-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
lazy_import('mjo.eja.eja_algebra', 'FiniteDimensionalEuclideanJordanAlgebra')
+lazy_import('mjo.eja.eja_subalgebra',
+ 'FiniteDimensionalEuclideanJordanElementSubalgebra')
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):
"""
sage: x.apply_univariate_polynomial(p)
0
+ The characteristic polynomials of the zero and unit elements
+ should be what we think they are in a subalgebra, too::
+
+ sage: J = RealCartesianProductEJA(3)
+ sage: p1 = J.one().characteristic_polynomial()
+ sage: q1 = J.zero().characteristic_polynomial()
+ sage: e0,e1,e2 = J.gens()
+ sage: A = (e0 + 2*e1 + 3*e2).subalgebra_generated_by() # dim 3
+ sage: p2 = A.one().characteristic_polynomial()
+ sage: q2 = A.zero().characteristic_polynomial()
+ sage: p1 == p2
+ True
+ sage: q1 == q2
+ True
+
"""
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
sage: x.is_invertible() == (x.det() != 0)
True
+ Ensure that the determinant is multiplicative on an associative
+ subalgebra as in Faraut and Koranyi's Proposition II.2.2::
+
+ sage: set_random_seed()
+ sage: J = random_eja().random_element().subalgebra_generated_by()
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: (x*y).det() == x.det()*y.det()
+ True
+
"""
P = self.parent()
r = P.rank()
# -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:
sage: J.one().is_invertible()
True
- The zero element is never invertible::
+ The zero element is never invertible in a non-trivial algebra::
sage: set_random_seed()
sage: J = random_eja()
- sage: J.zero().is_invertible()
+ sage: (not J.is_trivial()) and J.zero().is_invertible()
False
"""
- zero = self.parent().zero()
+ if self.is_zero():
+ if self.parent().is_trivial():
+ return True
+ else:
+ return False
+
+ # 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)
TESTS:
- The zero element should never be regular::
+ The zero element should never be regular, unless the parent
+ algebra has dimension one::
sage: set_random_seed()
sage: J = random_eja()
- sage: J.zero().is_regular()
- False
+ sage: J.dimension() == 1 or not J.zero().is_regular()
+ True
The unit element isn't regular unless the algebra happens to
consist of only its scalar multiples::
True
"""
- return self.span_of_powers().dimension()
+ if self.is_zero() and not self.parent().is_trivial():
+ # The minimal polynomial of zero in a nontrivial algebra
+ # is "t"; in a trivial algebra it's "1" by convention
+ # (it's an empty product).
+ return 1
+ return self.subalgebra_generated_by().dimension()
def left_matrix(self):
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)
0
"""
- V = self.span_of_powers()
- assoc_subalg = self.subalgebra_generated_by()
- # Mis-design warning: the basis used for span_of_powers()
- # and subalgebra_generated_by() must be the same, and in
- # the same order!
- elt = assoc_subalg(V.coordinates(self.vector()))
- return elt.operator().minimal_polynomial()
+ if self.is_zero():
+ # We would generate a zero-dimensional subalgebra
+ # where the minimal polynomial would be constant.
+ # That might be correct, but only if *this* algebra
+ # is trivial too.
+ if not self.parent().is_trivial():
+ # Pretty sure we know what the minimal polynomial of
+ # the zero operator is going to be. This ensures
+ # consistency of e.g. the polynomial variable returned
+ # in the "normal" case without us having to think about it.
+ return self.operator().minimal_polynomial()
+
+ A = self.subalgebra_generated_by()
+ return A(self).operator().minimal_polynomial()
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))
+ W = self.parent().natural_basis_space()
+ 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)])
return ( L*M + M*L - (self*other).operator() )
- def span_of_powers(self):
- """
- Return the vector space spanned by successive powers of
- this element.
- """
- # The dimension of the subalgebra can't be greater than
- # the big algebra, so just put everything into a list
- # and let span() get rid of the excess.
- #
- # We do the extra ambient_vector_space() in case we're messing
- # with polynomials and the direct parent is a module.
- V = self.parent().vector_space()
- return V.span( (self**d).vector() for d in xrange(V.dimension()) )
def subalgebra_generated_by(self):
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: set_random_seed()
sage: x = random_eja().random_element()
- sage: u = x.subalgebra_generated_by().random_element()
- sage: u.operator()(u) == u^2
+ sage: A = x.subalgebra_generated_by()
+ 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
+
"""
- # First get the subspace spanned by the powers of myself...
- V = self.span_of_powers()
- F = self.base_ring()
-
- # Now figure out the entries of the right-multiplication
- # matrix for the successive basis elements b0, b1,... of
- # that subspace.
- mats = []
- for b_right in V.basis():
- eja_b_right = self.parent()(b_right)
- b_right_rows = []
- # The first row of the right-multiplication matrix by
- # b1 is what we get if we apply that matrix to b1. The
- # second row of the right multiplication matrix by b1
- # is what we get when we apply that matrix to b2...
- #
- # IMPORTANT: this assumes that all vectors are COLUMN
- # vectors, unlike our superclass (which uses row vectors).
- for b_left in V.basis():
- eja_b_left = self.parent()(b_left)
- # Multiply in the original EJA, but then get the
- # coordinates from the subalgebra in terms of its
- # basis.
- this_row = V.coordinates((eja_b_left*eja_b_right).vector())
- b_right_rows.append(this_row)
- b_right_matrix = matrix(F, b_right_rows)
- mats.append(b_right_matrix)
-
- # It's an algebra of polynomials in one element, and EJAs
- # are power-associative.
- #
- # TODO: choose generator names intelligently.
- #
- # The rank is the highest possible degree of a minimal polynomial,
- # and is bounded above by the dimension. We know in this case that
- # there's an element whose minimal polynomial has the same degree
- # as the space's dimension, so that must be its rank too.
- return FiniteDimensionalEuclideanJordanAlgebra(
- F,
- mats,
- V.dimension(),
- assume_associative=True,
- names='f')
+ return FiniteDimensionalEuclideanJordanElementSubalgebra(self)
def subalgebra_idempotent(self):
if self.is_nilpotent():
raise ValueError("this only works with non-nilpotent elements!")
- V = self.span_of_powers()
J = self.subalgebra_generated_by()
- # Mis-design warning: the basis used for span_of_powers()
- # and subalgebra_generated_by() must be the same, and in
- # the same order!
- u = J(V.coordinates(self.vector()))
+ u = J(self)
# The image of the matrix of left-u^m-multiplication
# will be minimal for some natural number s...
s = 0
- minimal_dim = V.dimension()
- for i in xrange(1, V.dimension()):
+ minimal_dim = J.dimension()
+ for i in xrange(1, minimal_dim):
this_dim = (u**i).operator().matrix().image().dimension()
if this_dim < minimal_dim:
minimal_dim = this_dim
# Our FiniteDimensionalAlgebraElement superclass uses rows.
u_next = u**(s+1)
A = u_next.operator().matrix()
- c_coordinates = A.solve_right(u_next.vector())
+ c = J.from_vector(A.solve_right(u_next.to_vector()))
- # Now c_coordinates is the idempotent we want, but it's in
- # the coordinate system of the subalgebra.
- #
- # We need the basis for J, but as elements of the parent algebra.
- #
- basis = [self.parent(v) for v in V.basis()]
- return self.parent().linear_combination(zip(c_coordinates, basis))
+ # Now c is the idempotent we want, but it still lives in the subalgebra.
+ return c.superalgebra_element()
def trace(self):
# -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