X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;ds=inline;f=mjo%2Feja%2Feja_element.py;h=a4af4eaedbb4ce96c16aa31ab0e98c2fa4c5b6c7;hb=64a06cae592dafa4ad007110d5d7ea9ae62dcee5;hp=eee8f69bd76ddfba49e3cb4531f55d0e970ebd1d;hpb=bbd6d4c6e39870b0936949b510e70af2b5358f9e;p=sage.d.git

diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py
index eee8f69..a4af4ea 100644
--- a/mjo/eja/eja_element.py
+++ b/mjo/eja/eja_element.py
@@ -407,7 +407,8 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
         SETUP::
 
-            sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+            sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
+            ....:                                  JordanSpinEJA,
             ....:                                  random_eja)
 
         EXAMPLES:
@@ -473,6 +474,22 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             ....:    x.operator().inverse()(J.one()) == x.inverse() )
             True
 
+        Proposition II.2.4 in Faraut and Korányi gives a formula for
+        the inverse based on the characteristic polynomial and the
+        Cayley-Hamilton theorem for Euclidean Jordan algebras::
+
+            sage: set_random_seed()
+            sage: J = ComplexHermitianEJA(3)
+            sage: x = J.random_element()
+            sage: while not x.is_invertible():
+            ....:     x = J.random_element()
+            sage: r = J.rank()
+            sage: a = x.characteristic_polynomial().coefficients(sparse=False)
+            sage: expected  = (-1)^(r+1)/x.det()
+            sage: expected *= sum( a[i+1]*x^i for i in range(r) )
+            sage: x.inverse() == expected
+            True
+
         """
         if not self.is_invertible():
             raise ValueError("element is not invertible")
@@ -1023,12 +1040,82 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
 
 
+    def spectral_decomposition(self):
+        """
+        Return the unique spectral decomposition of this element.
+
+        ALGORITHM:
+
+        Following Faraut and Korányi's Theorem III.1.1, we restrict this
+        element's left-multiplication-by operator to the subalgebra it
+        generates. We then compute the spectral decomposition of that
+        operator, and the spectral projectors we get back must be the
+        left-multiplication-by operators for the idempotents we
+        seek. Thus applying them to the identity element gives us those
+        idempotents.
+
+        Since the eigenvalues are required to be distinct, we take
+        the spectral decomposition of the zero element to be zero
+        times the identity element of the algebra (which is idempotent,
+        obviously).
+
+        SETUP::
+
+            sage: from mjo.eja.eja_algebra import RealSymmetricEJA
+
+        EXAMPLES:
+
+        The spectral decomposition of the identity is ``1`` times itself,
+        and the spectral decomposition of zero is ``0`` times the identity::
+
+            sage: J = RealSymmetricEJA(3,AA)
+            sage: J.one()
+            e0 + e2 + e5
+            sage: J.one().spectral_decomposition()
+            [(1, e0 + e2 + e5)]
+            sage: J.zero().spectral_decomposition()
+            [(0, e0 + e2 + e5)]
+
+        TESTS::
+
+            sage: J = RealSymmetricEJA(4,AA)
+            sage: x = sum(J.gens())
+            sage: sd = x.spectral_decomposition()
+            sage: l0 = sd[0][0]
+            sage: l1 = sd[1][0]
+            sage: c0 = sd[0][1]
+            sage: c1 = sd[1][1]
+            sage: c0.inner_product(c1) == 0
+            True
+            sage: c0.is_idempotent()
+            True
+            sage: c1.is_idempotent()
+            True
+            sage: c0 + c1 == J.one()
+            True
+            sage: l0*c0 + l1*c1 == x
+            True
+
+        """
+        P = self.parent()
+        A = self.subalgebra_generated_by(orthonormalize_basis=True)
+        result = []
+        for (evalue, proj) in A(self).operator().spectral_decomposition():
+            result.append( (evalue, proj(A.one()).superalgebra_element()) )
+        return result
 
     def subalgebra_generated_by(self, orthonormalize_basis=False):
         """
         Return the associative subalgebra of the parent EJA generated
         by this element.
 
+        Since our parent algebra is unital, we want "subalgebra" to mean
+        "unital subalgebra" as well; thus the subalgebra that an element
+        generates will itself be a Euclidean Jordan algebra after
+        restricting the algebra operations appropriately. This is the
+        subalgebra that Faraut and Korányi work with in section II.2, for
+        example.
+
         SETUP::
 
             sage: from mjo.eja.eja_algebra import random_eja
@@ -1053,14 +1140,13 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: A(x^2) == A(x)*A(x)
             True
 
-        The subalgebra generated by the zero element is trivial::
+        By definition, the subalgebra generated by the zero element is the
+        one-dimensional algebra generated by the identity element::
 
             sage: set_random_seed()
             sage: A = random_eja().zero().subalgebra_generated_by()
-            sage: A
-            Euclidean Jordan algebra of dimension 0 over...
-            sage: A.one()
-            0
+            sage: A.dimension()
+            1
 
         """
         return FiniteDimensionalEuclideanJordanElementSubalgebra(self, orthonormalize_basis)