3. Implement the octonion simple EJA.
-4. Factor out the unit-norm basis (and operator symmetry) tests once
- all of the algebras pass.
+4. Override random_instance(), one(), et cetera in DirectSumEJA.
-5. Override inner_product(), _max_test_case_size(), et cetera in
- DirectSumEJA.
-
-6. Switch to QQ in *all* algebras for _charpoly_coefficients().
+5. Switch to QQ in *all* algebras for _charpoly_coefficients().
This only works when we know that the basis can be rationalized...
which is the case at least for the concrete EJAs we provide,
but not in general.
-7. Pass already_echelonized (default: False) and echelon_basis
+6. Pass already_echelonized (default: False) and echelon_basis
(default: None) into the subalgebra constructor. The value of
already_echelonized can be passed to V.span_of_basis() to save
some time, and usinf e.g. FreeModule_submodule_with_basis_field
This may require supporting "basis" as a list of basis vectors
(as opposed to superalgebra elements) in the subalgebra constructor.
+
+7. The inner product should be an *argument* to the main EJA
+ constructor. Afterwards, the basis normalization step should be
+ optional (and enabled by default) for ALL algebras, since any
+ algebra can have a nonstandard inner-product and its basis can be
+ normalized with respect to that inner- product. For example, the
+ HadamardEJA could be equipped with an inner- product that is twice
+ the usual one. Then for the basis to be orthonormal, we would need
+ to divide e.g. (1,0,0) by <(1,0,0),(1,0,0)> = 2 to normalize it.
+
+8. Pre-cache charpoly for some small algebras?
+
+9. Compute the scalar in the general natural_inner_product() for
+ matrices, so no overrides are necessary.
+
+10. Eliminate "natural representation" for non-matrix algebras.
+
+11. The main EJA element constructor is happy to convert between
+ e.g. HadamardEJA(3) and JordanSpinEJA(3).