"""
(A_of_x, x, xr, detA) = self._charpoly_matrix_system()
R = A_of_x.base_ring()
- if i >= self.rank():
+
+ if i == self.rank():
+ return R.one()
+ if i > self.rank():
# Guaranteed by theory
return R.zero()
SETUP::
- sage: from mjo.eja.eja_algebra import JordanSpinEJA
+ sage: from mjo.eja.eja_algebra import JordanSpinEJA, TrivialEJA
EXAMPLES:
sage: p(*xvec)
t^2 - 2*t + 1
+ By definition, the characteristic polynomial is a monic
+ degree-zero polynomial in a rank-zero algebra. Note that
+ Cayley-Hamilton is indeed satisfied since the polynomial
+ ``1`` evaluates to the identity element of the algebra on
+ any argument::
+
+ sage: J = TrivialEJA()
+ sage: J.characteristic_polynomial()
+ 1
+
"""
r = self.rank()
n = self.dimension()
- # The list of coefficient polynomials a_1, a_2, ..., a_n.
- a = [ self._charpoly_coeff(i) for i in range(n) ]
+ # The list of coefficient polynomials a_0, a_1, a_2, ..., a_n.
+ a = [ self._charpoly_coeff(i) for i in range(r+1) ]
# We go to a bit of trouble here to reorder the
# indeterminates, so that it's easier to evaluate the
S = PolynomialRing(S, R.variable_names())
t = S(t)
- # Note: all entries past the rth should be zero. The
- # coefficient of the highest power (x^r) is 1, but it doesn't
- # appear in the solution vector which contains coefficients
- # for the other powers (to make them sum to x^r).
- if (r < n):
- a[r] = 1 # corresponds to x^r
- else:
- # When the rank is equal to the dimension, trying to
- # assign a[r] goes out-of-bounds.
- a.append(1) # corresponds to x^r
-
return sum( a[k]*(t**k) for k in xrange(len(a)) )
SETUP::
- sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
+ sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
+ ....: TrivialEJA)
EXAMPLES::
sage: J.is_trivial()
False
+ ::
+
+ sage: J = TrivialEJA()
+ sage: J.is_trivial()
+ True
+
"""
return self.dimension() == 0
TESTS:
Ensure that every EJA that we know how to construct has a
- positive integer rank::
+ positive integer rank, unless the algebra is trivial in
+ which case its rank will be zero::
sage: set_random_seed()
- sage: r = random_eja().rank()
- sage: r in ZZ and r > 0
+ sage: J = random_eja()
+ sage: r = J.rank()
+ sage: r in ZZ
+ True
+ sage: r > 0 or (r == 0 and J.is_trivial())
True
"""
Beware, this will crash for "most instances" because the
constructor below looks wrong.
"""
+ if cls is TrivialEJA:
+ # The TrivialEJA class doesn't take an "n" argument because
+ # there's only one.
+ return cls(field)
+
n = ZZ.random_element(cls._max_test_case_size()) + 1
return cls(n, field, **kwargs)
return x.to_vector().inner_product(y.to_vector())
-def random_eja(field=QQ):
+def random_eja(field=QQ, nontrivial=False):
"""
Return a "random" finite-dimensional Euclidean Jordan Algebra.
- ALGORITHM:
-
- For now, we choose a random natural number ``n`` (greater than zero)
- and then give you back one of the following:
-
- * The cartesian product of the rational numbers ``n`` times; this is
- ``QQ^n`` with the Hadamard product.
-
- * The Jordan spin algebra on ``QQ^n``.
-
- * The ``n``-by-``n`` rational symmetric matrices with the symmetric
- product.
-
- * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
- in the space of ``2n``-by-``2n`` real symmetric matrices.
-
- * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
- in the space of ``4n``-by-``4n`` real symmetric matrices.
-
- Later this might be extended to return Cartesian products of the
- EJAs above.
-
SETUP::
sage: from mjo.eja.eja_algebra import random_eja
Euclidean Jordan algebra of dimension...
"""
- classname = choice(KnownRankEJA.__subclasses__())
+ eja_classes = KnownRankEJA.__subclasses__()
+ if nontrivial:
+ eja_classes.remove(TrivialEJA)
+ classname = choice(eja_classes)
return classname.random_instance(field=field)
"""
return x.to_vector().inner_product(y.to_vector())
+
+
+class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA):
+ """
+ The trivial Euclidean Jordan algebra consisting of only a zero element.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import TrivialEJA
+
+ EXAMPLES::
+
+ sage: J = TrivialEJA()
+ sage: J.dimension()
+ 0
+ sage: J.zero()
+ 0
+ sage: J.one()
+ 0
+ sage: 7*J.one()*12*J.one()
+ 0
+ sage: J.one().inner_product(J.one())
+ 0
+ sage: J.one().norm()
+ 0
+ sage: J.one().subalgebra_generated_by()
+ Euclidean Jordan algebra of dimension 0 over Rational Field
+ sage: J.rank()
+ 0
+
+ """
+ def __init__(self, field=QQ, **kwargs):
+ mult_table = []
+ fdeja = super(TrivialEJA, self)
+ # The rank is zero using my definition, namely the dimension of the
+ # largest subalgebra generated by any element.
+ return fdeja.__init__(field, mult_table, rank=0, **kwargs)
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