-Trace inner product tests:
+1. Add CartesianProductEJA.
- TESTS:
+2. Check the axioms in the constructor when check != False?
- The trace inner product is commutative::
+3. Add references and start citing them.
- 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
+4. Implement the octonion simple EJA.
- The trace inner product is bilinear::
+5. Factor out the Jordan axiom and norm tests once all of the
+ algebras pass.
- 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
-
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+6. Create Element subclasses for the matrix EJAs, and then override
+ their characteristic_polynomial() method to create a new algebra
+ over the rationals (with a non-normalized basis). We can then
+ compute the charpoly quickly by passing the natural representation
+ of the given element into the new algebra and computing its charpoly
+ there. (Relies on the theory to ensure that the charpolys are equal.)
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