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 QuadraticField
+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
from sage.structure.element import is_Matrix
from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
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 self.zero()
natural_basis = self.natural_basis()
- if elt not in natural_basis[0].matrix_space():
+ basis_space = natural_basis[0].matrix_space()
+ if elt not in basis_space:
raise ValueError("not a naturally-represented algebra element")
- # Thanks for nothing! Matrix spaces aren't vector
- # spaces in Sage, so we have to figure out its
- # natural-basis coordinates ourselves.
- V = VectorSpace(elt.base_ring(), elt.nrows()*elt.ncols())
+ # Thanks for nothing! Matrix spaces aren't vector spaces in
+ # Sage, so we have to figure out its natural-basis coordinates
+ # ourselves. We use the basis space's ring instead of the
+ # element's ring because the basis space might be an algebraic
+ # closure whereas the base ring of the 3-by-3 identity matrix
+ # could be QQ instead of QQbar.
+ V = VectorSpace(basis_space.base_ring(), elt.nrows()*elt.ncols())
W = V.span_of_basis( _mat2vec(s) for s in natural_basis )
coords = W.coordinate_vector(_mat2vec(elt))
return self.from_vector(coords)
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):
Finite family {0: e0, 1: e1, 2: e2}
sage: J.natural_basis()
(
- [1 0] [0 1] [0 0]
- [0 0], [1 0], [0 1]
+ [1 0] [ 0 1/2*sqrt2] [0 0]
+ [0 0], [1/2*sqrt2 0], [0 1]
)
::
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):
"""
sage: J = RealSymmetricEJA(2)
sage: J.vector_space()
- Vector space of dimension 3 over Rational Field
+ Vector space of dimension 3 over...
"""
return self.zero().to_vector().parent().ambient_vector_space()
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
if i == j:
Sij = Eij
else:
- # Beware, orthogonal but not normalized!
Sij = Eij + Eij.transpose()
S.append(Sij)
return tuple(S)
"""
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 _complex_hermitian_basis
sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
- sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
+ sage: field = QuadraticField(2, 'sqrt2')
+ sage: B = _complex_hermitian_basis(n, field)
+ sage: all( M.is_symmetric() for M in B)
True
"""
- F = QuadraticField(-1, 'I')
+ R = PolynomialRing(field, 'z')
+ z = R.gen()
+ F = NumberField(z**2 + 1, 'I', embedding=CLF(-1).sqrt())
I = F.gen()
# This is like the symmetric case, but we need to be careful:
S = []
for i in xrange(n):
for j in xrange(i+1):
- Eij = matrix(field, n, lambda k,l: k==i and l==j)
+ Eij = matrix(F, n, lambda k,l: k==i and l==j)
if i == j:
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)
- return tuple(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.
+ 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 _quaternion_hermitian_basis
sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
- sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) )
+ sage: B = _quaternion_hermitian_basis(n, QQ, False)
+ sage: all( M.is_symmetric() for M in B )
True
"""
EXAMPLES::
- sage: F = QuadraticField(-1,'i')
+ sage: F = QuadraticField(-1, 'i')
sage: x1 = F(4 - 2*i)
sage: x2 = F(1 + 2*i)
sage: x3 = F(-i)
field = M.base_ring()
blocks = []
for z in M.list():
- a = z.real()
- b = z.imag()
+ a = z.vector()[0] # real part, I guess
+ b = z.vector()[1] # imag part, I guess
blocks.append(matrix(field, 2, [[a,b],[-b,a]]))
# We can drop the imaginaries here.
if not n.mod(2).is_zero():
raise ValueError("the matrix 'M' must be a complex embedding")
- F = QuadraticField(-1, 'i')
+ field = M.base_ring() # This should already have sqrt2
+ R = PolynomialRing(field, 'z')
+ z = R.gen()
+ F = NumberField(z**2 + 1,'i', embedding=CLF(-1).sqrt())
i = F.gen()
# Go top-left to bottom-right (reading order), converting every
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.
sage: e0*e0
e0
sage: e1*e1
- e0 + e2
+ 1/2*e0 + 1/2*e2
sage: e2*e2
e2
sage: (x*y).inner_product(z) == y.inner_product(x*z)
True
+ Our basis is normalized with respect to the natural inner product::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: J = RealSymmetricEJA(n)
+ sage: all( b.norm() == 1 for b in J.gens() )
+ True
+
+ Left-multiplication operators are symmetric because they satisfy
+ the Jordan axiom::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: x = RealSymmetricEJA(n).random_element()
+ sage: x.operator().matrix().is_symmetric()
+ True
+
"""
- def __init__(self, n, field=QQ, **kwargs):
+ 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.
+ 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.__class__.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(RealSymmetricEJA, self)
natural_basis=S,
**kwargs)
- def inner_product(self, x, y):
- return _matrix_ip(x,y)
class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):
sage: (x*y).inner_product(z) == y.inner_product(x*z)
True
+ Our basis is normalized with respect to the natural inner product::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,4)
+ sage: J = ComplexHermitianEJA(n)
+ sage: all( b.norm() == 1 for b in J.gens() )
+ True
+
+ Left-multiplication operators are symmetric because they satisfy
+ the Jordan axiom::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5)
+ sage: x = ComplexHermitianEJA(n).random_element()
+ sage: x.operator().matrix().is_symmetric()
+ True
+
"""
- def __init__(self, n, field=QQ, **kwargs):
+ 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.
+ 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.__class__.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(ComplexHermitianEJA, self)
**kwargs)
- def inner_product(self, x, y):
- # Since a+bi on the diagonal is represented as
- #
- # a + bi = [ a b ]
- # [ -b a ],
- #
- # we'll double-count the "a" entries if we take the trace of
- # the embedding.
- return _matrix_ip(x,y)/2
-
+ @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())
projects / sage.d.git / blobdiff