-Trace inner product tests:
-
- TESTS:
-
- The trace inner product is commutative::
-
- sage: set_random_seed()
- sage: J = random_eja()
- sage: x = J.random_element(); y = J.random_element()
- sage: x.trace_inner_product(y) == y.trace_inner_product(x)
- True
-
- The trace inner product is bilinear::
-
- sage: set_random_seed()
- sage: J = random_eja()
- sage: x = J.random_element()
- sage: y = J.random_element()
- sage: z = J.random_element()
- sage: a = QQ.random_element();
- sage: actual = (a*(x+z)).trace_inner_product(y)
- sage: expected = a*x.trace_inner_product(y) + a*z.trace_inner_product(y)
- sage: actual == expected
- True
- sage: actual = x.trace_inner_product(a*(y+z))
- sage: expected = a*x.trace_inner_product(y) + a*x.trace_inner_product(z)
- sage: actual == expected
- True
-
- The trace inner product is associative::
-
- sage: pass
-
- The trace inner product satisfies the compatibility
- condition in the definition of a Euclidean Jordan algebra:
-
- sage: set_random_seed()
- sage: J = random_eja()
- sage: x = J.random_element()
- sage: y = J.random_element()
- sage: z = J.random_element()
- sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
- True
-
\ No newline at end of file
+1. Add CartesianProductEJA.
+
+2. Add references and start citing them.
+
+3. Implement the octonion simple EJA.
+
+4. Override random_instance(), one(), et cetera in DirectSumEJA.
+
+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.
+
+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 using e.g. FreeModule_submodule_with_basis_field
+ we may somehow be able to pass the echelon basis straight in to
+ save time.
+
+ 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. The main EJA element constructor is happy to convert between
+ e.g. HadamardEJA(3) and JordanSpinEJA(3).
+
+11. Figure out if CombinatorialFreeModule's use of IndexedGenerators
+ can be used to replace the matrix_basis().