from sage.misc.table import table
from sage.modules.free_module import FreeModule, VectorSpace
from sage.rings.integer_ring import ZZ
-from sage.rings.number_field.number_field import NumberField
+from sage.rings.number_field.number_field import NumberField, QuadraticField
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.rings.rational_field import QQ
from sage.rings.real_lazy import CLF, RLF
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()
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():
True
"""
- if (not x in self) or (not y in self):
- raise TypeError("arguments must live in this algebra")
- return x.trace_inner_product(y)
+ X = x.natural_representation()
+ Y = y.natural_representation()
+ return self.__class__.natural_inner_product(X,Y)
def is_trivial(self):
return self._natural_basis[0].matrix_space()
+ @staticmethod
+ def natural_inner_product(X,Y):
+ """
+ Compute the inner product of two naturally-represented elements.
+
+ For example in the real symmetric matrix EJA, this will compute
+ the trace inner-product of two n-by-n symmetric matrices. The
+ default should work for the real cartesian product EJA, the
+ Jordan spin EJA, and the real symmetric matrices. The others
+ will have to be overridden.
+ """
+ return (X.conjugate_transpose()*Y).trace()
+
+
@cached_method
def one(self):
"""
return fdeja.__init__(field, mult_table, rank=n, **kwargs)
def inner_product(self, x, y):
- return _usual_ip(x,y)
+ """
+ Faster to reimplement than to use natural representations.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
+
+ TESTS:
+
+ Ensure that this is the usual inner product for the algebras
+ over `R^n`::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: J = RealCartesianProductEJA(n)
+ 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)
+ True
+
+ """
+ return x.to_vector().inner_product(y.to_vector())
def random_eja():
sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
- sage: B = _real_symmetric_basis(n, QQbar)
+ sage: B = _real_symmetric_basis(n, QQ)
sage: all( M.is_symmetric() for M in B)
True
Sij = Eij
else:
Sij = Eij + Eij.transpose()
- # Now normalize it.
- Sij = Sij / _real_symmetric_matrix_ip(Sij,Sij).sqrt()
S.append(Sij)
return tuple(S)
sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
- sage: R = PolynomialRing(QQ, 'z')
- sage: z = R.gen()
- sage: field = NumberField(z**2 - 2, 'sqrt2', embedding=RLF(2).sqrt())
+ sage: field = QuadraticField(2, 'sqrt2')
sage: B = _complex_hermitian_basis(n, field)
sage: all( M.is_symmetric() for M in B)
True
Sij = _embed_complex_matrix(Eij)
S.append(Sij)
else:
- # Beware, orthogonal but not normalized! The second one
- # has a minus because it's conjugated.
+ # The second one has a minus because it's conjugated.
Sij_real = _embed_complex_matrix(Eij + Eij.transpose())
S.append(Sij_real)
Sij_imag = _embed_complex_matrix(I*Eij - I*Eij.transpose())
S.append(Sij_imag)
- # Normalize these with our inner product before handing them back.
- # And since we embedded them, we can drop back to the "field" that
- # we started with instead of the complex extension "F".
- return tuple( (s / _complex_hermitian_matrix_ip(s,s).sqrt()).change_ring(field)
- for s in S )
+ # Since we embedded these, we can drop back to the "field" that we
+ # started with instead of the complex extension "F".
+ return tuple( s.change_ring(field) for s in S )
-def _quaternion_hermitian_basis(n, field):
+def _quaternion_hermitian_basis(n, field, normalize):
"""
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)
+ sage: B = _quaternion_hermitian_basis(n, QQ, False)
sage: all( M.is_symmetric() for M in B )
True
EXAMPLES::
- sage: R = PolynomialRing(QQ, 'z')
- sage: z = R.gen()
- sage: F = NumberField(z**2 + 1, 'i', embedding=CLF(-1).sqrt())
+ sage: F = QuadraticField(-1, 'i')
sage: x1 = F(4 - 2*i)
sage: x2 = F(1 + 2*i)
sage: x3 = F(-i)
sage: set_random_seed()
sage: n = ZZ.random_element(5)
- sage: R = PolynomialRing(QQ, 'z')
- sage: z = R.gen()
- sage: F = NumberField(z**2 + 1, 'i', embedding=CLF(-1).sqrt())
+ sage: F = QuadraticField(-1, 'i')
sage: X = random_matrix(F, n)
sage: Y = random_matrix(F, n)
sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
Unembedding is the inverse of embedding::
sage: set_random_seed()
- sage: R = PolynomialRing(QQ, 'z')
- sage: z = R.gen()
- sage: F = NumberField(z**2 + 1, 'i', embedding=CLF(-1).sqrt())
+ sage: F = QuadraticField(-1, 'i')
sage: M = random_matrix(F, 3)
sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
True
if M.ncols() != n:
raise ValueError("the matrix 'M' must be square")
- R = PolynomialRing(QQ, 'z')
- z = R.gen()
- F = NumberField(z**2 + 1, 'i', embedding=CLF(-1).sqrt())
+ F = QuadraticField(-1, 'i')
i = F.gen()
blocks = []
return matrix(Q, n/4, elements)
-# The usual inner product on R^n.
-def _usual_ip(x,y):
- return x.to_vector().inner_product(y.to_vector())
-
# The inner product used for the real symmetric simple EJA.
# We keep it as a separate function because e.g. the complex
# algebra uses the same inner product, except divided by 2.
Y_mat = Y.natural_representation()
return (X_mat*Y_mat).trace()
-def _real_symmetric_matrix_ip(X,Y):
- return (X*Y).trace()
-
-def _complex_hermitian_matrix_ip(X,Y):
- # This takes EMBEDDED matrices.
- Xu = _unembed_complex_matrix(X)
- Yu = _unembed_complex_matrix(Y)
- # The trace need not be real; consider Xu = (i*I) and Yu = I.
- return ((Xu*Yu).trace()).vector()[0] # real part, I guess
class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra):
"""
True
"""
- def __init__(self, n, field=QQ, **kwargs):
- if n > 1:
+ def __init__(self, n, field=QQ, normalize_basis=True, **kwargs):
+ S = _real_symmetric_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.
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.__class__.natural_inner_product(s,s).sqrt())
+ for s in S )
+ S = tuple( s*c for (s,c) in zip(S,self._basis_normalizers) )
- S = _real_symmetric_basis(n, field)
Qs = _multiplication_table_from_matrix_basis(S)
fdeja = super(RealSymmetricEJA, self)
natural_basis=S,
**kwargs)
- def inner_product(self, x, y):
- X = x.natural_representation()
- Y = y.natural_representation()
- return _real_symmetric_matrix_ip(X,Y)
class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):
True
"""
- def __init__(self, n, field=QQ, **kwargs):
- if n > 1:
+ def __init__(self, n, field=QQ, normalize_basis=True, **kwargs):
+ S = _complex_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.
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.__class__.natural_inner_product(s,s).sqrt())
+ for s in S )
+ S = tuple( s*c for (s,c) in zip(S,self._basis_normalizers) )
- S = _complex_hermitian_basis(n, field)
Qs = _multiplication_table_from_matrix_basis(S)
fdeja = super(ComplexHermitianEJA, self)
**kwargs)
- def inner_product(self, x, y):
- X = x.natural_representation()
- Y = y.natural_representation()
- return _complex_hermitian_matrix_ip(X,Y)
-
+ @staticmethod
+ def natural_inner_product(X,Y):
+ Xu = _unembed_complex_matrix(X)
+ Yu = _unembed_complex_matrix(Y)
+ # 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):
"""
True
"""
- def __init__(self, n, field=QQ, **kwargs):
- S = _quaternion_hermitian_basis(n, field)
+ def __init__(self, n, field=QQ, normalize_basis=True, **kwargs):
+ S = _quaternion_hermitian_basis(n, field, normalize_basis)
Qs = _multiplication_table_from_matrix_basis(S)
fdeja = super(QuaternionHermitianEJA, self)
return fdeja.__init__(field, mult_table, rank=min(n,2), **kwargs)
def inner_product(self, x, y):
- return _usual_ip(x,y)
+ """
+ Faster to reimplement than to use natural representations.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import JordanSpinEJA
+
+ TESTS:
+
+ Ensure that this is the usual inner product for the algebras
+ over `R^n`::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: J = JordanSpinEJA(n)
+ 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)
+ True
+
+ """
+ return x.to_vector().inner_product(y.to_vector())