- 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
+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.