+ def _rank_computation(self):
+ r"""
+ Compute the rank of this algebra.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: JordanSpinEJA,
+ ....: RealSymmetricEJA,
+ ....: ComplexHermitianEJA,
+ ....: QuaternionHermitianEJA)
+
+ EXAMPLES::
+
+ sage: J = HadamardEJA(4)
+ sage: J._rank_computation() == J.rank()
+ True
+ sage: J = JordanSpinEJA(4)
+ sage: J._rank_computation() == J.rank()
+ True
+ sage: J = RealSymmetricEJA(3)
+ sage: J._rank_computation() == J.rank()
+ True
+ sage: J = ComplexHermitianEJA(2)
+ sage: J._rank_computation() == J.rank()
+ True
+ sage: J = QuaternionHermitianEJA(2)
+ sage: J._rank_computation() == J.rank()
+ True
+
+ """
+ n = self.dimension()
+ if n == 0:
+ return 0
+ elif n == 1:
+ return 1
+
+ var_names = [ "X" + str(z) for z in range(1,n+1) ]
+ R = PolynomialRing(self.base_ring(), var_names)
+ vars = R.gens()
+
+ 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(R, n, n, L_x_i_j)
+ x_powers = [ vars[k]*(L_x**k)*self.one().to_vector()
+ for k in range(n) ]
+
+ # Can assume n >= 2
+ M = matrix([x_powers[0]])
+ old_rank = 1
+
+ for d in range(1,n):
+ M = matrix(M.rows() + [x_powers[d]])
+ M.echelonize()
+ new_rank = M.rank()
+ if new_rank == old_rank:
+ return new_rank
+ else:
+ old_rank = new_rank
+
+ return n
+