-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 cartesian products to random_eja().
+
+2. Add references and start citing them.
+
+3. Implement the octonion simple EJA.
+
+4. Pre-cache charpoly for some small algebras?
+
+RealSymmetricEJA(4):
+
+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]
+
+5. Profile the construction of "large" matrix algebras (like the
+ 15-dimensional QuaternionHermitianAlgebra(3)) to find out why
+ they're so slow.
+
+6. We should compute whether or not the algebra is associative if it
+ is unknown. I guess the "associative" argument should be ternary
+ (True, False, None)? We should also figure out the correct
+ True/False values for the example classes, and of course add an
+ _is_associative() method.
+
+7. When field=RDF, subalgebra construction is failing because the
+ inner product isn't associative? Actually, it's the combination
+ of field=RDF and orthonormalize=True.