+ Vector space of dimension 3 over...
+
+ """
+ return self.zero().to_vector().parent().ambient_vector_space()
+
+
+
+class RationalBasisEJA(FiniteDimensionalEJA):
+ r"""
+ New class for algebras whose supplied basis elements have all rational entries.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import BilinearFormEJA
+
+ EXAMPLES:
+
+ The supplied basis is orthonormalized by default::
+
+ sage: B = matrix(QQ, [[1, 0, 0], [0, 25, -32], [0, -32, 41]])
+ sage: J = BilinearFormEJA(B)
+ sage: J.matrix_basis()
+ (
+ [1] [ 0] [ 0]
+ [0] [1/5] [32/5]
+ [0], [ 0], [ 5]
+ )
+
+ """
+ def __init__(self,
+ basis,
+ jordan_product,
+ inner_product,
+ field=AA,
+ check_field=True,
+ **kwargs):
+
+ if check_field:
+ # Abuse the check_field parameter to check that the entries of
+ # out basis (in ambient coordinates) are in the field QQ.
+ if not all( all(b_i in QQ for b_i in b.list()) for b in basis ):
+ raise TypeError("basis not rational")
+
+ self._rational_algebra = None
+ if field is not QQ:
+ # There's no point in constructing the extra algebra if this
+ # one is already rational.
+ #
+ # Note: the same Jordan and inner-products work here,
+ # because they are necessarily defined with respect to
+ # ambient coordinates and not any particular basis.
+ self._rational_algebra = FiniteDimensionalEJA(
+ basis,
+ jordan_product,
+ inner_product,
+ field=QQ,
+ orthonormalize=False,
+ check_field=False,
+ check_axioms=False)
+
+ super().__init__(basis,
+ jordan_product,
+ inner_product,
+ field=field,
+ check_field=check_field,
+ **kwargs)
+
+ @cached_method
+ def _charpoly_coefficients(self):
+ r"""
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (BilinearFormEJA,
+ ....: JordanSpinEJA)
+
+ EXAMPLES:
+
+ The base ring of the resulting polynomial coefficients is what
+ it should be, and not the rationals (unless the algebra was
+ already over the rationals)::
+
+ sage: J = JordanSpinEJA(3)
+ sage: J._charpoly_coefficients()
+ (X1^2 - X2^2 - X3^2, -2*X1)
+ sage: a0 = J._charpoly_coefficients()[0]
+ sage: J.base_ring()
+ Algebraic Real Field
+ sage: a0.base_ring()
+ Algebraic Real Field
+
+ """
+ if self._rational_algebra is None:
+ # There's no need to construct *another* algebra over the
+ # rationals if this one is already over the
+ # rationals. Likewise, if we never orthonormalized our
+ # basis, we might as well just use the given one.
+ return super()._charpoly_coefficients()
+
+ # Do the computation over the rationals. The answer will be
+ # the same, because all we've done is a change of basis.
+ # Then, change back from QQ to our real base ring
+ a = ( a_i.change_ring(self.base_ring())
+ for a_i in self._rational_algebra._charpoly_coefficients() )
+
+ if self._deortho_matrix is None:
+ # This can happen if our base ring was, say, AA and we
+ # chose not to (or didn't need to) orthonormalize. It's
+ # still faster to do the computations over QQ even if
+ # the numbers in the boxes stay the same.
+ return tuple(a)
+
+ # Otherwise, convert the coordinate variables back to the
+ # deorthonormalized ones.
+ R = self.coordinate_polynomial_ring()
+ from sage.modules.free_module_element import vector
+ X = vector(R, R.gens())
+ BX = self._deortho_matrix*X
+
+ subs_dict = { X[i]: BX[i] for i in range(len(X)) }
+ return tuple( a_i.subs(subs_dict) for a_i in a )
+
+class ConcreteEJA(RationalBasisEJA):
+ r"""
+ A class for the Euclidean Jordan algebras that we know by name.
+
+ These are the Jordan algebras whose basis, multiplication table,
+ rank, and so on are known a priori. More to the point, they are
+ the Euclidean Jordan algebras for which we are able to conjure up
+ a "random instance."
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import ConcreteEJA
+
+ TESTS:
+
+ Our basis is normalized with respect to the algebra's inner
+ product, unless we specify otherwise::
+
+ sage: set_random_seed()
+ sage: J = ConcreteEJA.random_instance()
+ sage: all( b.norm() == 1 for b in J.gens() )
+ True
+
+ Since our basis is orthonormal with respect to the algebra's 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: J = ConcreteEJA.random_instance()
+ sage: x = J.random_element()
+ sage: x.operator().is_self_adjoint()
+ True
+ """
+
+ @staticmethod
+ def _max_random_instance_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 ambiguous -- 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.
+
+ This method must be implemented in each subclass.
+ """
+ raise NotImplementedError
+
+ @classmethod
+ def random_instance(cls, *args, **kwargs):
+ """
+ Return a random instance of this type of algebra.
+
+ This method should be implemented in each subclass.
+ """
+ from sage.misc.prandom import choice
+ eja_class = choice(cls.__subclasses__())
+
+ # These all bubble up to the RationalBasisEJA superclass
+ # constructor, so any (kw)args valid there are also valid
+ # here.
+ return eja_class.random_instance(*args, **kwargs)
+
+
+class MatrixEJA:
+ @staticmethod
+ def dimension_over_reals():
+ r"""
+ The dimension of this matrix's base ring over the reals.
+
+ The reals are dimension one over themselves, obviously; that's
+ just `\mathbb{R}^{1}`. Likewise, the complex numbers `a + bi`
+ have dimension two. Finally, the quaternions have dimension
+ four over the reals.
+
+ This is used to determine the size of the matrix returned from
+ :meth:`real_embed`, among other things.
+ """
+ raise NotImplementedError
+
+ @classmethod
+ def real_embed(cls,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)
+
+ """
+ if M.ncols() != M.nrows():
+ raise ValueError("the matrix 'M' must be square")
+ return M
+
+
+ @classmethod
+ def real_unembed(cls,M):
+ """
+ The inverse of :meth:`real_embed`.
+ """
+ if M.ncols() != M.nrows():
+ raise ValueError("the matrix 'M' must be square")
+ if not ZZ(M.nrows()).mod(cls.dimension_over_reals()).is_zero():
+ raise ValueError("the matrix 'M' must be a real embedding")
+ return M
+
+ @staticmethod
+ def jordan_product(X,Y):
+ return (X*Y + Y*X)/2
+
+ @classmethod
+ def trace_inner_product(cls,X,Y):
+ r"""
+ Compute the trace inner-product of two real-embeddings.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (RealSymmetricEJA,
+ ....: ComplexHermitianEJA,
+ ....: QuaternionHermitianEJA)
+
+ EXAMPLES::
+
+ This gives the same answer as it would if we computed the trace
+ from the unembedded (original) matrices::
+
+ sage: set_random_seed()
+ sage: J = RealSymmetricEJA.random_instance()
+ sage: x,y = J.random_elements(2)
+ sage: Xe = x.to_matrix()
+ sage: Ye = y.to_matrix()
+ sage: X = J.real_unembed(Xe)
+ sage: Y = J.real_unembed(Ye)
+ sage: expected = (X*Y).trace()
+ sage: actual = J.trace_inner_product(Xe,Ye)
+ sage: actual == expected
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: J = ComplexHermitianEJA.random_instance()
+ sage: x,y = J.random_elements(2)
+ sage: Xe = x.to_matrix()
+ sage: Ye = y.to_matrix()
+ sage: X = J.real_unembed(Xe)
+ sage: Y = J.real_unembed(Ye)
+ sage: expected = (X*Y).trace().real()
+ sage: actual = J.trace_inner_product(Xe,Ye)
+ sage: actual == expected
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: J = QuaternionHermitianEJA.random_instance()
+ sage: x,y = J.random_elements(2)
+ sage: Xe = x.to_matrix()
+ sage: Ye = y.to_matrix()
+ sage: X = J.real_unembed(Xe)
+ sage: Y = J.real_unembed(Ye)
+ sage: expected = (X*Y).trace().coefficient_tuple()[0]
+ sage: actual = J.trace_inner_product(Xe,Ye)
+ sage: actual == expected
+ True
+
+ """
+ Xu = cls.real_unembed(X)
+ Yu = cls.real_unembed(Y)
+ tr = (Xu*Yu).trace()
+
+ try:
+ # Works in QQ, AA, RDF, et cetera.
+ return tr.real()
+ except AttributeError:
+ # A quaternion doesn't have a real() 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 RealMatrixEJA(MatrixEJA):
+ @staticmethod
+ def dimension_over_reals():
+ return 1
+
+
+class RealSymmetricEJA(ConcreteEJA, RealMatrixEJA):
+ """
+ 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, field=RDF, check_axioms=True)
+ Euclidean Jordan algebra of dimension 3 over Real Double Field
+ sage: RealSymmetricEJA(2, field=RR, check_axioms=True)
+ 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_random_instance_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).to_matrix()
+ sage: X = x.to_matrix()
+ sage: Y = y.to_matrix()
+ 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)
+
+ We can construct the (trivial) algebra of rank zero::
+
+ sage: RealSymmetricEJA(0)
+ Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
+
+ """
+ @classmethod
+ def _denormalized_basis(cls, n):
+ """
+ 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)
+ 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(ZZ, n, lambda k,l: k==i and l==j)
+ if i == j:
+ Sij = Eij
+ else:
+ Sij = Eij + Eij.transpose()
+ S.append(Sij)
+ return tuple(S)
+
+
+ @staticmethod
+ def _max_random_instance_size():
+ return 4 # Dimension 10
+
+ @classmethod
+ def random_instance(cls, **kwargs):
+ """
+ Return a random instance of this type of algebra.
+ """
+ n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ return cls(n, **kwargs)
+
+ def __init__(self, n, **kwargs):
+ # We know this is a valid EJA, but will double-check
+ # if the user passes check_axioms=True.
+ if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
+
+ super(RealSymmetricEJA, self).__init__(self._denormalized_basis(n),
+ self.jordan_product,
+ self.trace_inner_product,
+ **kwargs)
+
+ # TODO: this could be factored out somehow, but is left here
+ # because the MatrixEJA is not presently a subclass of the
+ # FDEJA class that defines rank() and one().
+ self.rank.set_cache(n)
+ idV = matrix.identity(ZZ, self.dimension_over_reals()*n)
+ self.one.set_cache(self(idV))
+
+
+
+class ComplexMatrixEJA(MatrixEJA):
+ # A manual dictionary-cache for the complex_extension() method,
+ # since apparently @classmethods can't also be @cached_methods.
+ _complex_extension = {}
+
+ @classmethod
+ def complex_extension(cls,field):
+ r"""
+ The complex field that we embed/unembed, as an extension
+ of the given ``field``.
+ """
+ if field in cls._complex_extension:
+ return cls._complex_extension[field]
+
+ # Sage doesn't know how to adjoin the complex "i" (the root of
+ # x^2 + 1) to a field in a general way. Here, we just enumerate
+ # all of the cases that I have cared to support so far.
+ if field is AA:
+ # Sage doesn't know how to embed AA into QQbar, i.e. how
+ # to adjoin sqrt(-1) to AA.
+ F = QQbar
+ elif not field.is_exact():
+ # RDF or RR
+ F = field.complex_field()
+ else:
+ # Works for QQ and... maybe some other fields.
+ R = PolynomialRing(field, 'z')
+ z = R.gen()
+ F = field.extension(z**2 + 1, 'I', embedding=CLF(-1).sqrt())
+
+ cls._complex_extension[field] = F
+ return F
+
+ @staticmethod
+ def dimension_over_reals():
+ return 2
+
+ @classmethod
+ def real_embed(cls,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 ComplexMatrixEJA
+
+ 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: ComplexMatrixEJA.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 = ZZ.random_element(3)
+ sage: F = QuadraticField(-1, 'I')
+ sage: X = random_matrix(F, n)
+ sage: Y = random_matrix(F, n)
+ sage: Xe = ComplexMatrixEJA.real_embed(X)
+ sage: Ye = ComplexMatrixEJA.real_embed(Y)
+ sage: XYe = ComplexMatrixEJA.real_embed(X*Y)
+ sage: Xe*Ye == XYe
+ True
+
+ """
+ super(ComplexMatrixEJA,cls).real_embed(M)
+ n = M.nrows()
+
+ # We don't need any adjoined elements...
+ field = M.base_ring().base_ring()
+
+ blocks = []
+ for z in M.list():
+ a = z.real()
+ b = z.imag()
+ blocks.append(matrix(field, 2, [ [ a, b],
+ [-b, a] ]))
+
+ return matrix.block(field, n, blocks)
+
+
+ @classmethod
+ def real_unembed(cls,M):
+ """
+ The inverse of _embed_complex_matrix().
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import ComplexMatrixEJA
+
+ EXAMPLES::
+
+ sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
+ ....: [-2, 1, -4, 3],
+ ....: [ 9, 10, 11, 12],
+ ....: [-10, 9, -12, 11] ])
+ sage: ComplexMatrixEJA.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 = ComplexMatrixEJA.real_embed(M)
+ sage: ComplexMatrixEJA.real_unembed(Me) == M
+ True
+
+ """
+ super(ComplexMatrixEJA,cls).real_unembed(M)
+ n = ZZ(M.nrows())
+ d = cls.dimension_over_reals()
+ F = cls.complex_extension(M.base_ring())
+ 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/d):
+ for j in range(n/d):
+ submat = M[d*k:d*k+d,d*j:d*j+d]
+ 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/d, elements)
+
+
+class ComplexHermitianEJA(ConcreteEJA, ComplexMatrixEJA):
+ """
+ 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, field=RDF, check_axioms=True)
+ Euclidean Jordan algebra of dimension 4 over Real Double Field
+ sage: ComplexHermitianEJA(2, field=RR, check_axioms=True)
+ 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_random_instance_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).to_matrix()
+ sage: X = x.to_matrix()
+ sage: Y = y.to_matrix()
+ 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)
+
+ We can construct the (trivial) algebra of rank zero::
+
+ sage: ComplexHermitianEJA(0)
+ Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
+
+ """
+
+ @classmethod
+ def _denormalized_basis(cls, n):
+ """
+ Returns a basis for the space of complex Hermitian n-by-n matrices.