also the left multiplication matrix and must be symmetric::
sage: set_random_seed()
+ sage: n = ZZ.random_element(1,10).abs()
+ sage: J = eja_rn(5)
+ sage: J.random_element().matrix().is_symmetric()
+ True
sage: J = eja_ln(5)
sage: J.random_element().matrix().is_symmetric()
True
def subalgebra_generated_by(self):
"""
- Return the subalgebra of the parent EJA generated by this element.
+ Return the associative subalgebra of the parent EJA generated
+ by this element.
+
+ TESTS::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,10).abs()
+ sage: J = eja_rn(n)
+ sage: x = J.random_element()
+ sage: x.subalgebra_generated_by().is_associative()
+ True
+ sage: J = eja_ln(n)
+ sage: x = J.random_element()
+ sage: x.subalgebra_generated_by().is_associative()
+ True
+
"""
# First get the subspace spanned by the powers of myself...
V = self.span_of_powers()
b_right_matrix = matrix(F, b_right_rows)
mats.append(b_right_matrix)
- return FiniteDimensionalEuclideanJordanAlgebra(F, mats)
+ # It's an algebra of polynomials in one element, and EJAs
+ # are power-associative.
+ return FiniteDimensionalEuclideanJordanAlgebra(F, mats, assume_associative=True)
def minimal_polynomial(self):
True
"""
- 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!
- subalg_self = assoc_subalg(V.coordinates(self.vector()))
- return subalg_self.matrix().minimal_polynomial()
+ # The element we're going to call "minimal_polynomial()" on.
+ # Either myself, interpreted as an element of a finite-
+ # dimensional algebra, or an element of an associative
+ # subalgebra.
+ elt = None
+
+ if self.parent().is_associative():
+ elt = FiniteDimensionalAlgebraElement(self.parent(), self)
+ else:
+ 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()))
+
+ # Recursive call, but should work since elt lives in an
+ # associative algebra.
+ return elt.minimal_polynomial()
+
+
+ def is_nilpotent(self):
+ """
+ Return whether or not some power of this element is zero.
+
+ The superclass method won't work unless we're in an
+ associative algebra, and we aren't. However, we generate
+ an assocoative subalgebra and we're nilpotent there if and
+ only if we're nilpotent here (probably).
+
+ TESTS:
+
+ The identity element is never nilpotent::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(2,10).abs()
+ sage: J = eja_rn(n)
+ sage: J.one().is_nilpotent()
+ False
+ sage: J = eja_ln(n)
+ sage: J.one().is_nilpotent()
+ False
+
+ The additive identity is always nilpotent::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(2,10).abs()
+ sage: J = eja_rn(n)
+ sage: J.zero().is_nilpotent()
+ True
+ sage: J = eja_ln(n)
+ sage: J.zero().is_nilpotent()
+ True
+
+ """
+ # The element we're going to call "is_nilpotent()" on.
+ # Either myself, interpreted as an element of a finite-
+ # dimensional algebra, or an element of an associative
+ # subalgebra.
+ elt = None
+
+ if self.parent().is_associative():
+ elt = FiniteDimensionalAlgebraElement(self.parent(), self)
+ else:
+ 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()))
+
+ # Recursive call, but should work since elt lives in an
+ # associative algebra.
+ return elt.is_nilpotent()
def characteristic_polynomial(self):