category = MagmaticAlgebras(field).FiniteDimensional()
category = category.WithBasis().Unital().Commutative()
+ if n <= 1:
+ # All zero- and one-dimensional algebras are just the real
+ # numbers with (some positive multiples of) the usual
+ # multiplication as its Jordan and inner-product.
+ associative = True
if associative is None:
# We should figure it out. As with check_axioms, we have to do
# this without the help of the _jordan_product_is_associative()
class MatrixEJA:
+ @staticmethod
+ def _denormalized_basis(A):
+ """
+ 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.hurwitz import (ComplexMatrixAlgebra,
+ ....: QuaternionMatrixAlgebra,
+ ....: OctonionMatrixAlgebra)
+ sage: from mjo.eja.eja_algebra import MatrixEJA
+
+ TESTS::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: A = MatrixSpace(QQ, n)
+ sage: B = MatrixEJA._denormalized_basis(A)
+ sage: all( M.is_hermitian() for M in B)
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: A = ComplexMatrixAlgebra(n, scalars=QQ)
+ sage: B = MatrixEJA._denormalized_basis(A)
+ sage: all( M.is_hermitian() for M in B)
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: A = QuaternionMatrixAlgebra(n, scalars=QQ)
+ sage: B = MatrixEJA._denormalized_basis(A)
+ sage: all( M.is_hermitian() for M in B )
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: A = OctonionMatrixAlgebra(n, scalars=QQ)
+ sage: B = MatrixEJA._denormalized_basis(A)
+ sage: all( M.is_hermitian() for M in B )
+ True
+
+ """
+ # These work for real MatrixSpace, whose monomials only have
+ # two coordinates (because the last one would always be "1").
+ es = A.base_ring().gens()
+ gen = lambda A,m: A.monomial(m[:2])
+
+ if hasattr(A, 'entry_algebra_gens'):
+ # We've got a MatrixAlgebra, and its monomials will have
+ # three coordinates.
+ es = A.entry_algebra_gens()
+ gen = lambda A,m: A.monomial(m)
+
+ basis = []
+ for i in range(A.nrows()):
+ for j in range(i+1):
+ if i == j:
+ E_ii = gen(A, (i,j,es[0]))
+ basis.append(E_ii)
+ else:
+ for e in es:
+ E_ij = gen(A, (i,j,e))
+ E_ij += E_ij.conjugate_transpose()
+ basis.append(E_ij)
+
+ return tuple( basis )
+
@staticmethod
def jordan_product(X,Y):
return (X*Y + Y*X)/2
r"""
A trace inner-product for matrices that aren't embedded in the
reals. It takes MATRICES as arguments, not EJA elements.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (RealSymmetricEJA,
+ ....: ComplexHermitianEJA,
+ ....: QuaternionHermitianEJA,
+ ....: OctonionHermitianEJA)
+
+ EXAMPLES::
+
+ sage: J = RealSymmetricEJA(2,field=QQ,orthonormalize=False)
+ sage: I = J.one().to_matrix()
+ sage: J.trace_inner_product(I, -I)
+ -2
+
+ ::
+
+ sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False)
+ sage: I = J.one().to_matrix()
+ sage: J.trace_inner_product(I, -I)
+ -2
+
+ ::
+
+ sage: J = QuaternionHermitianEJA(2,field=QQ,orthonormalize=False)
+ sage: I = J.one().to_matrix()
+ sage: J.trace_inner_product(I, -I)
+ -2
+
+ ::
+
+ sage: J = OctonionHermitianEJA(2,field=QQ,orthonormalize=False)
+ sage: I = J.one().to_matrix()
+ sage: J.trace_inner_product(I, -I)
+ -2
+
"""
- return (X*Y).trace().real()
+ tr = (X*Y).trace()
+ if hasattr(tr, 'coefficient'):
+ # Works for octonions, and has to come first because they
+ # also have a "real()" method that doesn't return an
+ # element of the scalar ring.
+ return tr.coefficient(0)
+ elif hasattr(tr, 'coefficient_tuple'):
+ # Works for quaternions.
+ return tr.coefficient_tuple()[0]
+
+ # Works for real and complex numbers.
+ return tr.real()
+
class RealSymmetricEJA(RationalBasisEJA, ConcreteEJA, MatrixEJA):
Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
"""
- @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,ZZ)
- 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 tuple(S)
-
-
@staticmethod
def _max_random_instance_size():
return 4 # Dimension 10
# if the user passes check_axioms=True.
if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
- associative = False
- if n <= 1:
- associative = True
-
- super().__init__(self._denormalized_basis(n,field),
+ A = MatrixSpace(field, n)
+ super().__init__(self._denormalized_basis(A),
self.jordan_product,
self.trace_inner_product,
field=field,
- associative=associative,
**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 = self.matrix_space().one()
- self.one.set_cache(self(idV))
+ self.one.set_cache(self(A.one()))
Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
"""
-
- @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: B = ComplexHermitianEJA._denormalized_basis(n,QQ)
- sage: all( M.is_hermitian() for M in B)
- True
-
- """
- from mjo.hurwitz import ComplexMatrixAlgebra
- A = ComplexMatrixAlgebra(n, scalars=field)
- es = A.entry_algebra_gens()
-
- basis = []
- for i in range(n):
- for j in range(i+1):
- if i == j:
- E_ii = A.monomial( (i,j,es[0]) )
- basis.append(E_ii)
- else:
- for e in es:
- E_ij = A.monomial( (i,j,e) )
- ec = e.conjugate()
- # If the conjugate has a negative sign in front
- # of it, (j,i,ec) won't be a monomial!
- if (j,i,ec) in A.indices():
- E_ij += A.monomial( (j,i,ec) )
- else:
- E_ij -= A.monomial( (j,i,-ec) )
- basis.append(E_ij)
-
- return tuple( basis )
-
- @staticmethod
- def trace_inner_product(X,Y):
- r"""
- SETUP::
-
- sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
-
- TESTS::
-
- sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False)
- sage: I = J.one().to_matrix()
- sage: J.trace_inner_product(I, -I)
- -2
-
- """
- return (X*Y).trace().real()
-
def __init__(self, n, field=AA, **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
- associative = False
- if n <= 1:
- associative = True
-
- super().__init__(self._denormalized_basis(n,field),
+ from mjo.hurwitz import ComplexMatrixAlgebra
+ A = ComplexMatrixAlgebra(n, scalars=field)
+ super().__init__(self._denormalized_basis(A),
self.jordan_product,
self.trace_inner_product,
field=field,
- associative=associative,
**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 = self.matrix_space().one()
- self.one.set_cache(self(idV))
+ self.one.set_cache(self(A.one()))
@staticmethod
def _max_random_instance_size():
Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
"""
- @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_hermitian() for M in B )
- True
-
- """
- from mjo.hurwitz import QuaternionMatrixAlgebra
- A = QuaternionMatrixAlgebra(n, scalars=field)
- es = A.entry_algebra_gens()
-
- basis = []
- for i in range(n):
- for j in range(i+1):
- if i == j:
- E_ii = A.monomial( (i,j,es[0]) )
- basis.append(E_ii)
- else:
- for e in es:
- E_ij = A.monomial( (i,j,e) )
- ec = e.conjugate()
- # If the conjugate has a negative sign in front
- # of it, (j,i,ec) won't be a monomial!
- if (j,i,ec) in A.indices():
- E_ij += A.monomial( (j,i,ec) )
- else:
- E_ij -= A.monomial( (j,i,-ec) )
- basis.append(E_ij)
-
- return tuple( basis )
-
-
- @staticmethod
- def trace_inner_product(X,Y):
- r"""
- Overload the superclass method because the quaternions are weird
- and we need to use ``coefficient_tuple()`` to get the realpart.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
-
- TESTS::
-
- sage: J = QuaternionHermitianEJA(2,field=QQ,orthonormalize=False)
- sage: I = J.one().to_matrix()
- sage: J.trace_inner_product(I, -I)
- -2
-
- """
- return (X*Y).trace().coefficient_tuple()[0]
-
def __init__(self, n, field=AA, **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
- associative = False
- if n <= 1:
- associative = True
-
- super().__init__(self._denormalized_basis(n,field),
+ from mjo.hurwitz import QuaternionMatrixAlgebra
+ A = QuaternionMatrixAlgebra(n, scalars=field)
+ super().__init__(self._denormalized_basis(A),
self.jordan_product,
self.trace_inner_product,
field=field,
- associative=associative,
**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 = self.matrix_space().one()
- self.one.set_cache(self(idV))
+ self.one.set_cache(self(A.one()))
@staticmethod
# if the user passes check_axioms=True.
if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
- super().__init__(self._denormalized_basis(n,field),
+ from mjo.hurwitz import OctonionMatrixAlgebra
+ A = OctonionMatrixAlgebra(n, scalars=field)
+ super().__init__(self._denormalized_basis(A),
self.jordan_product,
self.trace_inner_product,
field=field,
# because the MatrixEJA is not presently a subclass of the
# FDEJA class that defines rank() and one().
self.rank.set_cache(n)
- idV = self.matrix_space().one()
- self.one.set_cache(self(idV))
-
-
- @classmethod
- def _denormalized_basis(cls, n, field):
- """
- Returns a basis for the space of octonion Hermitian n-by-n
- matrices.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import OctonionHermitianEJA
-
- EXAMPLES::
-
- sage: B = OctonionHermitianEJA._denormalized_basis(3,QQ)
- sage: all( M.is_hermitian() for M in B )
- True
- sage: len(B)
- 27
-
- """
- from mjo.hurwitz import OctonionMatrixAlgebra
- A = OctonionMatrixAlgebra(n, scalars=field)
- es = A.entry_algebra_gens()
-
- basis = []
- for i in range(n):
- for j in range(i+1):
- if i == j:
- E_ii = A.monomial( (i,j,es[0]) )
- basis.append(E_ii)
- else:
- for e in es:
- E_ij = A.monomial( (i,j,e) )
- ec = e.conjugate()
- # If the conjugate has a negative sign in front
- # of it, (j,i,ec) won't be a monomial!
- if (j,i,ec) in A.indices():
- E_ij += A.monomial( (j,i,ec) )
- else:
- E_ij -= A.monomial( (j,i,-ec) )
- basis.append(E_ij)
-
- return tuple( basis )
-
- @staticmethod
- def trace_inner_product(X,Y):
- r"""
- The octonions don't know that the reals are embedded in them,
- so we have to take the e0 component ourselves.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import OctonionHermitianEJA
-
- TESTS::
-
- sage: J = OctonionHermitianEJA(2,field=QQ,orthonormalize=False)
- sage: I = J.one().to_matrix()
- sage: J.trace_inner_product(I, -I)
- -2
-
- """
- return (X*Y).trace().coefficient(0)
+ self.one.set_cache(self(A.one()))
class AlbertEJA(OctonionHermitianEJA):
projects / sage.d.git / commitdiff