product. This will be applied to ``basis`` to compute an
inner-product table (basically a matrix) for this algebra.
+ - ``matrix_space`` -- the space that your matrix basis lives in,
+ or ``None`` (the default). So long as your basis does not have
+ length zero you can omit this. But in trivial algebras, it is
+ required.
+
- ``field`` -- a subfield of the reals (default: ``AA``); the scalar
field for the algebra.
sage: basis = tuple(b.superalgebra_element() for b in A.basis())
sage: J.subalgebra(basis, orthonormalize=False).is_associative()
True
-
"""
Element = FiniteDimensionalEJAElement
jordan_product,
inner_product,
field=AA,
+ matrix_space=None,
orthonormalize=True,
associative=None,
cartesian_product=False,
basis = tuple(gram_schmidt(basis, inner_product))
# Save the (possibly orthonormalized) matrix basis for
- # later...
+ # later, as well as the space that its elements live in.
+ # In most cases we can deduce the matrix space, but when
+ # n == 0 (that is, there are no basis elements) we cannot.
self._matrix_basis = basis
+ if matrix_space is None:
+ self._matrix_space = self._matrix_basis[0].parent()
+ else:
+ self._matrix_space = matrix_space
# Now create the vector space for the algebra, which will have
# its own set of non-ambient coordinates (in terms of the
sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False)
sage: J.matrix_space()
- Full MatrixSpace of 4 by 4 dense matrices over Rational Field
+ Module of 2 by 2 matrices with entries in Algebraic Field over
+ the scalar ring Rational Field
sage: J = QuaternionHermitianEJA(1,field=QQ,orthonormalize=False)
sage: J.matrix_space()
Module of 1 by 1 matrices with entries in Quaternion
the scalar ring Rational Field
"""
- if self.is_trivial():
- return MatrixSpace(self.base_ring(), 0)
- else:
- return self.matrix_basis()[0].parent()
+ return self._matrix_space
@cached_method
jordan_product,
inner_product,
field=QQ,
+ matrix_space=self.matrix_space(),
associative=self.is_associative(),
orthonormalize=False,
check_field=False,
"""
@staticmethod
- def _max_random_instance_size():
+ def _max_random_instance_dimension():
+ r"""
+ The maximum dimension of any random instance. Ten dimensions seems
+ to be about the point where everything takes a turn for the
+ worse. And dimension ten (but not nine) allows the 4-by-4 real
+ Hermitian matrices, the 2-by-2 quaternion Hermitian matrices,
+ and the 2-by-2 octonion Hermitian matrices.
+ """
+ return 10
+
+ @staticmethod
+ def _max_random_instance_size(max_dimension):
"""
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 algebra when it is used in a random test case. This size
+ (which can be passed to the algebra's constructor) is itself
+ based on the ``max_dimension`` parameter.
This method must be implemented in each subclass.
"""
raise NotImplementedError
@classmethod
- def random_instance(cls, *args, **kwargs):
+ def random_instance(cls, max_dimension=None, *args, **kwargs):
"""
- Return a random instance of this type of algebra.
+ Return a random instance of this type of algebra whose dimension
+ is less than or equal to ``max_dimension``. If the dimension bound
+ is omitted, then the ``_max_random_instance_dimension()`` is used
+ to get a suitable bound.
This method should be implemented in each subclass.
"""
return eja_class.random_instance(*args, **kwargs)
-class MatrixEJA:
+class MatrixEJA(FiniteDimensionalEJA):
@staticmethod
def _denormalized_basis(A):
"""
return tr.real()
+ def __init__(self, matrix_space, **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
-class RealSymmetricEJA(RationalBasisEJA, ConcreteEJA, MatrixEJA):
+
+ super().__init__(self._denormalized_basis(matrix_space),
+ self.jordan_product,
+ self.trace_inner_product,
+ field=matrix_space.base_ring(),
+ matrix_space=matrix_space,
+ **kwargs)
+
+ self.rank.set_cache(matrix_space.nrows())
+ self.one.set_cache( self(matrix_space.one()) )
+
+class RealSymmetricEJA(MatrixEJA, RationalBasisEJA, ConcreteEJA):
"""
The rank-n simple EJA consisting of real symmetric n-by-n
matrices, the usual symmetric Jordan product, and the trace inner
"""
@staticmethod
- def _max_random_instance_size():
- return 4 # Dimension 10
+ def _max_random_instance_size(max_dimension):
+ # Obtained by solving d = (n^2 + n)/2.
+ return int(ZZ(8*max_dimension + 1).sqrt()/2 - 1/2)
@classmethod
- def random_instance(cls, **kwargs):
+ def random_instance(cls, max_dimension=None, **kwargs):
"""
Return a random instance of this type of algebra.
"""
- n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ if max_dimension is None:
+ max_dimension = cls._max_random_instance_dimension()
+ max_size = cls._max_random_instance_size(max_dimension) + 1
+ n = ZZ.random_element(max_size)
return cls(n, **kwargs)
def __init__(self, n, field=AA, **kwargs):
if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
A = MatrixSpace(field, n)
- super().__init__(self._denormalized_basis(A),
- self.jordan_product,
- self.trace_inner_product,
- field=field,
- **kwargs)
+ super().__init__(A, **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)
- self.one.set_cache(self(A.one()))
+ from mjo.eja.eja_cache import real_symmetric_eja_coeffs
+ a = real_symmetric_eja_coeffs(self)
+ if a is not None:
+ if self._rational_algebra is None:
+ self._charpoly_coefficients.set_cache(a)
+ else:
+ self._rational_algebra._charpoly_coefficients.set_cache(a)
-class ComplexHermitianEJA(RationalBasisEJA, ConcreteEJA, MatrixEJA):
+class ComplexHermitianEJA(MatrixEJA, RationalBasisEJA, ConcreteEJA):
"""
The rank-n simple EJA consisting of complex Hermitian n-by-n
matrices over the real numbers, the usual symmetric Jordan product,
EXAMPLES:
- In theory, our "field" can be any subfield of the reals::
+ In theory, our "field" can be any subfield of the reals, but we
+ can't use inexact real fields at the moment because SageMath
+ doesn't know how to convert their elements into complex numbers,
+ or even into algebraic 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
+ sage: QQbar(RDF(1))
+ Traceback (most recent call last):
+ ...
+ TypeError: Illegal initializer for algebraic number
+ sage: AA(RR(1))
+ Traceback (most recent call last):
+ ...
+ TypeError: Illegal initializer for algebraic number
+
+ This causes the following error when we try to scale a matrix of
+ complex numbers by an inexact real number::
+
+ sage: ComplexHermitianEJA(2,field=RR)
+ Traceback (most recent call last):
+ ...
+ TypeError: Unable to coerce entries (=(1.00000000000000,
+ -0.000000000000000)) to coefficients in Algebraic Real Field
TESTS:
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,
- **kwargs)
+ super().__init__(A, **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)
- self.one.set_cache(self(A.one()))
+ from mjo.eja.eja_cache import complex_hermitian_eja_coeffs
+ a = complex_hermitian_eja_coeffs(self)
+ if a is not None:
+ if self._rational_algebra is None:
+ self._charpoly_coefficients.set_cache(a)
+ else:
+ self._rational_algebra._charpoly_coefficients.set_cache(a)
@staticmethod
- def _max_random_instance_size():
- return 3 # Dimension 9
+ def _max_random_instance_size(max_dimension):
+ # Obtained by solving d = n^2.
+ return int(ZZ(max_dimension).sqrt())
@classmethod
- def random_instance(cls, **kwargs):
+ def random_instance(cls, max_dimension=None, **kwargs):
"""
Return a random instance of this type of algebra.
"""
- n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ if max_dimension is None:
+ max_dimension = cls._max_random_instance_dimension()
+ n = ZZ.random_element(cls._max_random_instance_size(max_dimension) + 1)
return cls(n, **kwargs)
-class QuaternionHermitianEJA(RationalBasisEJA, ConcreteEJA, MatrixEJA):
+class QuaternionHermitianEJA(MatrixEJA, RationalBasisEJA, ConcreteEJA):
r"""
The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
matrices, the usual symmetric Jordan product, and the
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,
- **kwargs)
+ super().__init__(A, **kwargs)
+
+ from mjo.eja.eja_cache import quaternion_hermitian_eja_coeffs
+ a = quaternion_hermitian_eja_coeffs(self)
+ if a is not None:
+ if self._rational_algebra is None:
+ self._charpoly_coefficients.set_cache(a)
+ else:
+ self._rational_algebra._charpoly_coefficients.set_cache(a)
- # 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)
- self.one.set_cache(self(A.one()))
@staticmethod
- def _max_random_instance_size():
+ def _max_random_instance_size(max_dimension):
r"""
The maximum rank of a random QuaternionHermitianEJA.
"""
- return 2 # Dimension 6
+ # Obtained by solving d = 2n^2 - n.
+ return int(ZZ(8*max_dimension + 1).sqrt()/4 + 1/4)
@classmethod
- def random_instance(cls, **kwargs):
+ def random_instance(cls, max_dimension=None, **kwargs):
"""
Return a random instance of this type of algebra.
"""
- n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ if max_dimension is None:
+ max_dimension = cls._max_random_instance_dimension()
+ n = ZZ.random_element(cls._max_random_instance_size(max_dimension) + 1)
return cls(n, **kwargs)
-class OctonionHermitianEJA(RationalBasisEJA, ConcreteEJA, MatrixEJA):
+class OctonionHermitianEJA(MatrixEJA, RationalBasisEJA, ConcreteEJA):
r"""
SETUP::
sage: from mjo.eja.eja_algebra import (FiniteDimensionalEJA,
....: OctonionHermitianEJA)
+ sage: from mjo.hurwitz import Octonions, OctonionMatrixAlgebra
EXAMPLES:
After a change-of-basis, the 2-by-2 algebra has the same
multiplication table as the ten-dimensional Jordan spin algebra::
- sage: b = OctonionHermitianEJA._denormalized_basis(2,QQ)
+ sage: A = OctonionMatrixAlgebra(2,Octonions(QQ),QQ)
+ sage: b = OctonionHermitianEJA._denormalized_basis(A)
sage: basis = (b[0] + b[9],) + b[1:9] + (b[0] - b[9],)
sage: jp = OctonionHermitianEJA.jordan_product
sage: ip = OctonionHermitianEJA.trace_inner_product
"""
@staticmethod
- def _max_random_instance_size():
+ def _max_random_instance_size(max_dimension):
r"""
The maximum rank of a random QuaternionHermitianEJA.
"""
- return 1 # Dimension 1
+ # There's certainly a formula for this, but with only four
+ # cases to worry about, I'm not that motivated to derive it.
+ if max_dimension >= 27:
+ return 3
+ elif max_dimension >= 10:
+ return 2
+ elif max_dimension >= 1:
+ return 1
+ else:
+ return 0
@classmethod
- def random_instance(cls, **kwargs):
+ def random_instance(cls, max_dimension=None, **kwargs):
"""
Return a random instance of this type of algebra.
"""
- n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ if max_dimension is None:
+ max_dimension = cls._max_random_instance_dimension()
+ n = ZZ.random_element(cls._max_random_instance_size(max_dimension) + 1)
return cls(n, **kwargs)
def __init__(self, n, field=AA, **kwargs):
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,
- **kwargs)
+ super().__init__(A, **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)
- self.one.set_cache(self(A.one()))
+ from mjo.eja.eja_cache import octonion_hermitian_eja_coeffs
+ a = octonion_hermitian_eja_coeffs(self)
+ if a is not None:
+ if self._rational_algebra is None:
+ self._charpoly_coefficients.set_cache(a)
+ else:
+ self._rational_algebra._charpoly_coefficients.set_cache(a)
class AlbertEJA(OctonionHermitianEJA):
(r0, r1, r2)
"""
def __init__(self, n, field=AA, **kwargs):
+ MS = MatrixSpace(field, n, 1)
+
if n == 0:
jordan_product = lambda x,y: x
inner_product = lambda x,y: x
else:
def jordan_product(x,y):
- P = x.parent()
- return P( xi*yi for (xi,yi) in zip(x,y) )
+ return MS( xi*yi for (xi,yi) in zip(x,y) )
def inner_product(x,y):
return (x.T*y)[0,0]
if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False
if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
- column_basis = tuple( b.column()
- for b in FreeModule(field, n).basis() )
+ column_basis = tuple( MS(b) for b in FreeModule(field, n).basis() )
super().__init__(column_basis,
jordan_product,
inner_product,
field=field,
+ matrix_space=MS,
associative=True,
**kwargs)
self.rank.set_cache(n)
- if n == 0:
- self.one.set_cache( self.zero() )
- else:
- self.one.set_cache( sum(self.gens()) )
+ self.one.set_cache( self.sum(self.gens()) )
@staticmethod
- def _max_random_instance_size():
+ def _max_random_instance_dimension():
r"""
- The maximum dimension of a random HadamardEJA.
+ There's no reason to go higher than five here. That's
+ enough to get the point across.
"""
return 5
+ @staticmethod
+ def _max_random_instance_size(max_dimension):
+ r"""
+ The maximum size (=dimension) of a random HadamardEJA.
+ """
+ return max_dimension
+
@classmethod
- def random_instance(cls, **kwargs):
+ def random_instance(cls, max_dimension=None, **kwargs):
"""
Return a random instance of this type of algebra.
"""
- n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ if max_dimension is None:
+ max_dimension = cls._max_random_instance_dimension()
+ n = ZZ.random_element(cls._max_random_instance_size(max_dimension) + 1)
return cls(n, **kwargs)
# verify things, we'll skip the rest of the checks.
if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
+ n = B.nrows()
+ MS = MatrixSpace(field, n, 1)
+
def inner_product(x,y):
return (y.T*B*x)[0,0]
def jordan_product(x,y):
- P = x.parent()
x0 = x[0,0]
xbar = x[1:,0]
y0 = y[0,0]
ybar = y[1:,0]
z0 = inner_product(y,x)
zbar = y0*xbar + x0*ybar
- return P([z0] + zbar.list())
+ return MS([z0] + zbar.list())
- n = B.nrows()
- column_basis = tuple( b.column()
- for b in FreeModule(field, n).basis() )
+ column_basis = tuple( MS(b) for b in FreeModule(field, n).basis() )
# TODO: I haven't actually checked this, but it seems legit.
associative = False
jordan_product,
inner_product,
field=field,
+ matrix_space=MS,
associative=associative,
**kwargs)
# one-dimensional ambient space (because the rank is bounded
# by the ambient dimension).
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():
+ def _max_random_instance_dimension():
r"""
- The maximum dimension of a random BilinearFormEJA.
+ There's no reason to go higher than five here. That's
+ enough to get the point across.
"""
return 5
+ @staticmethod
+ def _max_random_instance_size(max_dimension):
+ r"""
+ The maximum size (=dimension) of a random BilinearFormEJA.
+ """
+ return max_dimension
+
@classmethod
- def random_instance(cls, **kwargs):
+ def random_instance(cls, max_dimension=None, **kwargs):
"""
Return a random instance of this algebra.
"""
- n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ if max_dimension is None:
+ max_dimension = cls._max_random_instance_dimension()
+ n = ZZ.random_element(cls._max_random_instance_size(max_dimension) + 1)
if n.is_zero():
B = matrix.identity(ZZ, n)
return cls(B, **kwargs)
# can pass in a field!
super().__init__(B, *args, **kwargs)
- @staticmethod
- def _max_random_instance_size():
- r"""
- The maximum dimension of a random JordanSpinEJA.
- """
- return 5
-
@classmethod
- def random_instance(cls, **kwargs):
+ def random_instance(cls, max_dimension=None, **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)
+ if max_dimension is None:
+ max_dimension = cls._max_random_instance_dimension()
+ n = ZZ.random_element(cls._max_random_instance_size(max_dimension) + 1)
return cls(n, **kwargs)
0
"""
- def __init__(self, **kwargs):
+ def __init__(self, field=AA, **kwargs):
jordan_product = lambda x,y: x
- inner_product = lambda x,y: 0
+ inner_product = lambda x,y: field.zero()
basis = ()
+ MS = MatrixSpace(field,0)
# New defaults for keyword arguments
if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False
jordan_product,
inner_product,
associative=True,
+ field=field,
+ matrix_space=MS,
**kwargs)
# The rank is zero using my definition, namely the dimension of the
associative = all( f.is_associative() for f in factors )
- MS = self.matrix_space()
+ # Compute my matrix space. This category isn't perfect, but
+ # is good enough for what we need to do.
+ MS_cat = MagmaticAlgebras(field).FiniteDimensional().WithBasis()
+ MS_cat = MS_cat.Unital().CartesianProducts()
+ MS_factors = tuple( J.matrix_space() for J in factors )
+ from sage.sets.cartesian_product import CartesianProduct
+ MS = CartesianProduct(MS_factors, MS_cat)
+
basis = []
zero = MS.zero()
for i in range(m):
jordan_product,
inner_product,
field=field,
+ matrix_space=MS,
orthonormalize=False,
associative=associative,
cartesian_product=True,
check_field=False,
check_axioms=False)
+ self.rank.set_cache(sum(J.rank() for J in factors))
ones = tuple(J.one().to_matrix() for J in factors)
self.one.set_cache(self(ones))
- self.rank.set_cache(sum(J.rank() for J in factors))
def cartesian_factors(self):
# Copy/pasted from CombinatorialFreeModule_CartesianProduct.
sage: J2 = ComplexHermitianEJA(1)
sage: J = cartesian_product([J1,J2])
sage: J.one().to_matrix()[0]
- [1 0]
- [0 1]
+ +---+
+ | 1 |
+ +---+
sage: J.one().to_matrix()[1]
- [1 0]
- [0 1]
+ +---+
+ | 1 |
+ +---+
::
+----+
"""
- scalars = self.cartesian_factor(0).base_ring()
-
- # This category isn't perfect, but is good enough for what we
- # need to do.
- cat = MagmaticAlgebras(scalars).FiniteDimensional().WithBasis()
- cat = cat.Unital().CartesianProducts()
- factors = tuple( J.matrix_space() for J in self.cartesian_factors() )
-
- from sage.sets.cartesian_product import CartesianProduct
- return CartesianProduct(factors, cat)
+ return super().matrix_space()
@cached_method
RationalBasisEJA.CartesianProduct = RationalBasisCartesianProductEJA
-def random_eja(*args, **kwargs):
- J1 = ConcreteEJA.random_instance(*args, **kwargs)
+def random_eja(max_dimension=None, *args, **kwargs):
+ # Use the ConcreteEJA default as the total upper bound (regardless
+ # of any whether or not any individual factors set a lower limit).
+ if max_dimension is None:
+ max_dimension = ConcreteEJA._max_random_instance_dimension()
+ J1 = ConcreteEJA.random_instance(max_dimension, *args, **kwargs)
- # This might make Cartesian products appear roughly as often as
- # any other ConcreteEJA.
- if ZZ.random_element(len(ConcreteEJA.__subclasses__()) + 1) == 0:
- # Use random_eja() again so we can get more than two factors.
- J2 = random_eja(*args, **kwargs)
- J = cartesian_product([J1,J2])
- return J
- else:
+
+ # Roll the dice to see if we attempt a Cartesian product.
+ dice_roll = ZZ.random_element(len(ConcreteEJA.__subclasses__()) + 1)
+ new_max_dimension = max_dimension - J1.dimension()
+ if new_max_dimension == 0 or dice_roll != 0:
+ # If it's already as big as we're willing to tolerate, just
+ # return it and don't worry about Cartesian products.
return J1
+ else:
+ # Use random_eja() again so we can get more than two factors
+ # if the sub-call also Decides on a cartesian product.
+ J2 = random_eja(new_max_dimension, *args, **kwargs)
+ return cartesian_product([J1,J2])
projects / sage.d.git / blobdiff