The ``field`` we're given must be real with ``check_field=True``::
- sage: JordanSpinEJA(2,QQbar)
+ sage: JordanSpinEJA(2, field=QQbar)
Traceback (most recent call last):
...
ValueError: scalar field is not real
+ sage: JordanSpinEJA(2, field=QQbar, check_field=False)
+ Euclidean Jordan algebra of dimension 2 over Algebraic Field
The multiplication table must be square with ``check_axioms=True``::
if not all( len(l) == n for l in inner_product_table ):
raise ValueError(msg)
+ # Check commutativity of the Jordan product (symmetry of
+ # the multiplication table) and the commutativity of the
+ # inner-product (symmetry of the inner-product table)
+ # first if we're going to check them at all.. This has to
+ # be done before we define product_on_basis(), because
+ # that method assumes that self._multiplication_table is
+ # symmetric. And it has to be done before we build
+ # self._inner_product_matrix, because the process used to
+ # construct it assumes symmetry as well.
if not all( multiplication_table[j][i]
== multiplication_table[i][j]
for i in range(n)
for j in range(i+1) ):
raise ValueError("Jordan product is not commutative")
+
if not all( inner_product_table[j][i]
== inner_product_table[i][j]
for i in range(n)
for j in range(i+1) ):
raise ValueError("inner-product is not commutative")
+
self._matrix_basis = matrix_basis
if category is None:
# Pre-cache the fact that these are Hermitian (real symmetric,
# in fact) in case some e.g. matrix multiplication routine can
# take advantage of it.
- self._inner_product_matrix = matrix(field, inner_product_table)
+ ip_matrix_constructor = lambda i,j: inner_product_table[i][j] if j <= i else inner_product_table[j][i]
+ self._inner_product_matrix = matrix(field, n, ip_matrix_constructor)
self._inner_product_matrix._cache = {'hermitian': True}
self._inner_product_matrix.set_immutable()
sage: x = J.random_element()
sage: J(x.to_vector().column()) == x
True
+
"""
msg = "not an element of this algebra"
if elt == 0:
# 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.
+ #
+ # We pass check=False because the matrix basis is "guaranteed"
+ # to be linearly independent... right? Ha ha.
V = VectorSpace(self.base_ring(), elt.nrows()*elt.ncols())
- W = V.span_of_basis( _mat2vec(s) for s in self.matrix_basis() )
+ W = V.span_of_basis( (_mat2vec(s) for s in self.matrix_basis()),
+ check=False)
try:
coords = W.coordinate_vector(_mat2vec(elt))
# Used to check whether or not something is zero in an inexact
# ring. This number is sufficient to allow the construction of
- # QuaternionHermitianEJA(2, RDF) with check_axioms=True.
+ # QuaternionHermitianEJA(2, field=RDF) with check_axioms=True.
epsilon = 1e-16
for i in range(self.dimension()):
sage: J = HadamardEJA(2)
sage: J.coordinate_polynomial_ring()
Multivariate Polynomial Ring in X1, X2...
- sage: J = RealSymmetricEJA(3,QQ,orthonormalize=False)
+ sage: J = RealSymmetricEJA(3,field=QQ,orthonormalize=False)
sage: J.coordinate_polynomial_ring()
Multivariate Polynomial Ring in X1, X2, X3, X4, X5, X6...
"""
def __init__(self,
- field,
basis,
jordan_product,
inner_product,
+ field=AA,
orthonormalize=True,
prefix='e',
category=None,
check_field=True,
check_axioms=True):
+ if check_field:
+ # Abuse the check_field parameter to check that the entries of
+ # out basis (in ambient coordinates) are in the field QQ.
+ if not all( all(b_i in QQ for b_i in b.list()) for b in basis ):
+ raise TypeError("basis not rational")
+
+ # Temporary(?) hack to ensure that the matrix and vector bases
+ # are over the same ring.
+ basis = tuple( b.change_ring(field) for b in basis )
+
n = len(basis)
vector_basis = basis
# deorthonormalized basis. These can be used later to
# construct a deorthonormalized copy of this algebra over QQ
# in which several operations are much faster.
- self._deortho_multiplication_table = None
- self._deortho_inner_product_table = None
+ self._rational_algebra = None
if orthonormalize:
- # Compute the deorthonormalized tables before we orthonormalize
- # the given basis.
- W = V.span_of_basis( vector_basis )
+ if self.base_ring() is not QQ:
+ # There's no point in constructing the extra algebra if this
+ # one is already rational. If the original basis is rational
+ # but normalization would make it irrational, then this whole
+ # constructor will just fail anyway as it tries to stick an
+ # irrational number into a rational algebra.
+ #
+ # Note: the same Jordan and inner-products work here,
+ # because they are necessarily defined with respect to
+ # ambient coordinates and not any particular basis.
+ self._rational_algebra = RationalBasisEuclideanJordanAlgebra(
+ basis,
+ jordan_product,
+ inner_product,
+ field=QQ,
+ orthonormalize=False,
+ prefix=prefix,
+ category=category,
+ check_field=False,
+ check_axioms=False)
- # TODO: use symmetry
- self._deortho_multiplication_table = [ [0 for j in range(n)]
- for i in range(n) ]
- self._deortho_inner_product_table = [ [0 for j in range(n)]
- for i in range(n) ]
+ # Compute the deorthonormalized tables before we orthonormalize
+ # the given basis. The "check" parameter here guarantees that
+ # the basis is linearly-independent.
+ W = V.span_of_basis( vector_basis, check=check_axioms)
# Note: the Jordan and inner-products are defined in terms
# of the ambient basis. It's important that their arguments
# table is in terms of vectors
elt = _mat2vec(elt)
- # TODO: use symmetry
- elt = W.coordinate_vector(elt)
- self._deortho_multiplication_table[i][j] = elt
- self._deortho_multiplication_table[j][i] = elt
- self._deortho_inner_product_table[i][j] = ip
- self._deortho_inner_product_table[j][i] = ip
-
- if self._deortho_multiplication_table is not None:
- self._deortho_multiplication_table = tuple(map(tuple, self._deortho_multiplication_table))
- if self._deortho_inner_product_table is not None:
- self._deortho_inner_product_table = tuple(map(tuple, self._deortho_inner_product_table))
-
# We overwrite the name "vector_basis" in a second, but never modify it
# in place, to this effectively makes a copy of it.
deortho_vector_basis = vector_basis
else:
vector_basis = gram_schmidt(vector_basis, inner_product)
- W = V.span_of_basis( vector_basis )
-
# Normalize the "matrix" basis, too!
basis = vector_basis
if basis_is_matrices:
basis = tuple( map(_vec2mat,basis) )
- W = V.span_of_basis( vector_basis )
+ W = V.span_of_basis( vector_basis, check=check_axioms)
# Now "W" is the vector space of our algebra coordinates. The
# variables "X1", "X2",... refer to the entries of vectors in
# W. Thus to convert back and forth between the orthonormal
# coordinates and the given ones, we need to stick the original
# basis in W.
- U = V.span_of_basis( deortho_vector_basis )
+ U = V.span_of_basis( deortho_vector_basis, check=check_axioms)
self._deortho_matrix = matrix( U.coordinate_vector(q)
for q in vector_basis )
- # TODO: use symmetry
- mult_table = [ [0 for j in range(n)] for i in range(n) ]
- ip_table = [ [0 for j in range(n)] for i in range(n) ]
+ # If the superclass constructor is going to verify the
+ # symmetry of this table, it has better at least be
+ # square...
+ if check_axioms:
+ mult_table = [ [0 for j in range(n)] for i in range(n) ]
+ ip_table = [ [0 for j in range(n)] for i in range(n) ]
+ else:
+ mult_table = [ [0 for j in range(i+1)] for i in range(n) ]
+ ip_table = [ [0 for j in range(i+1)] for i in range(n) ]
# Note: the Jordan and inner-products are defined in terms
# of the ambient basis. It's important that their arguments
# table is in terms of vectors
elt = _mat2vec(elt)
- # TODO: use symmetry
elt = W.coordinate_vector(elt)
mult_table[i][j] = elt
- mult_table[j][i] = elt
ip_table[i][j] = ip
- ip_table[j][i] = ip
+ if check_axioms:
+ # The tables are square if we're verifying that they
+ # are commutative.
+ mult_table[j][i] = elt
+ ip_table[j][i] = ip
if basis_is_matrices:
for m in basis:
Algebraic Real Field
"""
- if self.base_ring() is QQ:
+ if self.base_ring() is QQ or self._rational_algebra is None:
# There's no need to construct *another* algebra over the
- # rationals if this one is already over the rationals.
+ # rationals if this one is already over the
+ # rationals. Likewise, if we never orthonormalized our
+ # basis, we might as well just use the given one.
superclass = super(RationalBasisEuclideanJordanAlgebra, self)
return superclass._charpoly_coefficients()
# Do the computation over the rationals. The answer will be
# the same, because all we've done is a change of basis.
- J = FiniteDimensionalEuclideanJordanAlgebra(QQ,
- self._deortho_multiplication_table,
- self._deortho_inner_product_table)
-
- # Change back from QQ to our real base ring
+ # Then, change back from QQ to our real base ring
a = ( a_i.change_ring(self.base_ring())
- for a_i in J._charpoly_coefficients() )
+ for a_i in self._rational_algebra._charpoly_coefficients() )
# Now convert the coordinate variables back to the
# deorthonormalized ones.
In theory, our "field" can be any subfield of the reals::
- sage: RealSymmetricEJA(2, RDF)
+ sage: RealSymmetricEJA(2, field=RDF)
Euclidean Jordan algebra of dimension 3 over Real Double Field
- sage: RealSymmetricEJA(2, RR)
+ sage: RealSymmetricEJA(2, field=RR)
Euclidean Jordan algebra of dimension 3 over Real Field with
53 bits of precision
"""
@classmethod
- def _denormalized_basis(cls, n, field):
+ def _denormalized_basis(cls, n):
"""
Return a basis for the space of real symmetric n-by-n matrices.
sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
- sage: B = RealSymmetricEJA._denormalized_basis(n,QQ)
+ sage: B = RealSymmetricEJA._denormalized_basis(n)
sage: all( M.is_symmetric() for M in B)
True
S = []
for i in range(n):
for j in range(i+1):
- Eij = matrix(field, n, lambda k,l: k==i and l==j)
+ Eij = matrix(ZZ, n, lambda k,l: k==i and l==j)
if i == j:
Sij = Eij
else:
Sij = Eij + Eij.transpose()
S.append(Sij)
- return S
+ return tuple(S)
@staticmethod
return 4 # Dimension 10
@classmethod
- def random_instance(cls, field=AA, **kwargs):
+ def random_instance(cls, **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)
+ return cls(n, **kwargs)
- def __init__(self, n, field=AA, **kwargs):
- basis = self._denormalized_basis(n, field)
- super(RealSymmetricEJA, self).__init__(field,
- basis,
+ def __init__(self, n, **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
+
+ super(RealSymmetricEJA, self).__init__(self._denormalized_basis(n),
self.jordan_product,
self.trace_inner_product,
**kwargs)
self.rank.set_cache(n)
- self.one.set_cache(self(matrix.identity(field,n)))
+ self.one.set_cache(self(matrix.identity(ZZ,n)))
class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra):
In theory, our "field" can be any subfield of the reals::
- sage: ComplexHermitianEJA(2, RDF)
+ sage: ComplexHermitianEJA(2, field=RDF)
Euclidean Jordan algebra of dimension 4 over Real Double Field
- sage: ComplexHermitianEJA(2, RR)
+ sage: ComplexHermitianEJA(2, field=RR)
Euclidean Jordan algebra of dimension 4 over Real Field with
53 bits of precision
"""
@classmethod
- def _denormalized_basis(cls, n, field):
+ def _denormalized_basis(cls, n):
"""
Returns a basis for the space of complex Hermitian n-by-n matrices.
sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
sage: field = QuadraticField(2, 'sqrt2')
- sage: B = ComplexHermitianEJA._denormalized_basis(n, field)
+ sage: B = ComplexHermitianEJA._denormalized_basis(n)
sage: all( M.is_symmetric() for M in B)
True
"""
+ field = ZZ
R = PolynomialRing(field, 'z')
z = R.gen()
F = field.extension(z**2 + 1, 'I')
- I = F.gen()
+ I = F.gen(1)
# This is like the symmetric case, but we need to be careful:
#
return tuple( s.change_ring(field) for s in S )
- def __init__(self, n, field=AA, **kwargs):
- basis = self._denormalized_basis(n,field)
- super(ComplexHermitianEJA, self).__init__(field,
- basis,
- self.jordan_product,
- self.trace_inner_product,
- **kwargs)
+ def __init__(self, n, **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
+
+ super(ComplexHermitianEJA, self).__init__(self._denormalized_basis(n),
+ self.jordan_product,
+ self.trace_inner_product,
+ **kwargs)
self.rank.set_cache(n)
# TODO: pre-cache the identity!
return 3 # Dimension 9
@classmethod
- def random_instance(cls, field=AA, **kwargs):
+ def random_instance(cls, **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)
+ return cls(n, **kwargs)
class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra):
@staticmethod
In theory, our "field" can be any subfield of the reals::
- sage: QuaternionHermitianEJA(2, RDF)
+ sage: QuaternionHermitianEJA(2, field=RDF)
Euclidean Jordan algebra of dimension 6 over Real Double Field
- sage: QuaternionHermitianEJA(2, RR)
+ sage: QuaternionHermitianEJA(2, field=RR)
Euclidean Jordan algebra of dimension 6 over Real Field with
53 bits of precision
"""
@classmethod
- def _denormalized_basis(cls, n, field):
+ def _denormalized_basis(cls, n):
"""
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 = QuaternionHermitianEJA._denormalized_basis(n,QQ)
+ sage: B = QuaternionHermitianEJA._denormalized_basis(n)
sage: all( M.is_symmetric() for M in B )
True
"""
+ field = ZZ
Q = QuaternionAlgebra(QQ,-1,-1)
I,J,K = Q.gens()
return tuple( s.change_ring(field) for s in S )
- def __init__(self, n, field=AA, **kwargs):
- basis = self._denormalized_basis(n,field)
- super(QuaternionHermitianEJA, self).__init__(field,
- basis,
+ def __init__(self, n, **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
+
+ super(QuaternionHermitianEJA, self).__init__(self._denormalized_basis(n),
self.jordan_product,
self.trace_inner_product,
**kwargs)
return 2 # Dimension 6
@classmethod
- def random_instance(cls, field=AA, **kwargs):
+ def random_instance(cls, **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)
+ return cls(n, **kwargs)
class HadamardEJA(ConcreteEuclideanJordanAlgebra):
(r0, r1, r2)
"""
- def __init__(self, n, field=AA, **kwargs):
- V = VectorSpace(field, n)
- basis = V.basis()
-
+ def __init__(self, n, **kwargs):
def jordan_product(x,y):
- return V([ xi*yi for (xi,yi) in zip(x,y) ])
+ P = x.parent()
+ return P(tuple( xi*yi for (xi,yi) in zip(x,y) ))
def inner_product(x,y):
return x.inner_product(y)
- super(HadamardEJA, self).__init__(field,
- basis,
+ # New defaults for keyword arguments. Don't orthonormalize
+ # because our basis is already orthonormal with respect to our
+ # inner-product. Don't check the axioms, because we know this
+ # is a valid EJA... but do double-check if the user passes
+ # check_axioms=True. Note: we DON'T override the "check_field"
+ # default here, because the user can pass in a field!
+ if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False
+ if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
+
+
+ standard_basis = FreeModule(ZZ, n).basis()
+ super(HadamardEJA, self).__init__(standard_basis,
jordan_product,
inner_product,
**kwargs)
return 5
@classmethod
- def random_instance(cls, field=AA, **kwargs):
+ def random_instance(cls, **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)
+ return cls(n, **kwargs)
class BilinearFormEJA(ConcreteEuclideanJordanAlgebra):
sage: actual == expected
True
"""
- def __init__(self, B, field=AA, **kwargs):
+ def __init__(self, B, **kwargs):
if not B.is_positive_definite():
raise ValueError("bilinear form is not positive-definite")
- n = B.nrows()
- V = VectorSpace(field, n)
-
def inner_product(x,y):
return (B*x).inner_product(y)
def jordan_product(x,y):
+ P = x.parent()
x0 = x[0]
xbar = x[1:]
y0 = y[0]
ybar = y[1:]
z0 = inner_product(x,y)
zbar = y0*xbar + x0*ybar
- return V([z0] + zbar.list())
+ return P((z0,) + tuple(zbar))
+
+ # 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
- super(BilinearFormEJA, self).__init__(field,
- V.basis(),
+ n = B.nrows()
+ standard_basis = FreeModule(ZZ, n).basis()
+ super(BilinearFormEJA, self).__init__(standard_basis,
jordan_product,
inner_product,
**kwargs)
return 5
@classmethod
- def random_instance(cls, field=AA, **kwargs):
+ def random_instance(cls, **kwargs):
"""
Return a random instance of this algebra.
"""
n = ZZ.random_element(cls._max_random_instance_size() + 1)
if n.is_zero():
- B = matrix.identity(field, n)
- return cls(B, field, **kwargs)
+ B = matrix.identity(ZZ, n)
+ return cls(B, **kwargs)
- B11 = matrix.identity(field,1)
- M = matrix.random(field, n-1)
- I = matrix.identity(field, n-1)
- alpha = field.zero()
+ B11 = matrix.identity(ZZ, 1)
+ M = matrix.random(ZZ, n-1)
+ I = matrix.identity(ZZ, n-1)
+ alpha = ZZ.zero()
while alpha.is_zero():
- alpha = field.random_element().abs()
+ alpha = ZZ.random_element().abs()
B22 = M.transpose()*M + alpha*I
from sage.matrix.special import block_matrix
B = block_matrix(2,2, [ [B11, ZZ(0) ],
[ZZ(0), B22 ] ])
- return cls(B, field, **kwargs)
+ return cls(B, **kwargs)
class JordanSpinEJA(BilinearFormEJA):
True
"""
- def __init__(self, n, field=AA, **kwargs):
- # This is a special case of the BilinearFormEJA with the identity
- # matrix as its bilinear form.
- B = matrix.identity(field, n)
- super(JordanSpinEJA, self).__init__(B, field, **kwargs)
+ def __init__(self, n, **kwargs):
+ # This is a special case of the BilinearFormEJA with the
+ # identity matrix as its bilinear form.
+ B = matrix.identity(ZZ, n)
+
+ # Don't orthonormalize because our basis is already
+ # orthonormal with respect to our inner-product.
+ if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False
+
+ # But also don't pass check_field=False here, because the user
+ # can pass in a field!
+ super(JordanSpinEJA, self).__init__(B, **kwargs)
@staticmethod
def _max_random_instance_size():
return 5
@classmethod
- def random_instance(cls, field=AA, **kwargs):
+ def random_instance(cls, **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)
- return cls(n, field, **kwargs)
+ return cls(n, **kwargs)
class TrivialEJA(ConcreteEuclideanJordanAlgebra):
0
"""
- def __init__(self, field=AA, **kwargs):
+ def __init__(self, **kwargs):
jordan_product = lambda x,y: x
- inner_product = lambda x,y: field(0)
+ inner_product = lambda x,y: 0
basis = ()
- super(TrivialEJA, self).__init__(field,
- basis,
+
+ # New defaults for keyword arguments
+ if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False
+ if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
+
+ super(TrivialEJA, self).__init__(basis,
jordan_product,
inner_product,
**kwargs)
self.one.set_cache( self.zero() )
@classmethod
- def random_instance(cls, field=AA, **kwargs):
+ def random_instance(cls, **kwargs):
# We don't take a "size" argument so the superclass method is
# inappropriate for us.
- return cls(field, **kwargs)
+ return cls(**kwargs)
class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra):
r"""
have the same base ring; an error is raised otherwise::
sage: set_random_seed()
- sage: J1 = random_eja(AA)
- sage: J2 = random_eja(QQ,orthonormalize=False)
+ sage: J1 = random_eja(field=AA)
+ sage: J2 = random_eja(field=QQ,orthonormalize=False)
sage: J = DirectSumEJA(J1,J2)
Traceback (most recent call last):
...
n2 = J2.dimension()
n = n1+n2
V = VectorSpace(field, n)
- mult_table = [ [ V.zero() for j in range(n) ]
+ mult_table = [ [ V.zero() for j in range(i+1) ]
for i in range(n) ]
for i in range(n1):
- for j in range(n1):
+ for j in range(i+1):
p = (J1.monomial(i)*J1.monomial(j)).to_vector()
mult_table[i][j] = V(p.list() + [field.zero()]*n2)
for i in range(n2):
- for j in range(n2):
+ for j in range(i+1):
p = (J2.monomial(i)*J2.monomial(j)).to_vector()
mult_table[n1+i][n1+j] = V([field.zero()]*n1 + p.list())
# TODO: build the IP table here from the two constituent IP
# matrices (it'll be block diagonal, I think).
- ip_table = None
+ ip_table = [ [ field.zero() for j in range(i+1) ]
+ for i in range(n) ]
super(DirectSumEJA, self).__init__(field,
mult_table,
ip_table,
EXAMPLES::
- sage: J1 = HadamardEJA(2,QQ)
- sage: J2 = JordanSpinEJA(3,QQ)
+ sage: J1 = HadamardEJA(2, field=QQ)
+ sage: J2 = JordanSpinEJA(3, field=QQ)
sage: J = DirectSumEJA(J1,J2)
sage: J.factors()
(Euclidean Jordan algebra of dimension 2 over Rational Field,
EXAMPLE::
- sage: J1 = HadamardEJA(3,QQ)
- sage: J2 = QuaternionHermitianEJA(2,QQ,orthonormalize=False)
+ sage: J1 = HadamardEJA(3,field=QQ)
+ sage: J2 = QuaternionHermitianEJA(2,field=QQ,orthonormalize=False)
sage: J = DirectSumEJA(J1,J2)
sage: x1 = J1.one()
sage: x2 = x1
projects / sage.d.git / blobdiff