+ sage: set_random_seed()
+ sage: x = random_eja().random_element()
+ sage: x.degree() == x.minimal_polynomial().degree()
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: x = random_eja().random_element()
+ sage: x.degree() == x.minimal_polynomial().degree()
+ True
+
+ The minimal polynomial and the characteristic polynomial coincide
+ and are known (see Alizadeh, Example 11.11) for all elements of
+ the spin factor algebra that aren't scalar multiples of the
+ identity::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(2,10).abs()
+ sage: J = JordanSpinSimpleEJA(n)
+ 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: actual = y.minimal_polynomial()
+ sage: x = SR.symbol('x', domain='real')
+ sage: expected = x^2 - 2*y0*x + (y0^2 - norm(y_bar)^2)
+ sage: bool(actual == expected)
+ True
+
+ """
+ # 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 = JordanSpinSimpleEJA(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 = JordanSpinSimpleEJA(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!")
+
+ 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()))
+
+ # 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()):
+ this_dim = (u**i).matrix().image().dimension()
+ if this_dim < minimal_dim:
+ minimal_dim = this_dim
+ s = i
+
+ # Now minimal_matrix should correspond to the smallest
+ # non-zero subspace in Baes's (or really, Koecher's)
+ # proposition.
+ #
+ # However, we need to restrict the matrix to work on the
+ # subspace... or do we? Can't we just solve, knowing that
+ # A(c) = u^(s+1) should have a solution in the big space,
+ # too?
+ #
+ # Beware, solve_right() means that we're using COLUMN vectors.
+ # Our FiniteDimensionalAlgebraElement superclass uses rows.
+ u_next = u**(s+1)
+ A = u_next.matrix()
+ c_coordinates = A.solve_right(u_next.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))
+
+
+ def trace(self):
+ """
+ Return my trace, the sum of my eigenvalues.
+
+ EXAMPLES::
+
+ sage: J = JordanSpinSimpleEJA(3)
+ sage: e0,e1,e2 = J.gens()
+ sage: x = e0 + e1 + e2
+ sage: x.trace()
+ 2
+
+ """
+ cs = self.characteristic_polynomial().coefficients(sparse=False)
+ if len(cs) >= 2:
+ return -1*cs[-2]
+ else:
+ raise ValueError('charpoly had fewer than 2 coefficients')
+
+
+ def trace_inner_product(self, other):
+ """
+ Return the trace inner product of myself and ``other``.
+ """
+ if not other in self.parent():
+ raise ArgumentError("'other' must live in the same algebra")
+
+ return (self*other).trace()