-1. Add CartesianProductEJA.
+1. Finish DirectSumEJA: add to_matrix(), random_instance(),
+ one()... methods. Make it subclass RationalBasisEuclideanJordanAlgebra.
+ This is not a general direct sum / cartesian product implementation,
+ it's used only with the other rationalbasis algebras (to make non-
+ simple EJAs out of the simple ones).
2. Add references and start citing them.
3. Implement the octonion simple EJA.
-4. Override random_instance(), one(), et cetera in DirectSumEJA.
+4. Pre-cache charpoly for some small algebras?
-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.
+RealSymmetricEJA(4):
-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
- we may somehow be able to pass the echelon basis straight in to
- save time.
+sage: F = J.base_ring()
+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]
- This may require supporting "basis" as a list of basis vectors
- (as opposed to superalgebra elements) in the subalgebra constructor.
+5. Compute the scalar in the general natural_inner_product() for
+ matrices, so no overrides are necessary.
-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.
+6. The main EJA element constructor is happy to convert between
+ e.g. HadamardEJA(3) and JordanSpinEJA(3).
-8. Use charpoly for inverse itself?
+7. Figure out if CombinatorialFreeModule's use of IndexedGenerators
+ can be used to replace the matrix_basis().
-9. Pre-cache charpoly for some small algebras?
+9. Add back the check_field=False and check_axioms=False parameters
+ for the EJAs we've constructed ourselves.
projects / sage.d.git / blobdiff