mjo/eja/TODO

   1 1. Finish CartesianProductEJA: add to_matrix(),
   2    random_instance(),... methods. This will require rethinking what a
   3    "matrix representation" and "matrix space" means for a cartesian
   4    product algebra. Do we want our matrix basis to consist of ordered
   5    pairs (or triples, or...)? Should the matrix_space() of the algebra
   6    be the cartesian product of the factors' matrix spaces? Can we just
   7    fix the matrix basis/space after we call the FDEJA initializer?
   8
   9 2. Add references and start citing them.
  10
  11 3. Implement the octonion simple EJA.
  12
  13 4. Pre-cache charpoly for some small algebras?
  14
  15 RealSymmetricEJA(4):
  16
  17 sage: F = J.base_ring()
  18 sage: a0 = (1/4)*X[4]**2*X[6]**2 - (1/2)*X[2]*X[5]*X[6]**2 - (1/2)*X[3]*X[4]*X[6]*X[7] + (F(2).sqrt()/2)*X[1]*X[5]*X[6]*X[7] + (1/4)*X[3]**2*X[7]**2 - (1/2)*X[0]*X[5]*X[7]**2 + (F(2).sqrt()/2)*X[2]*X[3]*X[6]*X[8] - (1/2)*X[1]*X[4]*X[6*X[8] - (1/2)*X[1]*X[3]*X[7]*X[8] + (F(2).sqrt()/2)*X[0]*X[4]*X[7]*X[8] + (1/4)*X[1]**2*X[8]**2 - (1/2)*X[0]*X[2]*X[8]**2 - (1/2)*X[2]*X[3]**2*X[9] + (F(2).sqrt()/2)*X[1]*X[3]*X[4]*X[9] - (1/2)*X[0]*X[4]**2*X[9] - (1/2)*X[1]**2*X[5]*X[9] + X[0]*X[2]*X[5]*X[9]
  19
  20 5. The main EJA element constructor is happy to convert between
  21    e.g. HadamardEJA(3) and JordanSpinEJA(3).
  22
  23 6. Profile the construction of "large" matrix algebras (like the
  24    15-dimensional QuaternionHermitianAlgebra(3)) to find out why
  25    they're so slow.