# 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.
+ basis = tuple( b.change_ring(field) for b in basis )
+
if check_axioms:
# Check commutativity of the Jordan and inner-products.
# we see in things like x = 1*e1 + 2*e2.
vector_basis = basis
+ def flatten(b):
+ # flatten a vector, matrix, or cartesian product of those
+ # things into a long list.
+ if cartesian_product:
+ return sum(( b_i.list() for b_i in b ), [])
+ else:
+ return b.list()
+
degree = 0
if n > 0:
- # Works on both column and square matrices...
- degree = len(basis[0].list())
+ degree = len(flatten(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(flatten(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(flatten(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(flatten(elt)))
self._multiplication_table[i][j] = self.from_vector(elt)
if not orthonormalize:
if not all( J.base_ring() == field for J in modules ):
raise ValueError("all factors must share the same base field")
- basis = tuple( b.to_vector().column() for b in self.basis() )
+ # 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()
+ )
- # 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()
+ # 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_mat, y_mat):
- x = self.from_vector(_mat2vec(x_mat))
- y = self.from_vector(_mat2vec(y_mat))
- return self.cartesian_inner_product(x,y)
+ 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.
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()
- )
-
def matrix_space(self):
r"""
Return the space that our matrix basis lives in as a Cartesian
FiniteDimensionalEJA.CartesianProduct = CartesianProductEJA
+
random_eja = ConcreteEJA.random_instance
+#def random_eja(*args, **kwargs):
+# from sage.categories.cartesian_product import cartesian_product
+# J1 = HadamardEJA(1, **kwargs)
+# J2 = RealSymmetricEJA(2, **kwargs)
+# J = cartesian_product([J1,J2])
+# return J
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