from sage.rings.all import (ZZ, QQ, AA, QQbar, RR, RLF, CLF,
PolynomialRing,
QuadraticField)
-from mjo.eja.eja_element import (CartesianProductEJAElement,
- FiniteDimensionalEJAElement)
+from mjo.eja.eja_element import FiniteDimensionalEJAElement
from mjo.eja.eja_operator import FiniteDimensionalEJAOperator
-from mjo.eja.eja_utils import _mat2vec
+from mjo.eja.eja_utils import _all2list, _mat2vec
class FiniteDimensionalEJA(CombinatorialFreeModule):
r"""
`(a,b)` into column matrices `(a,b)^{T}` after converting
`a` and `b` themselves.
- - jordan_product -- function of two elements (in matrix form)
- that returns their jordan product in this algebra; this will
- be applied to ``basis`` to compute a multiplication table for
- the algebra.
-
- - inner_product -- function of two elements (in matrix form) that
- returns their inner product. This will be applied to ``basis`` to
- compute an inner-product table (basically a matrix) for this algebra.
+ - jordan_product -- function of two ``basis`` elements (in
+ matrix form) that returns their jordan product, also in matrix
+ form; this will be applied to ``basis`` to compute a
+ multiplication table for the algebra.
+ - inner_product -- function of two ``basis`` elements (in matrix
+ form) that returns their inner product. This will be applied
+ to ``basis`` to compute an inner-product table (basically a
+ matrix) for this algebra.
"""
Element = FiniteDimensionalEJAElement
check_axioms=True,
prefix='e'):
+ # Keep track of whether or not the matrix basis consists of
+ # tuples, since we need special cases for them damned near
+ # everywhere. This is INDEPENDENT of whether or not the
+ # algebra is a cartesian product, since a subalgebra of a
+ # cartesian product will have a basis of tuples, but will not
+ # in general itself be a cartesian product algebra.
+ self._matrix_basis_is_cartesian = False
+ n = len(basis)
+ if n > 0:
+ if hasattr(basis[0], 'cartesian_factors'):
+ self._matrix_basis_is_cartesian = True
+
if check_field:
if not field.is_subring(RR):
# Note: this does return true for the real algebraic
# If the basis given to us wasn't over the field that it's
# supposed to be over, fix that. Or, you know, crash.
- basis = tuple( b.change_ring(field) for b in basis )
+ if not cartesian_product:
+ # The field for a cartesian product algebra comes from one
+ # of its factors and is the same for all factors, so
+ # there's no need to "reapply" it on product algebras.
+ if self._matrix_basis_is_cartesian:
+ # OK since if n == 0, the basis does not consist of tuples.
+ P = basis[0].parent()
+ basis = tuple( P(tuple(b_i.change_ring(field) for b_i in b))
+ for b in basis )
+ else:
+ basis = tuple( b.change_ring(field) for b in basis )
+
if check_axioms:
# Check commutativity of the Jordan and inner-products.
# Call the superclass constructor so that we can use its from_vector()
# method to build our multiplication table.
- n = len(basis)
- super().__init__(field,
- range(n),
- prefix=prefix,
- category=category,
- bracket=False)
+ CombinatorialFreeModule.__init__(self,
+ field,
+ range(n),
+ prefix=prefix,
+ category=category,
+ bracket=False)
# Now comes all of the hard work. We'll be constructing an
# ambient vector space V that our (vectorized) basis lives in,
degree = 0
if n > 0:
- # Works on both column and square matrices...
- degree = len(basis[0].list())
+ degree = len(_all2list(basis[0]))
# Build an ambient space that fits our matrix basis when
# written out as "long vectors."
# Save a copy of the un-orthonormalized basis for later.
# Convert it to ambient V (vector) coordinates while we're
# at it, because we'd have to do it later anyway.
- deortho_vector_basis = tuple( V(b.list()) for b in basis )
+ deortho_vector_basis = tuple( V(_all2list(b)) for b in basis )
from mjo.eja.eja_utils import gram_schmidt
basis = tuple(gram_schmidt(basis, inner_product))
# Now create the vector space for the algebra, which will have
# its own set of non-ambient coordinates (in terms of the
# supplied basis).
- vector_basis = tuple( V(b.list()) for b in basis )
+ vector_basis = tuple( V(_all2list(b)) for b in basis )
W = V.span_of_basis( vector_basis, check=check_axioms)
if orthonormalize:
# The jordan product returns a matrixy answer, so we
# have to convert it to the algebra coordinates.
elt = jordan_product(q_i, q_j)
- elt = W.coordinate_vector(V(elt.list()))
+ elt = W.coordinate_vector(V(_all2list(elt)))
self._multiplication_table[i][j] = self.from_vector(elt)
if not orthonormalize:
def product_on_basis(self, i, j):
+ r"""
+ Returns the Jordan product of the `i` and `j`th basis elements.
+
+ This completely defines the Jordan product on the algebra, and
+ is used direclty by our superclass machinery to implement
+ :meth:`product`.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import random_eja
+
+ TESTS::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: n = J.dimension()
+ sage: ei = J.zero()
+ sage: ej = J.zero()
+ sage: ei_ej = J.zero()*J.zero()
+ sage: if n > 0:
+ ....: i = ZZ.random_element(n)
+ ....: j = ZZ.random_element(n)
+ ....: ei = J.gens()[i]
+ ....: ej = J.gens()[j]
+ ....: ei_ej = J.product_on_basis(i,j)
+ sage: ei*ej == ei_ej
+ True
+
+ """
# We only stored the lower-triangular portion of the
# multiplication table.
if j <= i:
sage: y = J.random_element()
sage: (n == 1) or (x.inner_product(y) == (x*y).trace()/2)
True
+
"""
B = self._inner_product_matrix
return (B*x.to_vector()).inner_product(y.to_vector())
- def _is_commutative(self):
+ def is_associative(self):
r"""
- Whether or not this algebra's multiplication table is commutative.
+ Return whether or not this algebra's Jordan product is associative.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
+
+ EXAMPLES::
+
+ sage: J = ComplexHermitianEJA(3, field=QQ, orthonormalize=False)
+ sage: J.is_associative()
+ False
+ sage: x = sum(J.gens())
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
+ sage: A.is_associative()
+ True
- This method should of course always return ``True``, unless
- this algebra was constructed with ``check_axioms=False`` and
- passed an invalid multiplication table.
"""
- return all( self.product_on_basis(i,j) == self.product_on_basis(i,j)
- for i in range(self.dimension())
- for j in range(self.dimension()) )
+ return "Associative" in self.category().axioms()
def _is_jordanian(self):
r"""
We only check one arrangement of `x` and `y`, so for a
``True`` result to be truly true, you should also check
- :meth:`_is_commutative`. This method should of course always
+ :meth:`is_commutative`. This method should of course always
return ``True``, unless this algebra was constructed with
``check_axioms=False`` and passed an invalid multiplication table.
"""
...
ValueError: not an element of this algebra
+ Tuples work as well, provided that the matrix basis for the
+ algebra consists of them::
+
+ sage: J1 = HadamardEJA(3)
+ sage: J2 = RealSymmetricEJA(2)
+ sage: J = cartesian_product([J1,J2])
+ sage: J( (J1.matrix_basis()[1], J2.matrix_basis()[2]) )
+ e(0, 1) + e(1, 2)
+
TESTS:
Ensure that we can convert any element of the two non-matrix
sage: J(x.to_vector().column()) == x
True
+ We cannot coerce elements between algebras just because their
+ matrix representations are compatible::
+
+ sage: J1 = HadamardEJA(3)
+ sage: J2 = JordanSpinEJA(3)
+ sage: J2(J1.one())
+ Traceback (most recent call last):
+ ...
+ ValueError: not an element of this algebra
+ sage: J1(J2.zero())
+ Traceback (most recent call last):
+ ...
+ ValueError: not an element of this algebra
+
"""
msg = "not an element of this algebra"
- if elt == 0:
- # The superclass implementation of random_element()
- # needs to be able to coerce "0" into the algebra.
- return self.zero()
- elif elt in self.base_ring():
+ if elt in self.base_ring():
# Ensure that no base ring -> algebra coercion is performed
# by this method. There's some stupidity in sage that would
# otherwise propagate to this method; for example, sage thinks
raise ValueError(msg)
try:
+ # Try to convert a vector into a column-matrix...
elt = elt.column()
except (AttributeError, TypeError):
- # Try to convert a vector into a column-matrix
+ # and ignore failure, because we weren't really expecting
+ # a vector as an argument anyway.
pass
if elt not in self.matrix_space():
# closure whereas the base ring of the 3-by-3 identity matrix
# could be QQ instead of QQbar.
#
+ # And, we also have to handle Cartesian product bases (when
+ # the matrix basis consists of tuples) here. The "good news"
+ # is that we're already converting everything to long vectors,
+ # and that strategy works for tuples as well.
+ #
# 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()),
+ elt = _all2list(elt)
+ V = VectorSpace(self.base_ring(), len(elt))
+ W = V.span_of_basis( (V(_all2list(s)) for s in self.matrix_basis()),
check=False)
try:
- coords = W.coordinate_vector(_mat2vec(elt))
+ coords = W.coordinate_vector(V(elt))
except ArithmeticError: # vector is not in free module
raise ValueError(msg)
we think of them as matrices (including column vectors of the
appropriate size).
- Generally this will be an `n`-by-`1` column-vector space,
+ "By default" this will be an `n`-by-`1` column-matrix space,
except when the algebra is trivial. There it's `n`-by-`n`
(where `n` is zero), to ensure that two elements of the matrix
- space (empty matrices) can be multiplied.
+ space (empty matrices) can be multiplied. For algebras of
+ matrices, this returns the space in which their
+ real embeddings live.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
+ ....: JordanSpinEJA,
+ ....: QuaternionHermitianEJA,
+ ....: TrivialEJA)
+
+ EXAMPLES:
+
+ By default, the matrix representation is just a column-matrix
+ equivalent to the vector representation::
+
+ sage: J = JordanSpinEJA(3)
+ sage: J.matrix_space()
+ Full MatrixSpace of 3 by 1 dense matrices over Algebraic
+ Real Field
+
+ The matrix representation in the trivial algebra is
+ zero-by-zero instead of the usual `n`-by-one::
+
+ sage: J = TrivialEJA()
+ sage: J.matrix_space()
+ Full MatrixSpace of 0 by 0 dense matrices over Algebraic
+ Real Field
+
+ The matrix space for complex/quaternion Hermitian matrix EJA
+ is the space in which their real-embeddings live, not the
+ original complex/quaternion matrix space::
+
+ sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False)
+ sage: J.matrix_space()
+ Full MatrixSpace of 4 by 4 dense matrices over Rational Field
+ sage: J = QuaternionHermitianEJA(1,field=QQ,orthonormalize=False)
+ sage: J.matrix_space()
+ Full MatrixSpace of 4 by 4 dense matrices over Rational Field
- Matrix algebras override this with something more useful.
"""
if self.is_trivial():
return MatrixSpace(self.base_ring(), 0)
if not c.is_idempotent():
raise ValueError("element is not idempotent: %s" % c)
- from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra
-
# Default these to what they should be if they turn out to be
# trivial, because eigenspaces_left() won't return eigenvalues
# corresponding to trivial spaces (e.g. it returns only the
# eigenspace corresponding to lambda=1 if you take the
# decomposition relative to the identity element).
- trivial = FiniteDimensionalEJASubalgebra(self, ())
+ trivial = self.subalgebra(())
J0 = trivial # eigenvalue zero
J5 = VectorSpace(self.base_ring(), 0) # eigenvalue one-half
J1 = trivial # eigenvalue one
J5 = eigspace
else:
gens = tuple( self.from_vector(b) for b in eigspace.basis() )
- subalg = FiniteDimensionalEJASubalgebra(self,
- gens,
- check_axioms=False)
+ subalg = self.subalgebra(gens, check_axioms=False)
if eigval == 0:
J0 = subalg
elif eigval == 1:
return len(self._charpoly_coefficients())
+ def subalgebra(self, basis, **kwargs):
+ r"""
+ Create a subalgebra of this algebra from the given basis.
+ """
+ from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra
+ return FiniteDimensionalEJASubalgebra(self, basis, **kwargs)
+
+
def vector_space(self):
"""
Return the vector space that underlies this algebra.
return self.zero().to_vector().parent().ambient_vector_space()
- Element = FiniteDimensionalEJAElement
class RationalBasisEJA(FiniteDimensionalEJA):
r"""
if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
column_basis = tuple( b.column() for b in FreeModule(ZZ, n).basis() )
- super().__init__(column_basis, jordan_product, inner_product, **kwargs)
+ super().__init__(column_basis,
+ jordan_product,
+ inner_product,
+ associative=True,
+ **kwargs)
self.rank.set_cache(n)
if n == 0:
sage: J.rank() == J1.rank() + J2.rank()
True
+ The product algebra will be associative if and only if all of its
+ components are associative::
+
+ sage: J1 = HadamardEJA(2)
+ sage: J1.is_associative()
+ True
+ sage: J2 = HadamardEJA(3)
+ sage: J2.is_associative()
+ True
+ sage: J3 = RealSymmetricEJA(3)
+ sage: J3.is_associative()
+ False
+ sage: CP1 = cartesian_product([J1,J2])
+ sage: CP1.is_associative()
+ True
+ sage: CP2 = cartesian_product([J1,J3])
+ sage: CP2.is_associative()
+ False
+
TESTS:
All factors must share the same base field::
...
ValueError: all factors must share the same base field
- The "cached" Jordan and inner products are the componentwise
- ones::
-
- sage: set_random_seed()
- sage: J1 = random_eja()
- sage: J2 = random_eja()
- sage: J = cartesian_product([J1,J2])
- sage: x,y = J.random_elements(2)
- sage: x*y == J.cartesian_jordan_product(x,y)
- True
- sage: x.inner_product(y) == J.cartesian_inner_product(x,y)
- True
-
The cached unit element is the same one that would be computed::
sage: set_random_seed() # long time
True
"""
- def __init__(self, modules, **kwargs):
+ Element = FiniteDimensionalEJAElement
+
+
+ def __init__(self, algebras, **kwargs):
CombinatorialFreeModule_CartesianProduct.__init__(self,
- modules,
+ algebras,
**kwargs)
- field = modules[0].base_ring()
- if not all( J.base_ring() == field for J in modules ):
+ field = algebras[0].base_ring()
+ if not all( J.base_ring() == field for J in algebras ):
raise ValueError("all factors must share the same base field")
- basis = tuple( b.to_vector().column() for b in self.basis() )
+ associative = all( m.is_associative() for m in algebras )
- # Define jordan/inner products that operate on the basis.
- def jordan_product(x_mat,y_mat):
- x = self.from_vector(_mat2vec(x_mat))
- y = self.from_vector(_mat2vec(y_mat))
- return self.cartesian_jordan_product(x,y).to_vector().column()
+ # The definition of matrix_space() and self.basis() relies
+ # only on the stuff in the CFM_CartesianProduct class, which
+ # we've already initialized.
+ Js = self.cartesian_factors()
+ m = len(Js)
+ MS = self.matrix_space()
+ basis = tuple(
+ MS(tuple( self.cartesian_projection(i)(b).to_matrix()
+ for i in range(m) ))
+ for b in self.basis()
+ )
- def inner_product(x_mat, y_mat):
- x = self.from_vector(_mat2vec(x_mat))
- y = self.from_vector(_mat2vec(y_mat))
- return self.cartesian_inner_product(x,y)
+ # Define jordan/inner products that operate on that matrix_basis.
+ def jordan_product(x,y):
+ return MS(tuple(
+ (Js[i](x[i])*Js[i](y[i])).to_matrix() for i in range(m)
+ ))
+
+ def inner_product(x, y):
+ return sum(
+ Js[i](x[i]).inner_product(Js[i](y[i])) for i in range(m)
+ )
- # Use whatever category the superclass came up with. Usually
- # some join of the EJA and Cartesian product
- # categories. There's no need to check the field since it
- # already came from an EJA.
+ # There's no need to check the field since it already came
+ # from an EJA. Likewise the axioms are guaranteed to be
+ # satisfied, unless the guy writing this class sucks.
#
# If you want the basis to be orthonormalized, orthonormalize
# the factors.
inner_product,
field=field,
orthonormalize=False,
+ associative=associative,
cartesian_product=True,
check_field=False,
check_axioms=False)
- ones = tuple(J.one() for J in modules)
+ ones = tuple(J.one() for J in algebras)
self.one.set_cache(self._cartesian_product_of_elements(ones))
- self.rank.set_cache(sum(J.rank() for J in modules))
-
- # Now that everything else is ready, we clobber our computed
- # matrix basis with the "correct" one consisting of ordered
- # tuples. Since we didn't orthonormalize our basis, we can
- # create these from the basis that was handed to us; that is,
- # we don't need to use the one that the earlier __init__()
- # method came up with.
- m = len(self.cartesian_factors())
- MS = self.matrix_space()
- self._matrix_basis = tuple(
- MS(tuple( self.cartesian_projection(i)(b).to_matrix()
- for i in range(m) ))
- for b in self.basis()
- )
+ self.rank.set_cache(sum(J.rank() for J in algebras))
def matrix_space(self):
r"""
return FiniteDimensionalEJAOperator(Ji,self,Ei.matrix())
- def cartesian_jordan_product(self, x, y):
- r"""
- The componentwise Jordan product.
-
- We project ``x`` and ``y`` onto our factors, and add up the
- Jordan products from the subalgebras. This may still be useful
- after (if) the default Jordan product in the Cartesian product
- algebra is overridden.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import (HadamardEJA,
- ....: JordanSpinEJA)
-
- EXAMPLE::
-
- sage: J1 = HadamardEJA(3)
- sage: J2 = JordanSpinEJA(3)
- sage: J = cartesian_product([J1,J2])
- sage: x1 = J1.from_vector(vector(QQ,(1,2,1)))
- sage: y1 = J1.from_vector(vector(QQ,(1,0,2)))
- sage: x2 = J2.from_vector(vector(QQ,(1,2,3)))
- sage: y2 = J2.from_vector(vector(QQ,(1,1,1)))
- sage: z1 = J.from_vector(vector(QQ,(1,2,1,1,2,3)))
- sage: z2 = J.from_vector(vector(QQ,(1,0,2,1,1,1)))
- sage: (x1*y1).to_vector()
- (1, 0, 2)
- sage: (x2*y2).to_vector()
- (6, 3, 4)
- sage: J.cartesian_jordan_product(z1,z2).to_vector()
- (1, 0, 2, 6, 3, 4)
-
- """
- m = len(self.cartesian_factors())
- projections = ( self.cartesian_projection(i) for i in range(m) )
- products = ( P(x)*P(y) for P in projections )
- return self._cartesian_product_of_elements(tuple(products))
-
- def cartesian_inner_product(self, x, y):
- r"""
- The standard componentwise Cartesian inner-product.
-
- We project ``x`` and ``y`` onto our factors, and add up the
- inner-products from the subalgebras. This may still be useful
- after (if) the default inner product in the Cartesian product
- algebra is overridden.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import (HadamardEJA,
- ....: QuaternionHermitianEJA)
-
- EXAMPLE::
-
- sage: J1 = HadamardEJA(3,field=QQ)
- sage: J2 = QuaternionHermitianEJA(2,field=QQ,orthonormalize=False)
- sage: J = cartesian_product([J1,J2])
- sage: x1 = J1.one()
- sage: x2 = x1
- sage: y1 = J2.one()
- sage: y2 = y1
- sage: x1.inner_product(x2)
- 3
- sage: y1.inner_product(y2)
- 2
- sage: z1 = J._cartesian_product_of_elements((x1,y1))
- sage: z2 = J._cartesian_product_of_elements((x2,y2))
- sage: J.cartesian_inner_product(z1,z2)
- 5
-
- """
- m = len(self.cartesian_factors())
- projections = ( self.cartesian_projection(i) for i in range(m) )
- return sum( P(x).inner_product(P(y)) for P in projections )
-
-
- def _element_constructor_(self, elt):
- r"""
- Construct an element of this algebra from an ordered tuple.
-
- We just apply the element constructor from each of our factors
- to the corresponding component of the tuple, and package up
- the result.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import (HadamardEJA,
- ....: RealSymmetricEJA)
-
- EXAMPLES::
-
- sage: J1 = HadamardEJA(3)
- sage: J2 = RealSymmetricEJA(2)
- sage: J = cartesian_product([J1,J2])
- sage: J( (J1.matrix_basis()[1], J2.matrix_basis()[2]) )
- e(0, 1) + e(1, 2)
- """
- m = len(self.cartesian_factors())
- try:
- z = tuple( self.cartesian_factors()[i](elt[i]) for i in range(m) )
- return self._cartesian_product_of_elements(z)
- except:
- raise ValueError("not an element of this algebra")
-
- Element = CartesianProductEJAElement
-
FiniteDimensionalEJA.CartesianProduct = CartesianProductEJA
+
random_eja = ConcreteEJA.random_instance
projects / sage.d.git / blobdiff