+ sage: K = random_cone(max_ambient_dim=4)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: Z_of_K = Z_transformation_gens(K)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: Z_cone = Cone([ z.list() for z in Z_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: pi_cone.dim() == Z_cone.dim()
+ True
+ sage: pi_star = pi_cone.dual()
+ sage: z_star = Z_cone.dual()
+ sage: pi_star.linear_subspace() == z_star.linear_subspace()
+ True
+
+ The trivial cone, full space, and half-plane all give rise to the
+ expected dimensions::
+
+ sage: n = ZZ.random_element().abs()
+ sage: K = Cone([[0] * n], ToricLattice(n))
+ sage: K.is_trivial()
+ True
+ sage: L = ToricLattice(n^2)
+ sage: Z_of_K = Z_transformation_gens(K)
+ sage: Z_cone = Cone([z.list() for z in Z_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = Z_cone.dim()
+ sage: actual == n^2
+ True
+ sage: K = K.dual()
+ sage: K.is_full_space()
+ True
+ sage: Z_of_K = Z_transformation_gens(K)
+ sage: Z_cone = Cone([z.list() for z in Z_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = Z_cone.dim()
+ sage: actual == n^2
+ True
+ sage: K = Cone([(1,0),(0,1),(0,-1)])
+ sage: Z_of_K = Z_transformation_gens(K)
+ sage: Z_cone = Cone([z.list() for z in Z_of_K], check=False)
+ sage: Z_cone.dim() == 3
+ True
+
+ The Z-transformations of a permuted cone can be obtained by
+ conjugation::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: p = SymmetricGroup(K.lattice_dim()).random_element().matrix()
+ sage: pK = Cone([ p*k for k in K ], K.lattice(), check=False)
+ sage: Z_of_pK = Z_transformation_gens(pK)
+ sage: actual = Cone([t.list() for t in Z_of_pK],
+ ....: lattice=L,
+ ....: check=False)
+ sage: Z_of_K = Z_transformation_gens(K)
+ sage: expected = Cone([(p*t*p.inverse()).list() for t in Z_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual.is_equivalent(expected)
+ True
+
+ A transformation is a Z-transformation on a cone if and only if its
+ adjoint is a Z-transformation on the dual of that cone::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: F = K.lattice().vector_space().base_field()
+ sage: n = K.lattice_dim()
+ sage: L = ToricLattice(n**2)
+ sage: W = VectorSpace(F, n**2)
+ sage: Z_of_K = Z_transformation_gens(K)
+ sage: Z_of_K_star = Z_transformation_gens(K.dual())
+ sage: Z_cone = Cone([p.list() for p in Z_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: Z_star = Cone([p.list() for p in Z_of_K_star],
+ ....: lattice=L,
+ ....: check=False)
+ sage: M = MatrixSpace(F, n)
+ sage: L = M(Z_cone.random_element(ring=QQ).list())
+ sage: Z_star.contains(W(L.transpose().list()))
+ True
+
+ sage: L = W.random_element()
+ sage: L_star = W(M(L.list()).transpose().list())
+ sage: Z_cone.contains(L) == Z_star.contains(L_star)