+ sage: from mjo.eja.eja_algebra import random_eja, RealSymmetricEJA
+
+ EXAMPLES:
+
+ The canonical example comes from the symmetric matrices, which
+ decompose into diagonal and off-diagonal parts::
+
+ sage: J = RealSymmetricEJA(3)
+ sage: C = matrix(QQ, [ [1,0,0],
+ ....: [0,1,0],
+ ....: [0,0,0] ])
+ sage: c = J(C)
+ sage: J0,J5,J1 = J.peirce_decomposition(c)
+ sage: J0
+ Euclidean Jordan algebra of dimension 1...
+ sage: J5
+ Vector space of degree 6 and dimension 2...
+ sage: J1
+ Euclidean Jordan algebra of dimension 3...
+
+ TESTS:
+
+ Every algebra decomposes trivially with respect to its identity
+ element::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: J0,J5,J1 = J.peirce_decomposition(J.one())
+ sage: J0.dimension() == 0 and J5.dimension() == 0
+ True
+ sage: J1.superalgebra() == J and J1.dimension() == J.dimension()
+ True
+
+ The identity elements in the two subalgebras are the
+ projections onto their respective subspaces of the
+ superalgebra's identity element::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: if not J.is_trivial():
+ ....: while x.is_nilpotent():
+ ....: x = J.random_element()
+ sage: c = x.subalgebra_idempotent()
+ sage: J0,J5,J1 = J.peirce_decomposition(c)
+ sage: J1(c) == J1.one()
+ True
+ sage: J0(J.one() - c) == J0.one()
+ True
+
+ """
+ if not c.is_idempotent():
+ raise ValueError("element is not idempotent: %s" % c)
+
+ # Default these to what they should be if they turn out to be
+ # trivial, because eigenspaces_left() won't return eigenvalues
+ # corresponding to trivial spaces (e.g. it returns only the
+ # eigenspace corresponding to lambda=1 if you take the
+ # decomposition relative to the identity element).
+ trivial = FiniteDimensionalEuclideanJordanSubalgebra(self, ())
+ J0 = trivial # eigenvalue zero
+ J5 = VectorSpace(self.base_ring(), 0) # eigenvalue one-half
+ J1 = trivial # eigenvalue one
+
+ for (eigval, eigspace) in c.operator().matrix().right_eigenspaces():
+ if eigval == ~(self.base_ring()(2)):
+ J5 = eigspace
+ else:
+ gens = tuple( self.from_vector(b) for b in eigspace.basis() )
+ subalg = FiniteDimensionalEuclideanJordanSubalgebra(self, gens)
+ if eigval == 0:
+ J0 = subalg
+ elif eigval == 1:
+ J1 = subalg
+ else:
+ raise ValueError("unexpected eigenvalue: %s" % eigval)
+
+ return (J0, J5, J1)
+
+
+ def random_elements(self, count):
+ """
+ Return ``count`` random elements as a tuple.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import JordanSpinEJA
+
+ EXAMPLES::
+
+ sage: J = JordanSpinEJA(3)
+ sage: x,y,z = J.random_elements(3)
+ sage: all( [ x in J, y in J, z in J ])
+ True
+ sage: len( J.random_elements(10) ) == 10
+ True
+
+ """
+ return tuple( self.random_element() for idx in range(count) )
+
+ @classmethod
+ def random_instance(cls, field=AA, **kwargs):
+ """
+ Return a random instance of this type of algebra.
+
+ 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)
+
+ @cached_method
+ def _charpoly_coefficients(self):
+ r"""
+ The `r` polynomial coefficients of the "characteristic polynomial
+ of" function.
+ """
+ n = self.dimension()
+ var_names = [ "X" + str(z) for z in range(1,n+1) ]
+ R = PolynomialRing(self.base_ring(), var_names)
+ vars = R.gens()
+ F = R.fraction_field()
+
+ def L_x_i_j(i,j):
+ # From a result in my book, these are the entries of the
+ # basis representation of L_x.
+ return sum( vars[k]*self.monomial(k).operator().matrix()[i,j]
+ for k in range(n) )
+
+ L_x = matrix(F, n, n, L_x_i_j)
+ # Compute an extra power in case the rank is equal to
+ # the dimension (otherwise, we would stop at x^(r-1)).
+ x_powers = [ (L_x**k)*self.one().to_vector()
+ for k in range(n+1) ]
+ A = matrix.column(F, x_powers[:n])
+ AE = A.extended_echelon_form()
+ E = AE[:,n:]
+ A_rref = AE[:,:n]
+ r = A_rref.rank()
+ b = x_powers[r]
+
+ # The theory says that only the first "r" coefficients are
+ # nonzero, and they actually live in the original polynomial
+ # ring and not the fraction field. We negate them because
+ # in the actual characteristic polynomial, they get moved
+ # to the other side where x^r lives.
+ return -A_rref.solve_right(E*b).change_ring(R)[:r]
+
+ @cached_method
+ def rank(self):
+ r"""
+ Return the rank of this EJA.
+
+ This is a cached method because we know the rank a priori for
+ all of the algebras we can construct. Thus we can avoid the
+ expensive ``_charpoly_coefficients()`` call unless we truly
+ need to compute the whole characteristic polynomial.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: JordanSpinEJA,