- return tuple( self.random_element() for idx in range(count) )
-
-
- 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