eja: add another test for element inverses.

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

diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py
index a681ae29652f913246033affb3db74d85b52c9c0..5944c0779a8b7a63b1fa41897947cef4dbee83bb 100644 (file)
--- a/mjo/eja/eja_element.py
+++ b/mjo/eja/eja_element.py
@@ -1,3 +1,5 @@
+from itertools import izip
+

 from sage.matrix.constructor import matrix
 from sage.modules.free_module import VectorSpace
 from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement
 from sage.matrix.constructor import matrix
 from sage.modules.free_module import VectorSpace
 from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement


@@ -78,7 +80,7 @@ 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):


@@ -243,9 +245,8 @@ 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
 
         """


@@ -280,9 +281,7 @@ 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


@@ -292,9 +291,7 @@ 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()


@@ -306,10 +303,7 @@ 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()


@@ -323,10 +317,7 @@ 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()


@@ -388,8 +379,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
             sage: set_random_seed()
             sage: J = random_eja().random_element().subalgebra_generated_by()
 
             sage: set_random_seed()
             sage: J = random_eja().random_element().subalgebra_generated_by()

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

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


@@ -424,8 +414,7 @@ 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()


@@ -471,6 +460,17 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             ...
             ValueError: element is not invertible
 
             ...
             ValueError: element is not invertible
 

+        Proposition II.2.3 in Faraut and Koranyi says that the inverse
+        of an element is the inverse of its left-multiplication operator
+        applied to the algebra's identity, when that inverse exists::
+
+            sage: set_random_seed()
+            sage: J = random_eja()
+            sage: x = J.random_element()
+            sage: (not x.operator().is_invertible()) or (
+            ....:    x.operator().inverse()(J.one()) == x.inverse() )
+            True
+

         """
         if not self.is_invertible():
             raise ValueError("element is not invertible")
         """
         if not self.is_invertible():
             raise ValueError("element is not invertible")


@@ -651,8 +651,7 @@ 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


@@ -709,6 +708,7 @@ 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:


@@ -734,10 +734,12 @@ 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():


@@ -758,6 +760,21 @@ 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,normalize_basis=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
         """
         if self.is_zero():
             # We would generate a zero-dimensional subalgebra


@@ -826,7 +843,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         """
         B = self.parent().natural_basis()
         W = self.parent().natural_basis_space()
         """
         B = self.parent().natural_basis()
         W = self.parent().natural_basis_space()

-        return W.linear_combination(zip(B,self.to_vector()))
+        return W.linear_combination(izip(B,self.to_vector()))

 
 
     def norm(self):
 
 
     def norm(self):


@@ -869,8 +886,7 @@ 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


@@ -901,10 +917,9 @@ 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)])


@@ -921,8 +936,7 @@ 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()


@@ -939,7 +953,7 @@ 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
 


@@ -967,10 +981,10 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: not x.is_invertible() or (
             ....:   x.quadratic_representation(x.inverse())*Qx
             ....:   ==
             sage: not x.is_invertible() or (
             ....:   x.quadratic_representation(x.inverse())*Qx
             ....:   ==

-            ....:   2*x.operator()*Qex - Qx )
+            ....:   2*Lx*Qex - Qx )

             True
 
             True
 

-            sage: 2*x.operator()*Qex - Qx == Lxx
+            sage: 2*Lx*Qex - Qx == Lxx

             True
 
         Property 5:
             True
 
         Property 5:


@@ -1008,7 +1022,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
 
 
 
 
 

-    def subalgebra_generated_by(self):
+    def subalgebra_generated_by(self, orthonormalize_basis=False):

         """
         Return the associative subalgebra of the parent EJA generated
         by this element.
         """
         Return the associative subalgebra of the parent EJA generated
         by this element.


@@ -1024,9 +1038,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: set_random_seed()
             sage: x0 = random_eja().random_element()
             sage: A = x0.subalgebra_generated_by()
             sage: set_random_seed()
             sage: x0 = random_eja().random_element()
             sage: A = x0.subalgebra_generated_by()

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

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


@@ -1044,12 +1056,12 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: set_random_seed()
             sage: A = random_eja().zero().subalgebra_generated_by()
             sage: A
             sage: set_random_seed()
             sage: A = random_eja().zero().subalgebra_generated_by()
             sage: A

-            Euclidean Jordan algebra of dimension 0 over Rational Field
+            Euclidean Jordan algebra of dimension 0 over...

             sage: A.one()
             0
 
         """
             sage: A.one()
             0
 
         """

-        return FiniteDimensionalEuclideanJordanElementSubalgebra(self)
+        return FiniteDimensionalEuclideanJordanElementSubalgebra(self, orthonormalize_basis)

 
 
     def subalgebra_idempotent(self):
 
 
     def subalgebra_idempotent(self):


@@ -1137,7 +1149,7 @@ 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
 
         """


@@ -1161,22 +1173,16 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
         TESTS:
 
 
         TESTS:
 

-        The trace inner product is commutative::
+        The trace inner product is commutative, bilinear, and associative::

 
             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) )


@@ -1187,15 +1193,7 @@ 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: # associative

             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