+ sage: L = W.random_element()
+ sage: L_star = W(M(L.list()).transpose().list())
+ sage: Z_cone.contains(L) == Z_star.contains(L_star)
+ True
+ """
+ # Matrices are not vectors in Sage, so we have to convert them
+ # to vectors explicitly before we can find a basis. We need these
+ # two values to construct the appropriate "long vector" space.
+ F = K.lattice().base_field()
+ n = K.lattice_dim()
+
+ # These tensor products contain generators for the dual cone of
+ # the cross-positive transformations.
+ tensor_products = [ s.tensor_product(x)
+ for (x,s) in K.discrete_complementarity_set() ]
+
+ # Turn our matrices into long vectors...
+ W = VectorSpace(F, n**2)
+ vectors = [ W(m.list()) for m in tensor_products ]
+
+ check = True
+ if K.is_proper():
+ # All of the generators involved are extreme vectors and
+ # therefore minimal. If this cone is neither solid nor
+ # strictly convex, then the tensor product of ``s`` and ``x``
+ # is the same as that of ``-s`` and ``-x``. However, as a
+ # /set/, ``tensor_products`` may still be minimal.
+ check = False
+
+ # Create the dual cone of the cross-positive operators,
+ # expressed as long vectors.
+ Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension()), check=check)
+
+ # Now compute the desired cone from its dual...
+ Sigma_cone = Sigma_dual.dual()
+
+ # And finally convert its rays back to matrix representations.
+ # But first, make them negative, so we get Z-transformations and
+ # not cross-positive ones.
+ M = MatrixSpace(F, n)
+ return [ -M(v.list()) for v in Sigma_cone ]
+
+
+def Z_cone(K):
+ gens = Z_transformation_gens(K)
+ L = ToricLattice(K.lattice_dim()**2)
+ return Cone([ g.list() for g in gens ], lattice=L, check=False)
+
+def pi_cone(K):
+ gens = positive_operator_gens(K)
+ L = ToricLattice(K.lattice_dim()**2)
+ return Cone([ g.list() for g in gens ], lattice=L, check=False)