eja: add a test for the symmetry of the _real_symmetric_basis().

[sage.d.git] / mjo / eja / eja_algebra.py

diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py
index 06f6f531ac46305da59ec2b4290ba808602cea50..204a537df12d2ab4d3d27764c20c830ec324a68a 100644 (file)
--- a/mjo/eja/eja_algebra.py
+++ b/mjo/eja/eja_algebra.py
@@ -9,6 +9,7 @@ from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra
 from sage.categories.magmatic_algebras import MagmaticAlgebras
 from sage.combinat.free_module import CombinatorialFreeModule
 from sage.matrix.constructor import matrix
+from sage.matrix.matrix_space import MatrixSpace
 from sage.misc.cachefunc import cached_method
 from sage.misc.prandom import choice
 from sage.misc.table import table
@@ -514,11 +515,23 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
 
         """
         if self._natural_basis is None:
-            return tuple( b.to_vector().column() for b in self.basis() )
+            M = self.natural_basis_space()
+            return tuple( M(b.to_vector()) for b in self.basis() )
         else:
             return self._natural_basis
 
 
+    def natural_basis_space(self):
+        """
+        Return the matrix space in which this algebra's natural basis
+        elements live.
+        """
+        if self._natural_basis is None or len(self._natural_basis) == 0:
+            return MatrixSpace(self.base_ring(), self.dimension(), 1)
+        else:
+            return self._natural_basis[0].matrix_space()
+
+
     @cached_method
     def one(self):
         """
@@ -709,6 +722,17 @@ class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra):
         sage: RealCartesianProductEJA(3, prefix='r').gens()
         (r0, r1, r2)
 
+    Our inner product satisfies the Jordan axiom::
+
+        sage: set_random_seed()
+        sage: n = ZZ.random_element(1,5)
+        sage: J = RealCartesianProductEJA(n)
+        sage: x = J.random_element()
+        sage: y = J.random_element()
+        sage: z = J.random_element()
+        sage: (x*y).inner_product(z) == y.inner_product(x*z)
+        True
+
     """
     def __init__(self, n, field=QQ, **kwargs):
         V = VectorSpace(field, n)
@@ -773,9 +797,22 @@ def random_eja():
 
 
 
-def _real_symmetric_basis(n, field=QQ):
+def _real_symmetric_basis(n, field):
     """
     Return a basis for the space of real symmetric n-by-n matrices.
+
+    SETUP::
+
+        sage: from mjo.eja.eja_algebra import _real_symmetric_basis
+
+    TESTS::
+
+        sage: set_random_seed()
+        sage: n = ZZ.random_element(1,5)
+        sage: B = _real_symmetric_basis(n, QQbar)
+        sage: all( M.is_symmetric() for M in  B)
+        True
+
     """
     # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
     # coordinates.
@@ -792,7 +829,7 @@ def _real_symmetric_basis(n, field=QQ):
     return tuple(S)
 
 
-def _complex_hermitian_basis(n, field=QQ):
+def _complex_hermitian_basis(n, field):
     """
     Returns a basis for the space of complex Hermitian n-by-n matrices.
 
@@ -833,7 +870,7 @@ def _complex_hermitian_basis(n, field=QQ):
     return tuple(S)
 
 
-def _quaternion_hermitian_basis(n, field=QQ):
+def _quaternion_hermitian_basis(n, field):
     """
     Returns a basis for the space of quaternion Hermitian n-by-n matrices.
 
@@ -1200,9 +1237,20 @@ class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra):
         sage: RealSymmetricEJA(3, prefix='q').gens()
         (q0, q1, q2, q3, q4, q5)
 
+    Our inner product satisfies the Jordan axiom::
+
+        sage: set_random_seed()
+        sage: n = ZZ.random_element(1,5)
+        sage: J = RealSymmetricEJA(n)
+        sage: x = J.random_element()
+        sage: y = J.random_element()
+        sage: z = J.random_element()
+        sage: (x*y).inner_product(z) == y.inner_product(x*z)
+        True
+
     """
     def __init__(self, n, field=QQ, **kwargs):
-        S = _real_symmetric_basis(n, field=field)
+        S = _real_symmetric_basis(n, field)
         Qs = _multiplication_table_from_matrix_basis(S)
 
         fdeja = super(RealSymmetricEJA, self)
@@ -1258,9 +1306,20 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):
         sage: ComplexHermitianEJA(2, prefix='z').gens()
         (z0, z1, z2, z3)
 
+    Our inner product satisfies the Jordan axiom::
+
+        sage: set_random_seed()
+        sage: n = ZZ.random_element(1,5)
+        sage: J = ComplexHermitianEJA(n)
+        sage: x = J.random_element()
+        sage: y = J.random_element()
+        sage: z = J.random_element()
+        sage: (x*y).inner_product(z) == y.inner_product(x*z)
+        True
+
     """
     def __init__(self, n, field=QQ, **kwargs):
-        S = _complex_hermitian_basis(n)
+        S = _complex_hermitian_basis(n, field)
         Qs = _multiplication_table_from_matrix_basis(S)
 
         fdeja = super(ComplexHermitianEJA, self)
@@ -1324,9 +1383,20 @@ class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):
         sage: QuaternionHermitianEJA(2, prefix='a').gens()
         (a0, a1, a2, a3, a4, a5)
 
+    Our inner product satisfies the Jordan axiom::
+
+        sage: set_random_seed()
+        sage: n = ZZ.random_element(1,5)
+        sage: J = QuaternionHermitianEJA(n)
+        sage: x = J.random_element()
+        sage: y = J.random_element()
+        sage: z = J.random_element()
+        sage: (x*y).inner_product(z) == y.inner_product(x*z)
+        True
+
     """
     def __init__(self, n, field=QQ, **kwargs):
-        S = _quaternion_hermitian_basis(n)
+        S = _quaternion_hermitian_basis(n, field)
         Qs = _multiplication_table_from_matrix_basis(S)
 
         fdeja = super(QuaternionHermitianEJA, self)
@@ -1386,6 +1456,17 @@ class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra):
         sage: JordanSpinEJA(2, prefix='B').gens()
         (B0, B1)
 
+    Our inner product satisfies the Jordan axiom::
+
+        sage: set_random_seed()
+        sage: n = ZZ.random_element(1,5)
+        sage: J = JordanSpinEJA(n)
+        sage: x = J.random_element()
+        sage: y = J.random_element()
+        sage: z = J.random_element()
+        sage: (x*y).inner_product(z) == y.inner_product(x*z)
+        True
+
     """
     def __init__(self, n, field=QQ, **kwargs):
         V = VectorSpace(field, n)