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.
+
+
+SETUP::
+
+ sage: from mjo.eja.eja_algebra import random_eja
+
+EXAMPLES::
+
+ sage: random_eja()
+ Euclidean Jordan algebra of dimension...
+
"""
from itertools import repeat
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, AA, QQbar, RR, RLF, CLF,
coords = W.coordinate_vector(_mat2vec(elt))
return self.from_vector(coords)
- @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
-
def _repr_(self):
"""
Return a string representation of ``self``.
sage: actual == expected
True
+ Ensure that the cached unit element (often precomputed by
+ hand) agrees with the computed one::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: cached = J.one()
+ sage: J.one.clear_cache()
+ sage: J.one() == cached
+ True
+
"""
# We can brute-force compute the matrices of the operators
# that correspond to the basis elements of this algebra.
# 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,
+ # Now we use basic 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)
+ A = matrix(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))
+ # Now if there's an identity element in the algebra, this
+ # should work. We solve on the left to avoid having to
+ # transpose the matrix "A".
+ return self.from_vector(A.solve_left(b))
def peirce_decomposition(self, c):
return tuple( self.random_element(thorough)
for idx in range(count) )
- @classmethod
- def random_instance(cls, field=AA, **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)
@cached_method
def _charpoly_coefficients(self):
Element = FiniteDimensionalEuclideanJordanAlgebraElement
-
-def random_eja(field=AA):
- """
- 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...
-
- """
- classname = choice([TrivialEJA,
- HadamardEJA,
- JordanSpinEJA,
- RealSymmetricEJA,
- ComplexHermitianEJA,
- QuaternionHermitianEJA])
- return classname.random_instance(field=field)
-
-
-
-
class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra):
r"""
Algebras whose basis consists of vectors with rational
return tuple(map(lambda x: x.change_ring(self.base_ring()), a))
-class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra):
+class ConcreteEuclideanJordanAlgebra:
+ 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 ConcreteEuclideanJordanAlgebra
+
+ TESTS:
+
+ Our natural basis is normalized with respect to the natural inner
+ product unless we specify otherwise::
+
+ sage: set_random_seed()
+ sage: J = ConcreteEuclideanJordanAlgebra.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: J = ConcreteEuclideanJordanAlgebra.random_instance()
+ sage: x = J.random_element()
+ sage: x.operator().matrix().is_symmetric()
+ True
+
+ """
+
@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 _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, field=AA, **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__())
+ return eja_class.random_instance(field)
+
+
+class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra):
def __init__(self, field, basis, normalize_basis=True, **kwargs):
"""
# time to ensure that it isn't a generator expression.
basis = tuple(basis)
- if len(basis) > 1 and normalize_basis:
+ algebra_dim = len(basis)
+ if algebra_dim > 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.
natural_basis=basis,
**kwargs)
+ if algebra_dim == 0:
+ self.one.set_cache(self.zero())
+ else:
+ n = basis[0].nrows()
+ # The identity wrt (A,B) -> (AB + BA)/2 is independent of the
+ # details of this algebra.
+ self.one.set_cache(self(matrix.identity(field,n)))
+
@cached_method
def _charpoly_coefficients(self):
return M
-class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra):
+class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra,
+ ConcreteEuclideanJordanAlgebra):
"""
The rank-n simple EJA consisting of real symmetric n-by-n
matrices, the usual symmetric Jordan product, and the trace inner
The dimension of this algebra is `(n^2 + n) / 2`::
sage: set_random_seed()
- sage: n_max = RealSymmetricEJA._max_test_case_size()
+ 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
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
-
We can construct the (trivial) algebra of rank zero::
sage: RealSymmetricEJA(0)
@staticmethod
- def _max_test_case_size():
+ def _max_random_instance_size():
return 4 # Dimension 10
+ @classmethod
+ def random_instance(cls, field=AA, **kwargs):
+ """
+ Return a random instance of this type of algebra.
+ """
+ n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ return cls(n, field, **kwargs)
def __init__(self, n, field=AA, **kwargs):
basis = self._denormalized_basis(n, field)
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: n = ZZ.random_element(3)
sage: F = QuadraticField(-1, 'I')
sage: X = random_matrix(F, n)
sage: Y = random_matrix(F, n)
return RealMatrixEuclideanJordanAlgebra.natural_inner_product(X,Y)/2
-class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra):
+class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra,
+ ConcreteEuclideanJordanAlgebra):
"""
The rank-n simple EJA consisting of complex Hermitian n-by-n
matrices over the real numbers, the usual symmetric Jordan product,
The dimension of this algebra is `n^2`::
sage: set_random_seed()
- sage: n_max = ComplexHermitianEJA._max_test_case_size()
+ 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
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
-
We can construct the (trivial) algebra of rank zero::
sage: ComplexHermitianEJA(0)
**kwargs)
self.rank.set_cache(n)
+ @staticmethod
+ def _max_random_instance_size():
+ return 3 # Dimension 9
+
+ @classmethod
+ def random_instance(cls, field=AA, **kwargs):
+ """
+ Return a random instance of this type of algebra.
+ """
+ n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ return cls(n, field, **kwargs)
class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra):
@staticmethod
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: n = ZZ.random_element(2)
sage: Q = QuaternionAlgebra(QQ,-1,-1)
sage: X = random_matrix(Q, n)
sage: Y = random_matrix(Q, n)
return RealMatrixEuclideanJordanAlgebra.natural_inner_product(X,Y)/4
-class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra):
- """
+class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra,
+ ConcreteEuclideanJordanAlgebra):
+ r"""
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 dimension of this algebra is `2*n^2 - n`::
sage: set_random_seed()
- sage: n_max = QuaternionHermitianEJA._max_test_case_size()
+ sage: n_max = QuaternionHermitianEJA._max_random_instance_size()
sage: n = ZZ.random_element(1, n_max)
sage: J = QuaternionHermitianEJA(n)
sage: J.dimension() == 2*(n^2) - n
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
-
We can construct the (trivial) algebra of rank zero::
sage: QuaternionHermitianEJA(0)
**kwargs)
self.rank.set_cache(n)
+ @staticmethod
+ def _max_random_instance_size():
+ r"""
+ The maximum rank of a random QuaternionHermitianEJA.
+ """
+ return 2 # Dimension 6
+
+ @classmethod
+ def random_instance(cls, field=AA, **kwargs):
+ """
+ Return a random instance of this type of algebra.
+ """
+ n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ return cls(n, field, **kwargs)
+
-class HadamardEJA(RationalBasisEuclideanJordanAlgebra):
+class HadamardEJA(RationalBasisEuclideanJordanAlgebra,
+ ConcreteEuclideanJordanAlgebra):
"""
Return the Euclidean Jordan Algebra corresponding to the set
`R^n` under the Hadamard product.
**kwargs)
self.rank.set_cache(n)
+ if n == 0:
+ self.one.set_cache( self.zero() )
+ else:
+ self.one.set_cache( sum(self.gens()) )
+
+ @staticmethod
+ def _max_random_instance_size():
+ r"""
+ The maximum dimension of a random HadamardEJA.
+ """
+ return 5
+
+ @classmethod
+ def random_instance(cls, field=AA, **kwargs):
+ """
+ Return a random instance of this type of algebra.
+ """
+ n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ return cls(n, field, **kwargs)
+
+
def inner_product(self, x, y):
"""
Faster to reimplement than to use natural representations.
return x.to_vector().inner_product(y.to_vector())
-class BilinearFormEJA(RationalBasisEuclideanJordanAlgebra):
+class BilinearFormEJA(RationalBasisEuclideanJordanAlgebra,
+ ConcreteEuclideanJordanAlgebra):
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``.
+ (<Bx,y>,y_bar>, x0*y_bar + y0*x_bar)`` where `B = 1 \times B22` is
+ a symmetric positive-definite "bilinear form" matrix. Its
+ dimension is the size of `B`, and it has rank two in dimensions
+ larger than two. It reduces to the ``JordanSpinEJA`` when `B` is
+ the identity matrix of order ``n``.
+
+ We insist that the one-by-one upper-left identity block of `B` be
+ passed in as well so that we can be passed a matrix of size zero
+ to construct a trivial algebra.
SETUP::
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: B = matrix.identity(AA,3)
+ sage: J0 = BilinearFormEJA(B)
sage: J1 = JordanSpinEJA(3)
sage: J0.multiplication_table() == J0.multiplication_table()
True
+ An error is raised if the matrix `B` does not correspond to a
+ positive-definite bilinear form::
+
+ sage: B = matrix.random(QQ,2,3)
+ sage: J = BilinearFormEJA(B)
+ Traceback (most recent call last):
+ ...
+ ValueError: bilinear form is not positive-definite
+ sage: B = matrix.zero(QQ,3)
+ sage: J = BilinearFormEJA(B)
+ Traceback (most recent call last):
+ ...
+ ValueError: bilinear form is not positive-definite
+
TESTS:
We can create a zero-dimensional algebra::
- sage: J = BilinearFormEJA(0)
+ sage: B = matrix.identity(AA,0)
+ sage: J = BilinearFormEJA(B)
sage: J.basis()
Finite family {}
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: B11 = matrix.identity(QQ,1)
+ sage: B22 = M.transpose()*M
+ sage: B = block_matrix(2,2,[ [B11,0 ],
+ ....: [0, B22 ] ])
+ sage: J = BilinearFormEJA(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()))
sage: actual == expected
True
"""
- def __init__(self, n, field=AA, B=None, **kwargs):
- if B is None:
- self._B = matrix.identity(field, max(0,n-1))
- else:
- self._B = B
+ def __init__(self, B, field=AA, **kwargs):
+ self._B = B
+ n = B.nrows()
+
+ if not B.is_positive_definite():
+ raise ValueError("bilinear form is not positive-definite")
V = VectorSpace(field, n)
mult_table = [[V.zero() for j in range(n)] for i in range(n)]
xbar = x[1:]
y0 = y[0]
ybar = y[1:]
- z0 = x0*y0 + (self._B*xbar).inner_product(ybar)
+ z0 = (B*x).inner_product(y)
zbar = y0*xbar + x0*ybar
z = V([z0] + zbar.list())
mult_table[i][j] = z
**kwargs)
self.rank.set_cache(min(n,2))
+ if n == 0:
+ self.one.set_cache( self.zero() )
+ else:
+ self.one.set_cache( self.monomial(0) )
+
+ @staticmethod
+ def _max_random_instance_size():
+ r"""
+ The maximum dimension of a random BilinearFormEJA.
+ """
+ return 5
+
+ @classmethod
+ def random_instance(cls, field=AA, **kwargs):
+ """
+ Return a random instance of this algebra.
+ """
+ n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ if n.is_zero():
+ B = matrix.identity(field, n)
+ return cls(B, field, **kwargs)
+
+ B11 = matrix.identity(field,1)
+ M = matrix.random(field, n-1)
+ I = matrix.identity(field, n-1)
+ alpha = field.zero()
+ while alpha.is_zero():
+ alpha = field.random_element().abs()
+ B22 = M.transpose()*M + alpha*I
+
+ from sage.matrix.special import block_matrix
+ B = block_matrix(2,2, [ [B11, ZZ(0) ],
+ [ZZ(0), B22 ] ])
+
+ return cls(B, field, **kwargs)
+
def inner_product(self, x, y):
r"""
Half of the trace inner product.
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: J = BilinearFormEJA.random_instance()
+ sage: n = J.dimension()
sage: x = J.random_element()
sage: y = J.random_element()
- sage: x.inner_product(y) == (x*y).trace()/2
+ sage: (n == 1) or (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)
+ return (self._B*x.to_vector()).inner_product(y.to_vector())
class JordanSpinEJA(BilinearFormEJA):
def __init__(self, n, field=AA, **kwargs):
# This is a special case of the BilinearFormEJA with the identity
# matrix as its bilinear form.
- super(JordanSpinEJA, self).__init__(n, field, **kwargs)
+ B = matrix.identity(field, n)
+ super(JordanSpinEJA, self).__init__(B, field, **kwargs)
+ @staticmethod
+ def _max_random_instance_size():
+ r"""
+ The maximum dimension of a random JordanSpinEJA.
+ """
+ return 5
-class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra):
+ @classmethod
+ def random_instance(cls, field=AA, **kwargs):
+ """
+ Return a random instance of this type of algebra.
+
+ Needed here to override the implementation for ``BilinearFormEJA``.
+ """
+ n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ return cls(n, field, **kwargs)
+
+
+class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra,
+ ConcreteEuclideanJordanAlgebra):
"""
The trivial Euclidean Jordan algebra consisting of only a zero element.
# The rank is zero using my definition, namely the dimension of the
# largest subalgebra generated by any element.
self.rank.set_cache(0)
+ self.one.set_cache( self.zero() )
+ @classmethod
+ def random_instance(cls, field=AA, **kwargs):
+ # We don't take a "size" argument so the superclass method is
+ # inappropriate for us.
+ return cls(field, **kwargs)
class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra):
r"""
iota_left = lambda x: self.from_vector(I1*x.to_vector())
iota_right = lambda x: self.from_vector(I2*+x.to_vector())
return (iota_left, iota_right)
+
+ def inner_product(self, x, y):
+ r"""
+ The standard Cartesian inner-product.
+
+ We project ``x`` and ``y`` onto our factors, and add up the
+ inner-products from the subalgebras.
+
+ SETUP::
+
+
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: QuaternionHermitianEJA,
+ ....: DirectSumEJA)
+
+ EXAMPLE::
+
+ sage: J1 = HadamardEJA(3)
+ sage: J2 = QuaternionHermitianEJA(2,QQ,normalize_basis=False)
+ sage: J = DirectSumEJA(J1,J2)
+ sage: x1 = J1.one()
+ sage: x2 = x1
+ sage: y1 = J2.one()
+ sage: y2 = y1
+ sage: x1.inner_product(x2)
+ 3
+ sage: y1.inner_product(y2)
+ 2
+ sage: J.one().inner_product(J.one())
+ 5
+
+ """
+ (pi_left, pi_right) = self.projections()
+ x1 = pi_left(x)
+ x2 = pi_right(x)
+ y1 = pi_left(y)
+ y2 = pi_right(y)
+
+ return (x1.inner_product(y1) + x2.inner_product(y2))
+
+
+
+random_eja = ConcreteEuclideanJordanAlgebra.random_instance
projects / sage.d.git / blobdiff