eja: add orthogonality examples/tests for the Peirce decomposition.

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

diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py
index 4e66b2db251274d3091875a20056692f33849513..ed2e2587282e91d2672cbcbc851d37fcddf7ae74 100644 (file)
--- a/mjo/eja/eja_algebra.py
+++ b/mjo/eja/eja_algebra.py
@@ -236,10 +236,12 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
         return self._multiplication_table[i][j]
 
     @cached_method
-    def characteristic_polynomial(self):
+    def characteristic_polynomial_of(self):
         """
-        Return a characteristic polynomial that works for all elements
-        of this algebra.
+        Return the algebra's "characteristic polynomial of" function,
+        which is itself a multivariate polynomial that, when evaluated
+        at the coordinates of some algebra element, returns that
+        element's characteristic polynomial.
 
         The resulting polynomial has `n+1` variables, where `n` is the
         dimension of this algebra. The first `n` variables correspond to
@@ -259,7 +261,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
         Alizadeh, Example 11.11::
 
             sage: J = JordanSpinEJA(3)
-            sage: p = J.characteristic_polynomial(); p
+            sage: p = J.characteristic_polynomial_of(); p
             X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
             sage: xvec = J.one().to_vector()
             sage: p(*xvec)
@@ -272,7 +274,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
         any argument::
 
             sage: J = TrivialEJA()
-            sage: J.characteristic_polynomial()
+            sage: J.characteristic_polynomial_of()
             1
 
         """
@@ -579,6 +581,25 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
             Vector space of degree 6 and dimension 2...
             sage: J1
             Euclidean Jordan algebra of dimension 3...
+            sage: J0.one().natural_representation()
+            [0 0 0]
+            [0 0 0]
+            [0 0 1]
+            sage: orig_df = AA.options.display_format
+            sage: AA.options.display_format = 'radical'
+            sage: J.from_vector(J5.basis()[0]).natural_representation()
+            [          0           0 1/2*sqrt(2)]
+            [          0           0           0]
+            [1/2*sqrt(2)           0           0]
+            sage: J.from_vector(J5.basis()[1]).natural_representation()
+            [          0           0           0]
+            [          0           0 1/2*sqrt(2)]
+            [          0 1/2*sqrt(2)           0]
+            sage: AA.options.display_format = orig_df
+            sage: J1.one().natural_representation()
+            [1 0 0]
+            [0 1 0]
+            [0 0 0]
 
         TESTS:
 
@@ -593,9 +614,10 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
             sage: J1.superalgebra() == J and J1.dimension() == J.dimension()
             True
 
-        The identity elements in the two subalgebras are the
-        projections onto their respective subspaces of the
-        superalgebra's identity element::
+        The decomposition is into eigenspaces, and its components are
+        therefore necessarily orthogonal. Moreover, the identity
+        elements in the two subalgebras are the projections onto their
+        respective subspaces of the superalgebra's identity element::
 
             sage: set_random_seed()
             sage: J = random_eja()
@@ -605,6 +627,16 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
             ....:         x = J.random_element()
             sage: c = x.subalgebra_idempotent()
             sage: J0,J5,J1 = J.peirce_decomposition(c)
+            sage: ipsum = 0
+            sage: for (w,y,z) in zip(J0.basis(), J5.basis(), J1.basis()):
+            ....:     w = w.superalgebra_element()
+            ....:     y = J.from_vector(y)
+            ....:     z = z.superalgebra_element()
+            ....:     ipsum += w.inner_product(y).abs()
+            ....:     ipsum += w.inner_product(z).abs()
+            ....:     ipsum += y.inner_product(z).abs()
+            sage: ipsum
+            0
             sage: J1(c) == J1.one()
             True
             sage: J0(J.one() - c) == J0.one()