[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 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.misc.cachefunc import cached_method
from sage.misc.prandom import choice
from sage.misc.table import table
from sage.modules.free_module import FreeModule, VectorSpace
from sage.rings.integer_ring import ZZ
from sage.rings.number_field.number_field import QuadraticField
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.rings.rational_field import QQ
from sage.structure.element import is_Matrix

from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
from mjo.eja.eja_utils import _mat2vec

class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
    # This is an ugly hack needed to prevent the category framework
    # from implementing a coercion from our base ring (e.g. the
    # rationals) into the algebra. First of all -- such a coercion is
    # nonsense to begin with. But more importantly, it tries to do so
    # in the category of rings, and since our algebras aren't
    # associative they generally won't be rings.
    _no_generic_basering_coercion = True

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

            sage: from mjo.eja.eja_algebra import random_eja

        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

        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 = [ 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,
            ....:                                  RealCartesianProductEJA,
            ....:                                  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 = RealCartesianProductEJA(5)
            sage: x = J.random_element()
            sage: J(x.to_vector().column()) == x
            True
            sage: J = JordanSpinEJA(5)
            sage: x = J.random_element()
            sage: J(x.to_vector().column()) == x
            True

        """
        if elt == 0:
            # The superclass implementation of random_element()
            # needs to be able to coerce "0" into the algebra.
            return self.zero()

        natural_basis = self.natural_basis()
        if elt not in natural_basis[0].matrix_space():
            raise ValueError("not a naturally-represented algebra element")

        # 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()*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():
            # 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

        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

        """
        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.

        SETUP::

            sage: from mjo.eja.eja_algebra import random_eja

        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 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

        EXAMPLES::

            sage: J = ComplexHermitianEJA(3)
            sage: J.is_trivial()
            False
            sage: A = J.zero().subalgebra_generated_by()
            sage: A.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]  [0 0]
            [0 0], [1 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:
            return tuple( b.to_vector().column() for b in self.basis() )
        else:
            return self._natural_basis


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

        SETUP::

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

        EXAMPLES::

            sage: J = RealCartesianProductEJA(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 random_element(self):
        # Temporary workaround for https://trac.sagemath.org/ticket/28327
        if self.is_trivial():
            return self.zero()
        else:
            s = super(FiniteDimensionalEuclideanJordanAlgebra, self)
            return s.random_element()


    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(2).rank()
            2
            sage: ComplexHermitianEJA(2).rank()
            2
            sage: QuaternionHermitianEJA(2).rank()
            2
            sage: RealSymmetricEJA(5).rank()
            5
            sage: ComplexHermitianEJA(5).rank()
            5
            sage: QuaternionHermitianEJA(5).rank()
            5

        TESTS:

        Ensure that every EJA that we know how to construct has a
        positive integer rank::

            sage: set_random_seed()
            sage: r = random_eja().rank()
            sage: r in ZZ and r > 0
            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 Rational Field

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


    Element = FiniteDimensionalEuclideanJordanAlgebraElement


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.

    SETUP::

        sage: from mjo.eja.eja_algebra import RealCartesianProductEJA

    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

    TESTS:

    We can change the generator prefix::

        sage: RealCartesianProductEJA(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(RealCartesianProductEJA, self)
        return fdeja.__init__(field, mult_table, rank=n, **kwargs)

    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.

    SETUP::

        sage: from mjo.eja.eja_algebra import random_eja

    TESTS::

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

    """

    # 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.

    SETUP::

        sage: from mjo.eja.eja_algebra import _complex_hermitian_basis

    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.

    SETUP::

        sage: from mjo.eja.eja_algebra import _quaternion_hermitian_basis

    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 _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


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]]``.

    SETUP::

        sage: from mjo.eja.eja_algebra import _embed_complex_matrix

    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 matrix.block(field.base_ring(), n, blocks)


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

    SETUP::

        sage: from mjo.eja.eja_algebra import (_embed_complex_matrix,
        ....:                                  _unembed_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.

    SETUP::

        sage: from mjo.eja.eja_algebra import _embed_quaternion_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 matrix.block(quaternions.base_ring(), n, blocks)


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

    SETUP::

        sage: from mjo.eja.eja_algebra import (_embed_quaternion_matrix,
        ....:                                  _unembed_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.to_vector().inner_product(y.to_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.

    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
        e0 + e2
        sage: e2*e2
        e2

    TESTS:

    The dimension 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.dimension() == (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

    We can change the generator prefix::

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

    """
    def __init__(self, n, field=QQ, **kwargs):
        S = _real_symmetric_basis(n, field=field)
        Qs = _multiplication_table_from_matrix_basis(S)

        fdeja = super(RealSymmetricEJA, self)
        return fdeja.__init__(field,
                              Qs,
                              rank=n,
                              natural_basis=S,
                              **kwargs)

    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.

    SETUP::

        sage: from mjo.eja.eja_algebra import ComplexHermitianEJA

    TESTS:

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

        sage: set_random_seed()
        sage: n = ZZ.random_element(1,5)
        sage: J = ComplexHermitianEJA(n)
        sage: J.dimension() == 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

    We can change the generator prefix::

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

    """
    def __init__(self, n, field=QQ, **kwargs):
        S = _complex_hermitian_basis(n)
        Qs = _multiplication_table_from_matrix_basis(S)

        fdeja = super(ComplexHermitianEJA, self)
        return fdeja.__init__(field,
                              Qs,
                              rank=n,
                              natural_basis=S,
                              **kwargs)


    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.

    SETUP::

        sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA

    TESTS:

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

        sage: set_random_seed()
        sage: n = ZZ.random_element(1,5)
        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: 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

    We can change the generator prefix::

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

    """
    def __init__(self, n, field=QQ, **kwargs):
        S = _quaternion_hermitian_basis(n)
        Qs = _multiplication_table_from_matrix_basis(S)

        fdeja = super(QuaternionHermitianEJA, self)
        return fdeja.__init__(field,
                              Qs,
                              rank=n,
                              natural_basis=S,
                              **kwargs)

    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.

    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)

    """
    def __init__(self, n, field=QQ, **kwargs):
        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:]
                # z = x*y
                z0 = x.inner_product(y)
                zbar = y0*xbar + x0*ybar
                z = V([z0] + zbar.list())
                mult_table[i][j] = z

        # 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, self)
        return fdeja.__init__(field, mult_table, rank=min(n,2), **kwargs)

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