eja: implement a working minimal_polynomial().

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

diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py
index 8487e0ecea385d00f45c176bdc43b45d31d617ff..d460aa029e777f137e64d6028f34a2b55cce6101 100644 (file)
--- a/mjo/eja/euclidean_jordan_algebra.py
+++ b/mjo/eja/euclidean_jordan_algebra.py
@@ -108,8 +108,86 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
             return self.span_of_powers().dimension()
 
 
+        def subalgebra_generated_by(self):
+            """
+            Return the subalgebra of the parent EJA generated by this element.
+            """
+            # First get the subspace spanned by the powers of myself...
+            V = self.span_of_powers()
+            F = self.base_ring()
+
+            # Now figure out the entries of the right-multiplication
+            # matrix for the successive basis elements b0, b1,... of
+            # that subspace.
+            mats = []
+            for b_right in V.basis():
+                eja_b_right = self.parent()(b_right)
+                b_right_rows = []
+                # The first row of the right-multiplication matrix by
+                # b1 is what we get if we apply that matrix to b1. The
+                # second row of the right multiplication matrix by b1
+                # is what we get when we apply that matrix to b2...
+                for b_left in V.basis():
+                    eja_b_left = self.parent()(b_left)
+                    # Multiply in the original EJA, but then get the
+                    # coordinates from the subalgebra in terms of its
+                    # basis.
+                    this_row = V.coordinates((eja_b_left*eja_b_right).vector())
+                    b_right_rows.append(this_row)
+                b_right_matrix = matrix(F, b_right_rows)
+                mats.append(b_right_matrix)
+
+            return FiniteDimensionalEuclideanJordanAlgebra(F, mats)
+
+
         def minimal_polynomial(self):
-            return self.matrix().minimal_polynomial()
+            """
+            EXAMPLES::
+
+                sage: set_random_seed()
+                sage: n = ZZ.random_element(1,10).abs()
+                sage: J = eja_rn(n)
+                sage: x = J.random_element()
+                sage: x.degree() == x.minimal_polynomial().degree()
+                True
+
+            ::
+
+                sage: set_random_seed()
+                sage: n = ZZ.random_element(1,10).abs()
+                sage: J = eja_ln(n)
+                sage: x = J.random_element()
+                sage: x.degree() == x.minimal_polynomial().degree()
+                True
+
+            The minimal polynomial and the characteristic polynomial coincide
+            and are known (see Alizadeh, Example 11.11) for all elements of
+            the spin factor algebra that aren't scalar multiples of the
+            identity::
+
+                sage: set_random_seed()
+                sage: n = ZZ.random_element(2,10).abs()
+                sage: J = eja_ln(n)
+                sage: y = J.random_element()
+                sage: while y == y.coefficient(0)*J.one():
+                ....:     y = J.random_element()
+                sage: y0 = y.vector()[0]
+                sage: y_bar = y.vector()[1:]
+                sage: actual = y.minimal_polynomial()
+                sage: x = SR.symbol('x', domain='real')
+                sage: expected = x^2 - 2*y0*x + (y0^2 - norm(y_bar)^2)
+                sage: bool(actual == expected)
+                True
+
+            """
+            V = self.span_of_powers()
+            assoc_subalg = self.subalgebra_generated_by()
+            # Mis-design warning: the basis used for span_of_powers()
+            # and subalgebra_generated_by() must be the same, and in
+            # the same order!
+            subalg_self = assoc_subalg(V.coordinates(self.vector()))
+            return subalg_self.matrix().minimal_polynomial()
+
 
         def characteristic_polynomial(self):
             return self.matrix().characteristic_polynomial()