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_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 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.
- """
- raise NotImplementedError
-
def _repr_(self):
"""
Return a string representation of ``self``.
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.
- """
- n = ZZ.random_element(cls._max_random_instance_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."
+ """
+
@staticmethod
def _max_random_instance_size():
- # Play it safe, since this will be squared and the underlying
- # field can have dimension 4 (quaternions) too.
- return 2
+ """
+ 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):
"""
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
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_random_instance_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,
**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_random_instance_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
**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.
@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 =
@staticmethod
def _max_random_instance_size():
+ r"""
+ The maximum dimension of a random BilinearFormEJA.
+ """
return 5
@classmethod
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
+
@classmethod
def random_instance(cls, field=AA, **kwargs):
"""
return cls(n, field, **kwargs)
-class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra):
+class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra,
+ ConcreteEuclideanJordanAlgebra):
"""
The trivial Euclidean Jordan algebra consisting of only a zero element.
y2 = pi_right(y)
return (x1.inner_product(y1) + x2.inner_product(y2))
+
+
+
+random_eja = ConcreteEuclideanJordanAlgebra.random_instance
projects / sage.d.git / commitdiff