eja: speed up the computation of element powers.

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

diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py
index 4ed0a59216e7d07eaba9a6aec1b43e91902d3513..7c861834723344ea6403f9f5da289af8aa299ae7 100644 (file)
--- a/mjo/eja/eja_element.py
+++ b/mjo/eja/eja_element.py
@@ -25,69 +25,6 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
                       dir(self.__class__) )
 
 
                       dir(self.__class__) )
 
 

-    def __init__(self, A, elt):
-        """
-
-        SETUP::
-
-            sage: from mjo.eja.eja_algebra import (RealSymmetricEJA,
-            ....:                                  random_eja)
-
-        EXAMPLES:
-
-        The identity in `S^n` is converted to the identity in the EJA::
-
-            sage: J = RealSymmetricEJA(3)
-            sage: I = matrix.identity(QQ,3)
-            sage: J(I) == J.one()
-            True
-
-        This skew-symmetric matrix can't be represented in the EJA::
-
-            sage: J = RealSymmetricEJA(3)
-            sage: A = matrix(QQ,3, lambda i,j: i-j)
-            sage: J(A)
-            Traceback (most recent call last):
-            ...
-            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).to_vector() == v
-            True
-
-        """
-        # Goal: if we're given a matrix, and if it lives in our
-        # parent algebra's "natural ambient space," convert it
-        # into an algebra element.
-        #
-        # The catch is, we make a recursive call after converting
-        # the given matrix into a vector that lives in the algebra.
-        # This we need to try the parent class initializer first,
-        # to avoid recursing forever if we're given something that
-        # already fits into the algebra, but also happens to live
-        # in the parent's "natural ambient space" (this happens with
-        # vectors in R^n).
-        ifme = super(FiniteDimensionalEuclideanJordanAlgebraElement, self)
-        try:
-            ifme.__init__(A, elt)
-        except ValueError:
-            natural_basis = A.natural_basis()
-            if elt in natural_basis[0].matrix_space():
-                # Thanks for nothing! Matrix spaces aren't vector
-                # spaces in Sage, so we have to figure out its
-                # natural-basis coordinates ourselves.
-                V = VectorSpace(elt.base_ring(), elt.nrows()**2)
-                W = V.span( _mat2vec(s) for s in natural_basis )
-                coords =  W.coordinate_vector(_mat2vec(elt))
-                ifme.__init__(A, coords)

 
 
     def __pow__(self, n):
 
 
     def __pow__(self, n):


@@ -141,7 +78,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         elif n == 1:
             return self
         else:
         elif n == 1:
             return self
         else:

-            return (self.operator()**(n-1))(self)
+            return (self**(n-1))*self

 
 
     def apply_univariate_polynomial(self, p):
 
 
     def apply_univariate_polynomial(self, p):


@@ -228,6 +165,21 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: x.apply_univariate_polynomial(p)
             0
 
             sage: x.apply_univariate_polynomial(p)
             0
 

+        The characteristic polynomials of the zero and unit elements
+        should be what we think they are in a subalgebra, too::
+
+            sage: J = RealCartesianProductEJA(3)
+            sage: p1 = J.one().characteristic_polynomial()
+            sage: q1 = J.zero().characteristic_polynomial()
+            sage: e0,e1,e2 = J.gens()
+            sage: A = (e0 + 2*e1 + 3*e2).subalgebra_generated_by() # dim 3
+            sage: p2 = A.one().characteristic_polynomial()
+            sage: q2 = A.zero().characteristic_polynomial()
+            sage: p1 == p2
+            True
+            sage: q1 == q2
+            True
+

         """
         p = self.parent().characteristic_polynomial()
         return p(*self.to_vector())
         """
         p = self.parent().characteristic_polynomial()
         return p(*self.to_vector())


@@ -291,9 +243,8 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
             sage: set_random_seed()
             sage: J = random_eja()
 
             sage: set_random_seed()
             sage: J = random_eja()

-            sage: x = J.random_element()
-            sage: y = J.random_element()
-            sage: x.inner_product(y) in RR
+            sage: x,y = J.random_elements(2)
+            sage: x.inner_product(y) in RLF

             True
 
         """
             True
 
         """


@@ -328,9 +279,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         Test Lemma 1 from Chapter III of Koecher::
 
             sage: set_random_seed()
         Test Lemma 1 from Chapter III of Koecher::
 
             sage: set_random_seed()

-            sage: J = random_eja()
-            sage: u = J.random_element()
-            sage: v = J.random_element()
+            sage: u,v = random_eja().random_elements(2)

             sage: lhs = u.operator_commutes_with(u*v)
             sage: rhs = v.operator_commutes_with(u^2)
             sage: lhs == rhs
             sage: lhs = u.operator_commutes_with(u*v)
             sage: rhs = v.operator_commutes_with(u^2)
             sage: lhs == rhs


@@ -340,9 +289,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         Chapter III, or from Baes (2.3)::
 
             sage: set_random_seed()
         Chapter III, or from Baes (2.3)::
 
             sage: set_random_seed()

-            sage: J = random_eja()
-            sage: x = J.random_element()
-            sage: y = J.random_element()
+            sage: x,y = random_eja().random_elements(2)

             sage: Lx = x.operator()
             sage: Ly = y.operator()
             sage: Lxx = (x*x).operator()
             sage: Lx = x.operator()
             sage: Ly = y.operator()
             sage: Lxx = (x*x).operator()


@@ -354,10 +301,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         Baes (2.4)::
 
             sage: set_random_seed()
         Baes (2.4)::
 
             sage: set_random_seed()

-            sage: J = random_eja()
-            sage: x = J.random_element()
-            sage: y = J.random_element()
-            sage: z = J.random_element()
+            sage: x,y,z = random_eja().random_elements(3)

             sage: Lx = x.operator()
             sage: Ly = y.operator()
             sage: Lz = z.operator()
             sage: Lx = x.operator()
             sage: Ly = y.operator()
             sage: Lz = z.operator()


@@ -371,10 +315,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         Baes (2.5)::
 
             sage: set_random_seed()
         Baes (2.5)::
 
             sage: set_random_seed()

-            sage: J = random_eja()
-            sage: u = J.random_element()
-            sage: y = J.random_element()
-            sage: z = J.random_element()
+            sage: u,y,z = random_eja().random_elements(3)

             sage: Lu = u.operator()
             sage: Ly = y.operator()
             sage: Lz = z.operator()
             sage: Lu = u.operator()
             sage: Ly = y.operator()
             sage: Lz = z.operator()


@@ -431,6 +372,15 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: x.is_invertible() == (x.det() != 0)
             True
 
             sage: x.is_invertible() == (x.det() != 0)
             True
 

+        Ensure that the determinant is multiplicative on an associative
+        subalgebra as in Faraut and Koranyi's Proposition II.2.2::
+
+            sage: set_random_seed()
+            sage: J = random_eja().random_element().subalgebra_generated_by()
+            sage: x,y = J.random_elements(2)
+            sage: (x*y).det() == x.det()*y.det()
+            True
+

         """
         P = self.parent()
         r = P.rank()
         """
         P = self.parent()
         r = P.rank()


@@ -462,8 +412,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         Example 11.11::
 
             sage: set_random_seed()
         Example 11.11::
 
             sage: set_random_seed()

-            sage: n = ZZ.random_element(1,10)
-            sage: J = JordanSpinEJA(n)
+            sage: J = JordanSpinEJA.random_instance()

             sage: x = J.random_element()
             sage: while not x.is_invertible():
             ....:     x = J.random_element()
             sage: x = J.random_element()
             sage: while not x.is_invertible():
             ....:     x = J.random_element()


@@ -473,7 +422,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: coeff = ~(x0^2 - x_bar.inner_product(x_bar))
             sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
             sage: x_inverse = coeff*inv_vec
             sage: coeff = ~(x0^2 - x_bar.inner_product(x_bar))
             sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
             sage: x_inverse = coeff*inv_vec

-            sage: x.inverse() == J(x_inverse)
+            sage: x.inverse() == J.from_vector(x_inverse)

             True
 
         TESTS:
             True
 
         TESTS:


@@ -545,15 +494,23 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: J.one().is_invertible()
             True
 
             sage: J.one().is_invertible()
             True
 

-        The zero element is never invertible::
+        The zero element is never invertible in a non-trivial algebra::

 
             sage: set_random_seed()
             sage: J = random_eja()
 
             sage: set_random_seed()
             sage: J = random_eja()

-            sage: J.zero().is_invertible()
+            sage: (not J.is_trivial()) and J.zero().is_invertible()

             False
 
         """
             False
 
         """

-        zero = self.parent().zero()
+        if self.is_zero():
+            if self.parent().is_trivial():
+                return True
+            else:
+                return False
+
+        # In fact, we only need to know if the constant term is non-zero,
+        # so we can pass in the field's zero element instead.
+        zero = self.base_ring().zero()

         p = self.minimal_polynomial()
         return not (p(zero) == zero)
 
         p = self.minimal_polynomial()
         return not (p(zero) == zero)
 


@@ -681,8 +638,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         aren't multiples of the identity are regular::
 
             sage: set_random_seed()
         aren't multiples of the identity are regular::
 
             sage: set_random_seed()

-            sage: n = ZZ.random_element(1,10)
-            sage: J = JordanSpinEJA(n)
+            sage: J = JordanSpinEJA.random_instance()

             sage: x = J.random_element()
             sage: x == x.coefficient(0)*J.one() or x.degree() == 2
             True
             sage: x = J.random_element()
             sage: x == x.coefficient(0)*J.one() or x.degree() == 2
             True


@@ -706,6 +662,11 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             True
 
         """
             True
 
         """

+        if self.is_zero() and not self.parent().is_trivial():
+            # The minimal polynomial of zero in a nontrivial algebra
+            # is "t"; in a trivial algebra it's "1" by convention
+            # (it's an empty product).
+            return 1

         return self.subalgebra_generated_by().dimension()
 
 
         return self.subalgebra_generated_by().dimension()
 
 


@@ -734,6 +695,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         SETUP::
 
             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
         SETUP::
 
             sage: from mjo.eja.eja_algebra import (JordanSpinEJA,

+            ....:                                  RealSymmetricEJA,

             ....:                                  random_eja)
 
         TESTS:
             ....:                                  random_eja)
 
         TESTS:


@@ -759,10 +721,12 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         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
         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::
+        identity. We require the dimension of the algebra to be at least
+        two here so that said elements actually exist::

 
             sage: set_random_seed()
 
             sage: set_random_seed()

-            sage: n = ZZ.random_element(2,10)
+            sage: n_max = max(2, JordanSpinEJA._max_test_case_size())
+            sage: n = ZZ.random_element(2, n_max)

             sage: J = JordanSpinEJA(n)
             sage: y = J.random_element()
             sage: while y == y.coefficient(0)*J.one():
             sage: J = JordanSpinEJA(n)
             sage: y = J.random_element()
             sage: while y == y.coefficient(0)*J.one():


@@ -783,9 +747,36 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: x.apply_univariate_polynomial(p)
             0
 
             sage: x.apply_univariate_polynomial(p)
             0
 

+        The minimal polynomial is invariant under a change of basis,
+        and in particular, a re-scaling of the basis::
+
+            sage: set_random_seed()
+            sage: n_max = RealSymmetricEJA._max_test_case_size()
+            sage: n = ZZ.random_element(1, n_max)
+            sage: J1 = RealSymmetricEJA(n,QQ)
+            sage: J2 = RealSymmetricEJA(n,QQ,False)
+            sage: X = random_matrix(QQ,n)
+            sage: X = X*X.transpose()
+            sage: x1 = J1(X)
+            sage: x2 = J2(X)
+            sage: x1.minimal_polynomial() == x2.minimal_polynomial()
+            True
+

         """
         """

+        if self.is_zero():
+            # We would generate a zero-dimensional subalgebra
+            # where the minimal polynomial would be constant.
+            # That might be correct, but only if *this* algebra
+            # is trivial too.
+            if not self.parent().is_trivial():
+                # Pretty sure we know what the minimal polynomial of
+                # the zero operator is going to be. This ensures
+                # consistency of e.g. the polynomial variable returned
+                # in the "normal" case without us having to think about it.
+                return self.operator().minimal_polynomial()
+

         A = self.subalgebra_generated_by()
         A = self.subalgebra_generated_by()

-        return A.element_class(A,self).operator().minimal_polynomial()
+        return A(self).operator().minimal_polynomial()

 
 
 
 
 
 


@@ -808,7 +799,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
             sage: J = ComplexHermitianEJA(3)
             sage: J.one()
 
             sage: J = ComplexHermitianEJA(3)
             sage: J.one()

-            e0 + e5 + e8
+            e0 + e3 + e8

             sage: J.one().natural_representation()
             [1 0 0 0 0 0]
             [0 1 0 0 0 0]
             sage: J.one().natural_representation()
             [1 0 0 0 0 0]
             [0 1 0 0 0 0]


@@ -821,7 +812,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
             sage: J = QuaternionHermitianEJA(3)
             sage: J.one()
 
             sage: J = QuaternionHermitianEJA(3)
             sage: J.one()

-            e0 + e9 + e14
+            e0 + e5 + e14

             sage: J.one().natural_representation()
             [1 0 0 0 0 0 0 0 0 0 0 0]
             [0 1 0 0 0 0 0 0 0 0 0 0]
             sage: J.one().natural_representation()
             [1 0 0 0 0 0 0 0 0 0 0 0]
             [0 1 0 0 0 0 0 0 0 0 0 0]


@@ -838,10 +829,37 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
         """
         B = self.parent().natural_basis()
 
         """
         B = self.parent().natural_basis()

-        W = B[0].matrix_space()
+        W = self.parent().natural_basis_space()

         return W.linear_combination(zip(B,self.to_vector()))
 
 
         return W.linear_combination(zip(B,self.to_vector()))
 
 

+    def norm(self):
+        """
+        The norm of this element with respect to :meth:`inner_product`.
+
+        SETUP::
+
+            sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+            ....:                                  RealCartesianProductEJA)
+
+        EXAMPLES::
+
+            sage: J = RealCartesianProductEJA(2)
+            sage: x = sum(J.gens())
+            sage: x.norm()
+            sqrt(2)
+
+        ::
+
+            sage: J = JordanSpinEJA(4)
+            sage: x = sum(J.gens())
+            sage: x.norm()
+            2
+
+        """
+        return self.inner_product(self).sqrt()
+
+

     def operator(self):
         """
         Return the left-multiplication-by-this-element
     def operator(self):
         """
         Return the left-multiplication-by-this-element


@@ -855,8 +873,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
             sage: set_random_seed()
             sage: J = random_eja()
 
             sage: set_random_seed()
             sage: J = random_eja()

-            sage: x = J.random_element()
-            sage: y = J.random_element()
+            sage: x,y = J.random_elements(2)

             sage: x.operator()(y) == x*y
             True
             sage: y.operator()(x) == x*y
             sage: x.operator()(y) == x*y
             True
             sage: y.operator()(x) == x*y


@@ -864,10 +881,12 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
         """
         P = self.parent()
 
         """
         P = self.parent()

+        left_mult_by_self = lambda y: self*y
+        L = P.module_morphism(function=left_mult_by_self, codomain=P)

         return FiniteDimensionalEuclideanJordanAlgebraOperator(
                  P,
                  P,
         return FiniteDimensionalEuclideanJordanAlgebraOperator(
                  P,
                  P,

-                 self.to_matrix() )
+                 L.matrix() )

 
 
     def quadratic_representation(self, other=None):
 
 
     def quadratic_representation(self, other=None):


@@ -885,10 +904,9 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         Alizadeh's Example 11.12::
 
             sage: set_random_seed()
         Alizadeh's Example 11.12::
 
             sage: set_random_seed()

-            sage: n = ZZ.random_element(1,10)
-            sage: J = JordanSpinEJA(n)
-            sage: x = J.random_element()
+            sage: x = JordanSpinEJA.random_instance().random_element()

             sage: x_vec = x.to_vector()
             sage: x_vec = x.to_vector()

+            sage: n = x_vec.degree()

             sage: x0 = x_vec[0]
             sage: x_bar = x_vec[1:]
             sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
             sage: x0 = x_vec[0]
             sage: x_bar = x_vec[1:]
             sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])


@@ -905,8 +923,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
             sage: set_random_seed()
             sage: J = random_eja()
 
             sage: set_random_seed()
             sage: J = random_eja()

-            sage: x = J.random_element()
-            sage: y = J.random_element()
+            sage: x,y = J.random_elements(2)

             sage: Lx = x.operator()
             sage: Lxx = (x*x).operator()
             sage: Qx = x.quadratic_representation()
             sage: Lx = x.operator()
             sage: Lxx = (x*x).operator()
             sage: Qx = x.quadratic_representation()


@@ -923,7 +940,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
         Property 2 (multiply on the right for :trac:`28272`):
 
 
         Property 2 (multiply on the right for :trac:`28272`):
 

-            sage: alpha = QQ.random_element()
+            sage: alpha = J.base_ring().random_element()

             sage: (alpha*x).quadratic_representation() == Qx*(alpha^2)
             True
 
             sage: (alpha*x).quadratic_representation() == Qx*(alpha^2)
             True
 


@@ -1001,11 +1018,15 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
             sage: from mjo.eja.eja_algebra import random_eja
 
 
             sage: from mjo.eja.eja_algebra import random_eja
 

-        TESTS::
+        TESTS:
+
+        This subalgebra, being composed of only powers, is associative::

 
             sage: set_random_seed()
 
             sage: set_random_seed()

-            sage: x = random_eja().random_element()
-            sage: x.subalgebra_generated_by().is_associative()
+            sage: x0 = random_eja().random_element()
+            sage: A = x0.subalgebra_generated_by()
+            sage: x,y,z = A.random_elements(3)
+            sage: (x*y)*z == x*(y*z)

             True
 
         Squaring in the subalgebra should work the same as in
             True
 
         Squaring in the subalgebra should work the same as in


@@ -1017,6 +1038,15 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: A(x^2) == A(x)*A(x)
             True
 
             sage: A(x^2) == A(x)*A(x)
             True
 

+        The subalgebra generated by the zero element is trivial::
+
+            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
+

         """
         return FiniteDimensionalEuclideanJordanElementSubalgebra(self)
 
         """
         return FiniteDimensionalEuclideanJordanElementSubalgebra(self)
 


@@ -1046,7 +1076,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             raise ValueError("this only works with non-nilpotent elements!")
 
         J = self.subalgebra_generated_by()
             raise ValueError("this only works with non-nilpotent elements!")
 
         J = self.subalgebra_generated_by()

-        u = J.from_vector(self.to_vector())
+        u = J(self)

 
         # The image of the matrix of left-u^m-multiplication
         # will be minimal for some natural number s...
 
         # The image of the matrix of left-u^m-multiplication
         # will be minimal for some natural number s...


@@ -1071,7 +1101,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         # Our FiniteDimensionalAlgebraElement superclass uses rows.
         u_next = u**(s+1)
         A = u_next.operator().matrix()
         # Our FiniteDimensionalAlgebraElement superclass uses rows.
         u_next = u**(s+1)
         A = u_next.operator().matrix()

-        c = J(A.solve_right(u_next.to_vector()))
+        c = J.from_vector(A.solve_right(u_next.to_vector()))

 
         # Now c is the idempotent we want, but it still lives in the subalgebra.
         return c.superalgebra_element()
 
         # Now c is the idempotent we want, but it still lives in the subalgebra.
         return c.superalgebra_element()


@@ -1106,7 +1136,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
             sage: set_random_seed()
             sage: J = random_eja()
 
             sage: set_random_seed()
             sage: J = random_eja()

-            sage: J.random_element().trace() in J.base_ring()
+            sage: J.random_element().trace() in RLF

             True
 
         """
             True
 
         """


@@ -1130,22 +1160,17 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
         TESTS:
 
 
         TESTS:
 

-        The trace inner product is commutative::
+        The trace inner product is commutative, bilinear, and satisfies
+        the Jordan axiom:

 
             sage: set_random_seed()
             sage: J = random_eja()
 
             sage: set_random_seed()
             sage: J = random_eja()

-            sage: x = J.random_element(); y = J.random_element()
+            sage: x,y,z = J.random_elements(3)
+            sage: # commutative

             sage: x.trace_inner_product(y) == y.trace_inner_product(x)
             True
             sage: x.trace_inner_product(y) == y.trace_inner_product(x)
             True

-
-        The trace inner product is bilinear::
-
-            sage: set_random_seed()
-            sage: J = random_eja()
-            sage: x = J.random_element()
-            sage: y = J.random_element()
-            sage: z = J.random_element()
-            sage: a = QQ.random_element();
+            sage: # bilinear
+            sage: a = J.base_ring().random_element();

             sage: actual = (a*(x+z)).trace_inner_product(y)
             sage: expected = ( a*x.trace_inner_product(y) +
             ....:              a*z.trace_inner_product(y) )
             sage: actual = (a*(x+z)).trace_inner_product(y)
             sage: expected = ( a*x.trace_inner_product(y) +
             ....:              a*z.trace_inner_product(y) )


@@ -1156,15 +1181,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             ....:              a*x.trace_inner_product(z) )
             sage: actual == expected
             True
             ....:              a*x.trace_inner_product(z) )
             sage: actual == expected
             True

-
-        The trace inner product satisfies the compatibility
-        condition in the definition of a Euclidean Jordan algebra::
-
-            sage: set_random_seed()
-            sage: J = random_eja()
-            sage: x = J.random_element()
-            sage: y = J.random_element()
-            sage: z = J.random_element()
+            sage: # jordan axiom

             sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
             True
 
             sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
             True
 


@@ -1173,3 +1190,30 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             raise TypeError("'other' must live in the same algebra")
 
         return (self*other).trace()
             raise TypeError("'other' must live in the same algebra")
 
         return (self*other).trace()

+
+
+    def trace_norm(self):
+        """
+        The norm of this element with respect to :meth:`trace_inner_product`.
+
+        SETUP::
+
+            sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+            ....:                                  RealCartesianProductEJA)
+
+        EXAMPLES::
+
+            sage: J = RealCartesianProductEJA(2)
+            sage: x = sum(J.gens())
+            sage: x.trace_norm()
+            sqrt(2)
+
+        ::
+
+            sage: J = JordanSpinEJA(4)
+            sage: x = sum(J.gens())
+            sage: x.trace_norm()
+            2*sqrt(2)
+
+        """
+        return self.trace_inner_product(self).sqrt()