eja: add and update a few docstrings and tests.

diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py
index cd39704407deda564f857677a3439bb382f2bcb7..f88a5523bf7e63bc15608479bff6f05f63736936 100644 (file)
--- a/mjo/eja/eja_algebra.py
+++ b/mjo/eja/eja_algebra.py
@@ -481,7 +481,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
 
             SETUP::
 
-                sage: from mjo.eja.eja_algebra import RealSymmetricEJA
+                sage: from mjo.eja.eja_algebra import (RealSymmetricEJA,
+                ....:                                  random_eja)
 
             EXAMPLES:
 
@@ -501,6 +502,18 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
                 ...
                 ArithmeticError: vector is not in free module
 
+            TESTS:
+
+            Ensure that we can convert any element of the parent's
+            underlying vector space back into an algebra element whose
+            vector representation is what we started with::
+
+                sage: set_random_seed()
+                sage: J = random_eja()
+                sage: v = J.vector_space().random_element()
+                sage: J(v).vector() == v
+                True
+
             """
             # Goal: if we're given a matrix, and if it lives in our
             # parent algebra's "natural ambient space," convert it
@@ -533,21 +546,21 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
             Jordan algebras are always power-associative; see for
             example Faraut and Koranyi, Proposition II.1.2 (ii).
 
-            .. WARNING:
-
-                We have to override this because our superclass uses row vectors
-                instead of column vectors! We, on the other hand, assume column
-                vectors everywhere.
+            We have to override this because our superclass uses row
+            vectors instead of column vectors! We, on the other hand,
+            assume column vectors everywhere.
 
             SETUP::
 
                 sage: from mjo.eja.eja_algebra import random_eja
 
-            EXAMPLES::
+            TESTS:
+
+            The definition of `x^2` is the unambiguous `x*x`::
 
                 sage: set_random_seed()
                 sage: x = random_eja().random_element()
-                sage: x.operator()(x) == (x^2)
+                sage: x*x == (x^2)
                 True
 
             A few examples of power-associativity::
@@ -653,6 +666,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
                 sage: J.zero().characteristic_polynomial()
                 t^3
 
+            TESTS:
+
             The characteristic polynomial of an element should evaluate
             to zero on that element::
 
@@ -770,8 +785,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
                 sage: lhs == rhs
                 True
 
-            Test the first polarization identity from my notes, Koecher Chapter
-            III, or from Baes (2.3)::
+            Test the first polarization identity from my notes, Koecher
+            Chapter III, or from Baes (2.3)::
 
                 sage: set_random_seed()
                 sage: J = random_eja()
@@ -954,9 +969,6 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
             """
             Return whether or not this element is invertible.
 
-            We can't use the superclass method because it relies on
-            the algebra being associative.
-
             ALGORITHM:
 
             The usual way to do this is to check if the determinant is
@@ -966,6 +978,9 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
             whether or not the paren't algebra's zero element is a root
             of this element's minimal polynomial.
 
+            Beware that we can't use the superclass method, because it
+            relies on the algebra being associative.
+
             SETUP::
 
                 sage: from mjo.eja.eja_algebra import random_eja