+ # 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 quadratic_representation(self):
+ """
+ Return the quadratic representation of this element.
+
+ EXAMPLES:
+
+ The explicit form in the spin factor algebra is given by
+ Alizadeh's Example 11.12::
+
+ sage: n = ZZ.random_element(1,10).abs()
+ sage: J = eja_ln(n)
+ sage: x = J.random_element()
+ sage: x_vec = x.vector()
+ sage: x0 = x_vec[0]
+ sage: x_bar = x_vec[1:]
+ sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
+ sage: B = 2*x0*x_bar.row()
+ sage: C = 2*x0*x_bar.column()
+ sage: D = identity_matrix(QQ, n-1)
+ sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
+ sage: D = D + 2*x_bar.tensor_product(x_bar)
+ sage: Q = block_matrix(2,2,[A,B,C,D])
+ sage: Q == x.quadratic_representation()
+ True
+
+ """
+ return 2*(self.matrix()**2) - (self**2).matrix()
+
+
+ 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.
+ V = self.vector().parent()
+ return V.span( (self**d).vector() for d in xrange(V.dimension()) )
+
+
+ def subalgebra_generated_by(self):
+ """
+ Return the associative subalgebra of the parent EJA generated
+ by this element.
+
+ TESTS::
+
+ sage: set_random_seed()
+ sage: x = random_eja().random_element()
+ sage: x.subalgebra_generated_by().is_associative()
+ True
+
+ Squaring in the subalgebra should be the same thing as
+ squaring in the superalgebra::
+
+ sage: set_random_seed()
+ sage: x = random_eja().random_element()
+ sage: u = x.subalgebra_generated_by().random_element()
+ sage: u.matrix()*u.vector() == (u**2).vector()
+ True
+
+ """
+ # 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.
+ return FiniteDimensionalEuclideanJordanAlgebra(F, mats, assume_associative=True, names='f')
+
+
+ def subalgebra_idempotent(self):
+ """
+ Find an idempotent in the associative subalgebra I generate
+ using Proposition 2.3.5 in Baes.
+
+ TESTS::
+
+ sage: set_random_seed()
+ sage: J = eja_rn(5)
+ sage: c = J.random_element().subalgebra_idempotent()
+ sage: c^2 == c
+ True
+ sage: J = eja_ln(5)
+ sage: c = J.random_element().subalgebra_idempotent()
+ sage: c^2 == c
+ True
+
+ """
+ if self.is_nilpotent():
+ raise ValueError("this only works with non-nilpotent elements!")
+