self._rank = rank
self._natural_basis = natural_basis
+ # TODO: HACK for the charpoly.. needs redesign badly.
+ self._basis_normalizers = None
+
if category is None:
category = MagmaticAlgebras(field).FiniteDimensional()
category = category.WithBasis().Unital()
vector representations) back and forth faithfully::
sage: set_random_seed()
- sage: J = RealCartesianProductEJA(5)
+ sage: J = RealCartesianProductEJA.random_instance()
sage: x = J.random_element()
sage: J(x.to_vector().column()) == x
True
- sage: J = JordanSpinEJA(5)
+ sage: J = JordanSpinEJA.random_instance()
sage: x = J.random_element()
sage: J(x.to_vector().column()) == x
True
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``.
return V.span_of_basis(b)
+
@cached_method
def _charpoly_coeff(self, i):
"""
store the trace/determinant (a_{r-1} and a_{0} respectively)
separate from the entire characteristic polynomial.
"""
+ if self._basis_normalizers is not None:
+ # Must be a matrix class?
+ # WARNING/TODO: this whole mess is mis-designed.
+ n = self.natural_basis_space().nrows()
+ field = self.base_ring().base_ring() # yeeeeaaaahhh
+ J = self.__class__(n, field, False)
+ (_,x,_,_) = J._charpoly_matrix_system()
+ p = J._charpoly_coeff(i)
+ # p might be missing some vars, have to substitute "optionally"
+ pairs = zip(x.base_ring().gens(), self._basis_normalizers)
+ substitutions = { v: v*c for (v,c) in pairs }
+ return p.subs(substitutions)
+
(A_of_x, x, xr, detA) = self._charpoly_matrix_system()
R = A_of_x.base_ring()
if i >= self.rank():
"""
X = x.natural_representation()
Y = y.natural_representation()
- return self.__class__.natural_inner_product(X,Y)
+ return self.natural_inner_product(X,Y)
def is_trivial(self):
return s.random_element()
+ @classmethod
+ def random_instance(cls, field=QQ, **kwargs):
+ """
+ Return a random instance of this type of algebra.
+
+ In subclasses for algebras that we know how to construct, this
+ is a shortcut for constructing test cases and examples.
+ """
+ if cls is FiniteDimensionalEuclideanJordanAlgebra:
+ # Red flag! But in theory we could do this I guess. The
+ # only finite-dimensional exceptional EJA is the
+ # octononions. So, we could just create an EJA from an
+ # associative matrix algebra (generated by a subset of
+ # elements) with the symmetric product. Or, we could punt
+ # to random_eja() here, override it in our subclasses, and
+ # not worry about it.
+ raise NotImplementedError
+
+ n = ZZ.random_element(1, cls._max_test_case_size())
+ return cls(n, field, **kwargs)
+
+
def rank(self):
"""
Return the rank of this EJA.
Our inner product satisfies the Jordan axiom::
sage: set_random_seed()
- sage: n = ZZ.random_element(1,5)
- sage: J = RealCartesianProductEJA(n)
+ sage: J = RealCartesianProductEJA.random_instance()
sage: x = J.random_element()
sage: y = J.random_element()
sage: z = J.random_element()
over `R^n`::
sage: set_random_seed()
- sage: n = ZZ.random_element(1,5)
- sage: J = RealCartesianProductEJA(n)
+ sage: J = RealCartesianProductEJA.random_instance()
sage: x = J.random_element()
sage: y = J.random_element()
sage: X = x.natural_representation()
sage: Y = y.natural_representation()
- sage: x.inner_product(y) == J.__class__.natural_inner_product(X,Y)
+ sage: x.inner_product(y) == J.natural_inner_product(X,Y)
True
"""
Euclidean Jordan algebra of dimension...
"""
-
- # The max_n component lets us choose different upper bounds on the
- # value "n" that gets passed to the constructor. This is needed
- # because e.g. R^{10} is reasonable to test, while the Hermitian
- # 10-by-10 quaternion matrices are not.
- (constructor, max_n) = choice([(RealCartesianProductEJA, 6),
- (JordanSpinEJA, 6),
- (RealSymmetricEJA, 5),
- (ComplexHermitianEJA, 4),
- (QuaternionHermitianEJA, 3)])
- n = ZZ.random_element(1, max_n)
- return constructor(n, field=QQ)
+ classname = choice([RealCartesianProductEJA,
+ JordanSpinEJA,
+ RealSymmetricEJA,
+ ComplexHermitianEJA,
+ QuaternionHermitianEJA])
+ return classname.random_instance()
-def _quaternion_hermitian_basis(n, field, normalize):
+def _quaternion_hermitian_basis(n, field):
"""
Returns a basis for the space of quaternion Hermitian n-by-n matrices.
sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
- sage: B = _quaternion_hermitian_basis(n, QQ, False)
+ sage: B = _quaternion_hermitian_basis(n, QQ)
sage: all( M.is_symmetric() for M in B )
True
SETUP::
- sage: from mjo.eja.eja_algebra import _embed_complex_matrix
+ sage: from mjo.eja.eja_algebra import (_embed_complex_matrix,
+ ....: ComplexHermitianEJA)
EXAMPLES::
Embedding is a homomorphism (isomorphism, in fact)::
sage: set_random_seed()
- sage: n = ZZ.random_element(5)
+ sage: n_max = ComplexHermitianEJA._max_test_case_size()
+ sage: n = ZZ.random_element(n_max)
sage: F = QuadraticField(-1, 'i')
sage: X = random_matrix(F, n)
sage: Y = random_matrix(F, n)
SETUP::
- sage: from mjo.eja.eja_algebra import _embed_quaternion_matrix
+ sage: from mjo.eja.eja_algebra import (_embed_quaternion_matrix,
+ ....: QuaternionHermitianEJA)
EXAMPLES::
Embedding is a homomorphism (isomorphism, in fact)::
sage: set_random_seed()
- sage: n = ZZ.random_element(5)
+ sage: n_max = QuaternionHermitianEJA._max_test_case_size()
+ sage: n = ZZ.random_element(n_max)
sage: Q = QuaternionAlgebra(QQ,-1,-1)
sage: X = random_matrix(Q, n)
sage: Y = random_matrix(Q, n)
if not n.mod(4).is_zero():
raise ValueError("the matrix 'M' must be a complex embedding")
- Q = QuaternionAlgebra(QQ,-1,-1)
+ # Use the base ring of the matrix to ensure that its entries can be
+ # multiplied by elements of the quaternion algebra.
+ field = M.base_ring()
+ Q = QuaternionAlgebra(field,-1,-1)
i,j,k = Q.gens()
# Go top-left to bottom-right (reading order), converting every
raise ValueError('bad on-diagonal submatrix')
if submat[0,1] != -submat[1,0].conjugate():
raise ValueError('bad off-diagonal submatrix')
- z = submat[0,0].real() + submat[0,0].imag()*i
- z += submat[0,1].real()*j + submat[0,1].imag()*k
+ z = submat[0,0].vector()[0] # real part
+ z += submat[0,0].vector()[1]*i # imag part
+ z += submat[0,1].vector()[0]*j # real part
+ z += submat[0,1].vector()[1]*k # imag part
elements.append(z)
return matrix(Q, n/4, elements)
The dimension of this algebra is `(n^2 + n) / 2`::
sage: set_random_seed()
- sage: n = ZZ.random_element(1,5)
+ sage: n_max = RealSymmetricEJA._max_test_case_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: n = ZZ.random_element(1,5)
- sage: J = RealSymmetricEJA(n)
+ sage: J = RealSymmetricEJA.random_instance()
sage: x = J.random_element()
sage: y = J.random_element()
sage: actual = (x*y).natural_representation()
Our inner product satisfies the Jordan axiom::
sage: set_random_seed()
- sage: n = ZZ.random_element(1,5)
- sage: J = RealSymmetricEJA(n)
+ sage: J = RealSymmetricEJA.random_instance()
sage: x = J.random_element()
sage: y = J.random_element()
sage: z = J.random_element()
sage: (x*y).inner_product(z) == y.inner_product(x*z)
True
- Our basis is normalized with respect to the natural inner product::
+ Our natural basis is normalized with respect to the natural inner
+ product unless we specify otherwise::
sage: set_random_seed()
- sage: n = ZZ.random_element(1,5)
- sage: J = RealSymmetricEJA(n)
+ sage: J = RealSymmetricEJA.random_instance()
sage: all( b.norm() == 1 for b in J.gens() )
True
- Left-multiplication operators are symmetric because they satisfy
- the Jordan axiom::
+ 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: n = ZZ.random_element(1,5)
- sage: x = RealSymmetricEJA(n).random_element()
+ sage: x = RealSymmetricEJA.random_instance().random_element()
sage: x.operator().matrix().is_symmetric()
True
p = z**2 - 2
if p.is_irreducible():
field = NumberField(p, 'sqrt2', embedding=RLF(2).sqrt())
- S = [ s.change_ring(field) for s in S ]
- self._basis_denormalizers = tuple(
- self.__class__.natural_inner_product(s,s).sqrt()
- for s in S )
- S = tuple( s/c for (s,c) in zip(S,self._basis_denormalizers) )
+ S = [ s.change_ring(field) for s in S ]
+ self._basis_normalizers = tuple(
+ ~(self.natural_inner_product(s,s).sqrt()) for s in S )
+ S = tuple( s*c for (s,c) in zip(S,self._basis_normalizers) )
Qs = _multiplication_table_from_matrix_basis(S)
natural_basis=S,
**kwargs)
+ @staticmethod
+ def _max_test_case_size():
+ return 5
class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):
The dimension of this algebra is `n^2`::
sage: set_random_seed()
- sage: n = ZZ.random_element(1,5)
+ sage: n_max = ComplexHermitianEJA._max_test_case_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: n = ZZ.random_element(1,5)
- sage: J = ComplexHermitianEJA(n)
+ sage: J = ComplexHermitianEJA.random_instance()
sage: x = J.random_element()
sage: y = J.random_element()
sage: actual = (x*y).natural_representation()
Our inner product satisfies the Jordan axiom::
sage: set_random_seed()
- sage: n = ZZ.random_element(1,5)
- sage: J = ComplexHermitianEJA(n)
+ sage: J = ComplexHermitianEJA.random_instance()
sage: x = J.random_element()
sage: y = J.random_element()
sage: z = J.random_element()
sage: (x*y).inner_product(z) == y.inner_product(x*z)
True
- Our basis is normalized with respect to the natural inner product::
+ Our natural basis is normalized with respect to the natural inner
+ product unless we specify otherwise::
sage: set_random_seed()
- sage: n = ZZ.random_element(1,4)
- sage: J = ComplexHermitianEJA(n)
+ sage: J = ComplexHermitianEJA.random_instance()
sage: all( b.norm() == 1 for b in J.gens() )
True
- Left-multiplication operators are symmetric because they satisfy
- the Jordan axiom::
+ 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: n = ZZ.random_element(1,5)
- sage: x = ComplexHermitianEJA(n).random_element()
+ sage: x = ComplexHermitianEJA.random_instance().random_element()
sage: x.operator().matrix().is_symmetric()
True
p = z**2 - 2
if p.is_irreducible():
field = NumberField(p, 'sqrt2', embedding=RLF(2).sqrt())
- S = [ s.change_ring(field) for s in S ]
- self._basis_denormalizers = tuple(
- self.__class__.natural_inner_product(s,s).sqrt()
- for s in S )
- S = tuple( s/c for (s,c) in zip(S,self._basis_denormalizers) )
+ S = [ s.change_ring(field) for s in S ]
+ self._basis_normalizers = tuple(
+ ~(self.natural_inner_product(s,s).sqrt()) for s in S )
+ S = tuple( s*c for (s,c) in zip(S,self._basis_normalizers) )
Qs = _multiplication_table_from_matrix_basis(S)
**kwargs)
+ @staticmethod
+ def _max_test_case_size():
+ return 4
+
@staticmethod
def natural_inner_product(X,Y):
Xu = _unembed_complex_matrix(X)
# The trace need not be real; consider Xu = (i*I) and Yu = I.
return ((Xu*Yu).trace()).vector()[0] # real part, I guess
+
+
class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):
"""
The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
TESTS:
- The dimension of this algebra is `n^2`::
+ The dimension of this algebra is `2*n^2 - n`::
sage: set_random_seed()
- sage: n = ZZ.random_element(1,5)
+ sage: n_max = QuaternionHermitianEJA._max_test_case_size()
+ sage: n = ZZ.random_element(1, n_max)
sage: J = QuaternionHermitianEJA(n)
sage: J.dimension() == 2*(n^2) - n
True
The Jordan multiplication is what we think it is::
sage: set_random_seed()
- sage: n = ZZ.random_element(1,5)
- sage: J = QuaternionHermitianEJA(n)
+ sage: J = QuaternionHermitianEJA.random_instance()
sage: x = J.random_element()
sage: y = J.random_element()
sage: actual = (x*y).natural_representation()
Our inner product satisfies the Jordan axiom::
sage: set_random_seed()
- sage: n = ZZ.random_element(1,5)
- sage: J = QuaternionHermitianEJA(n)
+ sage: J = QuaternionHermitianEJA.random_instance()
sage: x = J.random_element()
sage: y = J.random_element()
sage: z = J.random_element()
sage: (x*y).inner_product(z) == y.inner_product(x*z)
True
+ 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
+
"""
def __init__(self, n, field=QQ, normalize_basis=True, **kwargs):
- S = _quaternion_hermitian_basis(n, field, normalize_basis)
+ S = _quaternion_hermitian_basis(n, field)
+
+ if n > 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.
+ R = PolynomialRing(field, 'z')
+ z = R.gen()
+ p = z**2 - 2
+ if p.is_irreducible():
+ field = NumberField(p, 'sqrt2', embedding=RLF(2).sqrt())
+ S = [ s.change_ring(field) for s in S ]
+ self._basis_normalizers = tuple(
+ ~(self.natural_inner_product(s,s).sqrt()) for s in S )
+ S = tuple( s*c for (s,c) in zip(S,self._basis_normalizers) )
+
Qs = _multiplication_table_from_matrix_basis(S)
fdeja = super(QuaternionHermitianEJA, self)
natural_basis=S,
**kwargs)
- def inner_product(self, x, y):
- # Since a+bi+cj+dk on the diagonal is represented as
- #
- # a + bi +cj + dk = [ a b c d]
- # [ -b a -d c]
- # [ -c d a -b]
- # [ -d -c b a],
- #
- # we'll quadruple-count the "a" entries if we take the trace of
- # the embedding.
- return _matrix_ip(x,y)/4
+ @staticmethod
+ def _max_test_case_size():
+ return 3
+
+ @staticmethod
+ def natural_inner_product(X,Y):
+ Xu = _unembed_quaternion_matrix(X)
+ Yu = _unembed_quaternion_matrix(Y)
+ # The trace need not be real; consider Xu = (i*I) and Yu = I.
+ # The result will be a quaternion algebra element, which doesn't
+ # have a "vector" method, but does have coefficient_tuple() method
+ # that returns the coefficients of 1, i, j, and k -- in that order.
+ return ((Xu*Yu).trace()).coefficient_tuple()[0]
+
class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra):
Our inner product satisfies the Jordan axiom::
sage: set_random_seed()
- sage: n = ZZ.random_element(1,5)
- sage: J = JordanSpinEJA(n)
+ sage: J = JordanSpinEJA.random_instance()
sage: x = J.random_element()
sage: y = J.random_element()
sage: z = J.random_element()
over `R^n`::
sage: set_random_seed()
- sage: n = ZZ.random_element(1,5)
- sage: J = JordanSpinEJA(n)
+ sage: J = JordanSpinEJA.random_instance()
sage: x = J.random_element()
sage: y = J.random_element()
sage: X = x.natural_representation()
sage: Y = y.natural_representation()
- sage: x.inner_product(y) == J.__class__.natural_inner_product(X,Y)
+ sage: x.inner_product(y) == J.natural_inner_product(X,Y)
True
"""
projects / sage.d.git / blobdiff