1 1. Add CartesianProductEJA.
3 2. Add references and start citing them.
5 3. Implement the octonion simple EJA.
7 4. Override random_instance(), one(), et cetera in DirectSumEJA.
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,
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 usinf e.g. FreeModule_submodule_with_basis_field
18 we may somehow be able to pass the echelon basis straight in to
21 This may require supporting "basis" as a list of basis vectors
22 (as opposed to superalgebra elements) in the subalgebra constructor.
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.
33 8. Pre-cache charpoly for some small algebras?
35 9. Compute the scalar in the general natural_inner_product() for
36 matrices, so no overrides are necessary.
38 10. Eliminate "natural representation" for non-matrix algebras.