-1. Finish CartesianProductEJA: add to_matrix(), random_instance(),...
- methods. I guess we should create a separate class hierarchy for
- Cartesian products of RationalBasisEJA? That way we get fast
- charpoly and random_instance() defined...
+1. Add cartesian products to random_eja().
2. Add references and start citing them.
-3. Implement the octonion simple EJA.
+3. Implement the octonion simple EJA. We don't actually need octonions
+ for this to work, only their real embedding (some 8x8 monstrosity).
4. Pre-cache charpoly for some small algebras?
5. Profile the construction of "large" matrix algebras (like the
15-dimensional QuaternionHermitianAlgebra(3)) to find out why
they're so slow.
+
+6. Instead of storing a basis multiplication matrix, just make
+ product_on_basis() a cached method and manually cache its
+ entries. The cython cached method lookup should be faster than a
+ python-based matrix lookup anyway.
+
+7. What the ever-loving fuck is this shit?
+
+ sage: O = Octonions(QQ)
+ sage: e0 = O.monomial(0)
+ sage: e0*[[[[]]]]
+ [[[[]]]]*e0
+
+8. Factor out a class for matrices with real embeddings (i.e. not the
+ octonions).
+
+9. In fact, could my octonion matrix algebra be generalized for any
+ algebra of matrices over the reals whose entries are not real? Then
+ we wouldn't need real embeddings at all. They might even be fricking
+ vector spaces if I did that...