TODO

   1 1. Add CartesianProductEJA.
   2
   3 2. Add references and start citing them.
   4
   5 3. Implement the octonion simple EJA.
   6
   7 4. Override random_instance(), one(), et cetera in DirectSumEJA.
   8
   9 5. Switch to QQ in *all* algebras for _charpoly_coefficients().
  10    This only works when we know that the basis can be rationalized...
  11    which is the case at least for the concrete EJAs we provide,
  12    but not in general.
  13
  14 6. Pass already_echelonized (default: False) and echelon_basis
  15    (default: None) into the subalgebra constructor. The value of
  16    already_echelonized can be passed to V.span_of_basis() to save
  17    some time, and using e.g. FreeModule_submodule_with_basis_field
  18    we may somehow be able to pass the echelon basis straight in to
  19    save time.
  20
  21    This may require supporting "basis" as a list of basis vectors
  22    (as opposed to superalgebra elements) in the subalgebra constructor.
  23
  24 7. The inner product should be an *argument* to the main EJA
  25    constructor.  Afterwards, the basis normalization step should be
  26    optional (and enabled by default) for ALL algebras, since any
  27    algebra can have a nonstandard inner-product and its basis can be
  28    normalized with respect to that inner- product. For example, the
  29    HadamardEJA could be equipped with an inner- product that is twice
  30    the usual one. Then for the basis to be orthonormal, we would need
  31    to divide e.g. (1,0,0) by <(1,0,0),(1,0,0)> = 2 to normalize it.
  32
  33 8. Pre-cache charpoly for some small algebras?
  34
  35 RealSymmetricEJA(4):
  36
  37 sage: F = J.base_ring()
  38 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]
  39
  40 9. Compute the scalar in the general natural_inner_product() for
  41    matrices, so no overrides are necessary.
  42
  43 10. The main EJA element constructor is happy to convert between
  44     e.g. HadamardEJA(3) and JordanSpinEJA(3).
  45
  46 11. Figure out if CombinatorialFreeModule's use of IndexedGenerators
  47     can be used to replace the matrix_basis().
  48
  49 12. Don't pass in an n-by-n multiplication/i-p table since only the
  50     lower-left half is used.
  51
  52 13. Move the "field" argument to a keyword after basis, jp, and ip.
  53
  54 14. Instead of storing the multiplication and inner-product tables in
  55     RationalBasisEuclideanJordanAlgebra, why not just create the
  56     algebra over QQ in the constructor and save that? They're globally
  57     unique, so we won't wind up with multiple copies.