-from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_element import FiniteDimensionalAlgebraElement
from sage.matrix.constructor import matrix
from sage.modules.free_module import VectorSpace
+from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement
# TODO: make this unnecessary somehow.
from sage.misc.lazy_import import lazy_import
from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator
from mjo.eja.eja_utils import _mat2vec
-class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraElement):
+class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
"""
An element of a Euclidean Jordan algebra.
"""
dir(self.__class__) )
- def __init__(self, A, elt=None):
- """
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import (RealSymmetricEJA,
- ....: random_eja)
- EXAMPLES:
-
- The identity in `S^n` is converted to the identity in the EJA::
-
- sage: J = RealSymmetricEJA(3)
- sage: I = matrix.identity(QQ,3)
- sage: J(I) == J.one()
- True
-
- This skew-symmetric matrix can't be represented in the EJA::
-
- sage: J = RealSymmetricEJA(3)
- sage: A = matrix(QQ,3, lambda i,j: i-j)
- sage: J(A)
- Traceback (most recent call last):
- ...
- ArithmeticError: vector is not in free module
-
- TESTS:
-
- Ensure that we can convert any element of the parent's
- underlying vector space back into an algebra element whose
- vector representation is what we started with::
-
- sage: set_random_seed()
- sage: J = random_eja()
- sage: v = J.vector_space().random_element()
- sage: J(v).vector() == v
- True
-
- """
- # Goal: if we're given a matrix, and if it lives in our
- # parent algebra's "natural ambient space," convert it
- # into an algebra element.
- #
- # The catch is, we make a recursive call after converting
- # the given matrix into a vector that lives in the algebra.
- # This we need to try the parent class initializer first,
- # to avoid recursing forever if we're given something that
- # already fits into the algebra, but also happens to live
- # in the parent's "natural ambient space" (this happens with
- # vectors in R^n).
- try:
- FiniteDimensionalAlgebraElement.__init__(self, A, elt)
- except ValueError:
- natural_basis = A.natural_basis()
- if elt in natural_basis[0].matrix_space():
- # 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()**2)
- W = V.span( _mat2vec(s) for s in natural_basis )
- coords = W.coordinates(_mat2vec(elt))
- FiniteDimensionalAlgebraElement.__init__(self, A, coords)
def __pow__(self, n):
"""
"""
p = self.parent().characteristic_polynomial()
- return p(*self.vector())
+ return p(*self.to_vector())
def inner_product(self, other):
sage: y = vector(QQ,[4,5,6])
sage: x.inner_product(y)
32
- sage: J(x).inner_product(J(y))
+ sage: J.from_vector(x).inner_product(J.from_vector(y))
32
The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
# -1 to ensure that _charpoly_coeff(0) is really what
# appears in front of t^{0} in the charpoly. However,
# we want (-1)^r times THAT for the determinant.
- return ((-1)**r)*p(*self.vector())
+ return ((-1)**r)*p(*self.to_vector())
def inverse(self):
sage: x = J.random_element()
sage: while not x.is_invertible():
....: x = J.random_element()
- sage: x_vec = x.vector()
+ sage: x_vec = x.to_vector()
sage: x0 = x_vec[0]
sage: x_bar = x_vec[1:]
sage: coeff = ~(x0^2 - x_bar.inner_product(x_bar))
sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
sage: x_inverse = coeff*inv_vec
- sage: x.inverse() == J(x_inverse)
+ sage: x.inverse() == J.from_vector(x_inverse)
True
TESTS:
False
"""
- zero = self.parent().zero()
+ # In fact, we only need to know if the constant term is non-zero,
+ # so we can pass in the field's zero element instead.
+ zero = self.base_ring().zero()
p = self.minimal_polynomial()
return not (p(zero) == zero)
sage: y = J.random_element()
sage: while y == y.coefficient(0)*J.one():
....: y = J.random_element()
- sage: y0 = y.vector()[0]
- sage: y_bar = y.vector()[1:]
+ sage: y0 = y.to_vector()[0]
+ sage: y_bar = y.to_vector()[1:]
sage: actual = y.minimal_polynomial()
sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
sage: J = ComplexHermitianEJA(3)
sage: J.one()
- e0 + e5 + e8
+ e0 + e3 + e8
sage: J.one().natural_representation()
[1 0 0 0 0 0]
[0 1 0 0 0 0]
sage: J = QuaternionHermitianEJA(3)
sage: J.one()
- e0 + e9 + e14
+ e0 + e5 + e14
sage: J.one().natural_representation()
[1 0 0 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0 0]
"""
B = self.parent().natural_basis()
W = B[0].matrix_space()
- return W.linear_combination(zip(self.vector(), B))
+ return W.linear_combination(zip(B,self.to_vector()))
def operator(self):
"""
P = self.parent()
- fda_elt = FiniteDimensionalAlgebraElement(P, self)
+ left_mult_by_self = lambda y: self*y
+ L = P.module_morphism(function=left_mult_by_self, codomain=P)
return FiniteDimensionalEuclideanJordanAlgebraOperator(
P,
P,
- fda_elt.matrix().transpose() )
+ L.matrix() )
def quadratic_representation(self, other=None):
sage: n = ZZ.random_element(1,10)
sage: J = JordanSpinEJA(n)
sage: x = J.random_element()
- sage: x_vec = x.vector()
+ sage: x_vec = x.to_vector()
sage: x0 = x_vec[0]
sage: x_bar = x_vec[1:]
sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
sage: from mjo.eja.eja_algebra import random_eja
- TESTS::
+ TESTS:
+
+ This subalgebra, being composed of only powers, is associative::
sage: set_random_seed()
- sage: x = random_eja().random_element()
- sage: x.subalgebra_generated_by().is_associative()
+ sage: x0 = random_eja().random_element()
+ sage: A = x0.subalgebra_generated_by()
+ sage: x = A.random_element()
+ sage: y = A.random_element()
+ sage: z = A.random_element()
+ sage: (x*y)*z == x*(y*z)
True
Squaring in the subalgebra should work the same as in
sage: A(x^2) == A(x)*A(x)
True
+ The subalgebra generated by the zero element is trivial::
+
+ sage: set_random_seed()
+ sage: A = random_eja().zero().subalgebra_generated_by()
+ sage: A
+ Euclidean Jordan algebra of dimension 0 over Rational Field
+ sage: A.one()
+ 0
+
"""
return FiniteDimensionalEuclideanJordanElementSubalgebra(self)
# Our FiniteDimensionalAlgebraElement superclass uses rows.
u_next = u**(s+1)
A = u_next.operator().matrix()
- c = J(A.solve_right(u_next.vector()))
+ c = J.from_vector(A.solve_right(u_next.to_vector()))
# Now c is the idempotent we want, but it still lives in the subalgebra.
return c.superalgebra_element()
# -1 to ensure that _charpoly_coeff(r-1) is really what
# appears in front of t^{r-1} in the charpoly. However,
# we want the negative of THAT for the trace.
- return -p(*self.vector())
+ return -p(*self.to_vector())
def trace_inner_product(self, other):
projects / sage.d.git / blobdiff