eja: add random_elements() method to get multiple random elements at once.

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

diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py
index 90c236af8ef4dd46784007cb4927d00ba6b4e33e..8a17c25d4a0d52d98c063f856d888207e2ca74ea 100644 (file)
--- a/mjo/eja/eja_element.py
+++ b/mjo/eja/eja_element.py
@@ -243,8 +243,7 @@ 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.inner_product(y) in RR
             True
 
             sage: x.inner_product(y) in RR
             True
 


@@ -280,9 +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


@@ -292,9 +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()


@@ -306,10 +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()


@@ -323,10 +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()


@@ -388,8 +377,7 @@ 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 +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()


@@ -651,8 +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


@@ -709,6 +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:


@@ -734,10 +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():


@@ -758,6 +747,21 @@ 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,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


@@ -869,8 +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


@@ -901,10 +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)])


@@ -921,8 +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()


@@ -1024,9 +1025,7 @@ 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
 


@@ -1161,21 +1160,16 @@ 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: # 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) +
             sage: a = J.base_ring().random_element();
             sage: actual = (a*(x+z)).trace_inner_product(y)
             sage: expected = ( a*x.trace_inner_product(y) +


@@ -1187,15 +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