        its real embedding.

        SETUP::

            sage: from mjo.eja.eja_algebra import ComplexHermitianEJA

        TESTS:

        This gives the same answer as the slow, default method implemented
        in :class:`MatrixEuclideanJordanAlgebra`::

            sage: set_random_seed()
            sage: J = ComplexHermitianEJA.random_instance()
            sage: x,y = J.random_elements(2)
            sage: Xe = x.natural_representation()
            sage: Ye = y.natural_representation()
            sage: X = ComplexHermitianEJA.real_unembed(Xe)
            sage: Y = ComplexHermitianEJA.real_unembed(Ye)
            sage: expected = (X*Y).trace().vector()[0]
            sage: actual = ComplexHermitianEJA.natural_inner_product(Xe,Ye)
            sage: actual == expected
            True

        """
        return RealMatrixEuclideanJordanAlgebra.natural_inner_product(X,Y)/2


class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA):
    """
    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.

    SETUP::

        sage: from mjo.eja.eja_algebra import ComplexHermitianEJA

    EXAMPLES:

    In theory, our "field" can be any subfield of the reals::

        sage: ComplexHermitianEJA(2, AA)
        Euclidean Jordan algebra of dimension 4 over Algebraic Real Field
        sage: ComplexHermitianEJA(2, RR)
        Euclidean Jordan algebra of dimension 4 over Real Field with
        53 bits of precision

    TESTS:

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

        sage: set_random_seed()
        sage: n_max = ComplexHermitianEJA._max_test_case_size()
        sage: n = ZZ.random_element(1, n_max)
        sage: J = ComplexHermitianEJA(n)
        sage: J.dimension() == n^2
        True

    The Jordan multiplication is what we think it is::

        sage: set_random_seed()
        sage: J = ComplexHermitianEJA.random_instance()
        sage: x,y = J.random_elements(2)
        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

    We can change the generator prefix::

        sage: ComplexHermitianEJA(2, prefix='z').gens()
        (z0, z1, z2, z3)

    Our natural basis is normalized with respect to the natural inner
    product unless we specify otherwise::

        sage: set_random_seed()
        sage: J = ComplexHermitianEJA.random_instance()
        sage: all( b.norm() == 1 for b in J.gens() )
        True

    Since our natural basis is normalized with respect to the natural
    inner product, and since we know that this algebra is an EJA, any
    left-multiplication operator's matrix will be symmetric because
    natural->EJA basis representation is an isometry and within the EJA
    the operator is self-adjoint by the Jordan axiom::

        sage: set_random_seed()
        sage: x = ComplexHermitianEJA.random_instance().random_element()
        sage: x.operator().matrix().is_symmetric()
        True

    """

    @classmethod
    def _denormalized_basis(cls, n, field):
        """
        Returns a basis for the space of complex Hermitian n-by-n matrices.

        Why do we embed these? Basically, because all of numerical linear
        algebra assumes that you're working with vectors consisting of `n`
        entries from a field and scalars from the same field. There's no way
        to tell SageMath that (for example) the vectors contain complex
        numbers, while the scalar field is real.

        SETUP::

            sage: from mjo.eja.eja_algebra import ComplexHermitianEJA

        TESTS::

            sage: set_random_seed()
            sage: n = ZZ.random_element(1,5)
            sage: field = QuadraticField(2, 'sqrt2')
            sage: B = ComplexHermitianEJA._denormalized_basis(n, field)
            sage: all( M.is_symmetric() for M in  B)
            True

        """
        R = PolynomialRing(field, 'z')
        z = R.gen()
        F = field.extension(z**2 + 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 range(n):
            for j in range(i+1):
                Eij = matrix(F, n, lambda k,l: k==i and l==j)
                if i == j:
                    Sij = cls.real_embed(Eij)
                    S.append(Sij)
                else:
                    # The second one has a minus because it's conjugated.
                    Sij_real = cls.real_embed(Eij + Eij.transpose())
                    S.append(Sij_real)
                    Sij_imag = cls.real_embed(I*Eij - I*Eij.transpose())
                    S.append(Sij_imag)

        # Since we embedded these, we can drop back to the "field" that we
        # started with instead of the complex extension "F".
        return ( s.change_ring(field) for s in S )


    def __init__(self, n, field=QQ, **kwargs):
        basis = self._denormalized_basis(n,field)
        super(ComplexHermitianEJA,self).__init__(field, basis, n, **kwargs)


class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra):
    @staticmethod
    def real_embed(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.

        SETUP::

            sage: from mjo.eja.eja_algebra import \
            ....:   QuaternionMatrixEuclideanJordanAlgebra

        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: QuaternionMatrixEuclideanJordanAlgebra.real_embed(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_max = QuaternionMatrixEuclideanJordanAlgebra._max_test_case_size()
            sage: n = ZZ.random_element(n_max)
            sage: Q = QuaternionAlgebra(QQ,-1,-1)
            sage: X = random_matrix(Q, n)
            sage: Y = random_matrix(Q, n)
            sage: Xe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X)
            sage: Ye = QuaternionMatrixEuclideanJordanAlgebra.real_embed(Y)
            sage: XYe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X*Y)
            sage: Xe*Ye == XYe
            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]
            cplxM = matrix(F, 2, [[ a + b*i, c + d*i],
                                 [-c + d*i, a - b*i]])
            realM = ComplexMatrixEuclideanJordanAlgebra.real_embed(cplxM)
            blocks.append(realM)

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



    @staticmethod
    def real_unembed(M):
        """
        The inverse of _embed_quaternion_matrix().

        SETUP::

            sage: from mjo.eja.eja_algebra import \
            ....:   QuaternionMatrixEuclideanJordanAlgebra

        EXAMPLES::

            sage: M = matrix(QQ, [[ 1,  2,  3,  4],
            ....:                 [-2,  1, -4,  3],
            ....:                 [-3,  4,  1, -2],
            ....:                 [-4, -3,  2,  1]])
            sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(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: Me = QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
            sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(Me) == 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 quaternion embedding")

        # Use the base ring of the matrix to ensure that its entries can be
        # multiplied by elements of the quaternion algebra.
        field = M.base_ring()
        Q = QuaternionAlgebra(field,-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 range(n/4):
            for m in range(n/4):
                submat = ComplexMatrixEuclideanJordanAlgebra.real_unembed(
                    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].vector()[0]   # real part
                z += submat[0,0].vector()[1]*i # imag part
                z += submat[0,1].vector()[0]*j # real part
                z += submat[0,1].vector()[1]*k # imag part
                elements.append(z)

        return matrix(Q, n/4, elements)


    @classmethod
    def natural_inner_product(cls,X,Y):
        """
        Compute a natural inner product in this algebra directly from
        its real embedding.

        SETUP::

            sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA

        TESTS:

        This gives the same answer as the slow, default method implemented
        in :class:`MatrixEuclideanJordanAlgebra`::

            sage: set_random_seed()
            sage: J = QuaternionHermitianEJA.random_instance()
            sage: x,y = J.random_elements(2)
            sage: Xe = x.natural_representation()
            sage: Ye = y.natural_representation()
            sage: X = QuaternionHermitianEJA.real_unembed(Xe)
            sage: Y = QuaternionHermitianEJA.real_unembed(Ye)
            sage: expected = (X*Y).trace().coefficient_tuple()[0]
            sage: actual = QuaternionHermitianEJA.natural_inner_product(Xe,Ye)
            sage: actual == expected
            True

        """
        return RealMatrixEuclideanJordanAlgebra.natural_inner_product(X,Y)/4


class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra,
                             KnownRankEJA):
    """
    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.

    SETUP::

        sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA

    EXAMPLES:

    In theory, our "field" can be any subfield of the reals::

        sage: QuaternionHermitianEJA(2, AA)
        Euclidean Jordan algebra of dimension 6 over Algebraic Real Field
        sage: QuaternionHermitianEJA(2, RR)
        Euclidean Jordan algebra of dimension 6 over Real Field with
        53 bits of precision

    TESTS:

    The dimension of this algebra is `2*n^2 - n`::

        sage: set_random_seed()
        sage: n_max = QuaternionHermitianEJA._max_test_case_size()
        sage: n = ZZ.random_element(1, n_max)
        sage: J = QuaternionHermitianEJA(n)
        sage: J.dimension() == 2*(n^2) - n
        True

    The Jordan multiplication is what we think it is::

        sage: set_random_seed()
        sage: J = QuaternionHermitianEJA.random_instance()
        sage: x,y = J.random_elements(2)
        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

    We can change the generator prefix::

        sage: QuaternionHermitianEJA(2, prefix='a').gens()
        (a0, a1, a2, a3, a4, a5)

    Our natural basis is normalized with respect to the natural inner
    product unless we specify otherwise::

        sage: set_random_seed()
        sage: J = QuaternionHermitianEJA.random_instance()
        sage: all( b.norm() == 1 for b in J.gens() )
        True

    Since our natural basis is normalized with respect to the natural
    inner product, and since we know that this algebra is an EJA, any
    left-multiplication operator's matrix will be symmetric because
    natural->EJA basis representation is an isometry and within the EJA
    the operator is self-adjoint by the Jordan axiom::

        sage: set_random_seed()
        sage: x = QuaternionHermitianEJA.random_instance().random_element()
        sage: x.operator().matrix().is_symmetric()
        True

    """
    @classmethod
    def _denormalized_basis(cls, n, field):
        """
        Returns a basis for the space of quaternion Hermitian n-by-n matrices.

        Why do we embed these? Basically, because all of numerical
        linear algebra assumes that you're working with vectors consisting
        of `n` entries from a field and scalars from the same field. There's
        no way to tell SageMath that (for example) the vectors contain
        complex numbers, while the scalar field is real.

        SETUP::

            sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA

        TESTS::

            sage: set_random_seed()
            sage: n = ZZ.random_element(1,5)
            sage: B = QuaternionHermitianEJA._denormalized_basis(n,QQ)
            sage: all( M.is_symmetric() for M in B )
            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 range(n):
            for j in range(i+1):
                Eij = matrix(Q, n, lambda k,l: k==i and l==j)
                if i == j:
                    Sij = cls.real_embed(Eij)
                    S.append(Sij)
                else:
                    # The second, third, and fourth ones have a minus
                    # because they're conjugated.
                    Sij_real = cls.real_embed(Eij + Eij.transpose())
                    S.append(Sij_real)
                    Sij_I = cls.real_embed(I*Eij - I*Eij.transpose())
                    S.append(Sij_I)
                    Sij_J = cls.real_embed(J*Eij - J*Eij.transpose())
                    S.append(Sij_J)
                    Sij_K = cls.real_embed(K*Eij - K*Eij.transpose())
                    S.append(Sij_K)

        # Since we embedded these, we can drop back to the "field" that we
        # started with instead of the quaternion algebra "Q".
        return ( s.change_ring(field) for s in S )


    def __init__(self, n, field=QQ, **kwargs):
        basis = self._denormalized_basis(n,field)
        super(QuaternionHermitianEJA,self).__init__(field, basis, n, **kwargs)


class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA):
    r"""
    The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
    with the half-trace inner product and jordan product ``x*y =
    (x0*y0 + <B*x_bar,y_bar>, x0*y_bar + y0*x_bar)`` where ``B`` is a
    symmetric positive-definite "bilinear form" matrix. It has
    dimension `n` over the reals, and reduces to the ``JordanSpinEJA``
    when ``B`` is the identity matrix of order ``n-1``.

    SETUP::

        sage: from mjo.eja.eja_algebra import (BilinearFormEJA,
        ....:                                  JordanSpinEJA)

    EXAMPLES:

    When no bilinear form is specified, the identity matrix is used,
    and the resulting algebra is the Jordan spin algebra::

        sage: J0 = BilinearFormEJA(3)
        sage: J1 = JordanSpinEJA(3)
        sage: J0.multiplication_table() == J0.multiplication_table()
        True

    TESTS:

    We can create a zero-dimensional algebra::

        sage: J = BilinearFormEJA(0)
        sage: J.basis()
        Finite family {}

    We can check the multiplication condition given in the Jordan, von
    Neumann, and Wigner paper (and also discussed on my "On the
    symmetry..." paper). Note that this relies heavily on the standard
    choice of basis, as does anything utilizing the bilinear form matrix::

        sage: set_random_seed()
        sage: n = ZZ.random_element(5)
        sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
        sage: B = M.transpose()*M
        sage: J = BilinearFormEJA(n, B=B)
        sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis()
        sage: V = J.vector_space()
        sage: sis = [ J.from_vector(V([0] + (M.inverse()*ei).list()))
        ....:         for ei in eis ]
        sage: actual = [ sis[i]*sis[j]
        ....:            for i in range(n-1)
        ....:            for j in range(n-1) ]
        sage: expected = [ J.one() if i == j else J.zero()
        ....:              for i in range(n-1)
        ....:              for j in range(n-1) ]
        sage: actual == expected
        True
    """
    def __init__(self, n, field=QQ, B=None, **kwargs):
        if B is None:
            self._B = matrix.identity(field, max(0,n-1))
        else:
            self._B = B

        V = VectorSpace(field, n)
        mult_table = [[V.zero() for j in range(n)] for i in range(n)]
        for i in range(n):
            for j in range(n):
                x = V.gen(i)
                y = V.gen(j)
                x0 = x[0]
                xbar = x[1:]
                y0 = y[0]
                ybar = y[1:]
                z0 = x0*y0 + (self._B*xbar).inner_product(ybar)
                zbar = y0*xbar + x0*ybar
                z = V([z0] + zbar.list())
                mult_table[i][j] = z

        # The rank of this algebra is two, unless we're in a
        # one-dimensional ambient space (because the rank is bounded
        # by the ambient dimension).
        fdeja = super(BilinearFormEJA, self)
        return fdeja.__init__(field, mult_table, rank=min(n,2), **kwargs)

    def inner_product(self, x, y):
        r"""
        Half of the trace inner product.

        This is defined so that the special case of the Jordan spin
        algebra gets the usual inner product.

        SETUP::

            sage: from mjo.eja.eja_algebra import BilinearFormEJA

        TESTS:

        Ensure that this is one-half of the trace inner-product when
        the algebra isn't just the reals (when ``n`` isn't one). This
        is in Faraut and Koranyi, and also my "On the symmetry..."
        paper::

            sage: set_random_seed()
            sage: n = ZZ.random_element(2,5)
            sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
            sage: B = M.transpose()*M
            sage: J = BilinearFormEJA(n, B=B)
            sage: x = J.random_element()
            sage: y = J.random_element()
            sage: x.inner_product(y) == (x*y).trace()/2
            True

        """
        xvec = x.to_vector()
        xbar = xvec[1:]
        yvec = y.to_vector()
        ybar = yvec[1:]
        return x[0]*y[0] + (self._B*xbar).inner_product(ybar)


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

    SETUP::

        sage: from mjo.eja.eja_algebra import JordanSpinEJA

    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

    We can change the generator prefix::

        sage: JordanSpinEJA(2, prefix='B').gens()
        (B0, B1)

    TESTS:

        Ensure that we have the usual inner product on `R^n`::

            sage: set_random_seed()
            sage: J = JordanSpinEJA.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

    """
    def __init__(self, n, field=QQ, **kwargs):
        # This is a special case of the BilinearFormEJA with the identity
        # matrix as its bilinear form.
        return super(JordanSpinEJA, self).__init__(n, field, **kwargs)


class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA):
    """
    The trivial Euclidean Jordan algebra consisting of only a zero element.

    SETUP::

        sage: from mjo.eja.eja_algebra import TrivialEJA

    EXAMPLES::

        sage: J = TrivialEJA()
        sage: J.dimension()
        0
        sage: J.zero()
        0
        sage: J.one()
        0
        sage: 7*J.one()*12*J.one()
        0
        sage: J.one().inner_product(J.one())
        0
        sage: J.one().norm()
        0
        sage: J.one().subalgebra_generated_by()
        Euclidean Jordan algebra of dimension 0 over Rational Field
        sage: J.rank()
        0

    """
    def __init__(self, field=QQ, **kwargs):
        mult_table = []
        fdeja = super(TrivialEJA, self)
        # The rank is zero using my definition, namely the dimension of the
        # largest subalgebra generated by any element.
        return fdeja.__init__(field, mult_table, rank=0, **kwargs)