+ sage: K = Cone([(1,),(-1,)])
+ sage: K.is_full_space()
+ True
+ sage: Z_transformation_gens(K)
+ [[-1], [1]]
+
+ sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
+ sage: K.is_full_space()
+ True
+ sage: Z_transformation_gens(K)
+ [
+ [-1 0] [1 0] [ 0 -1] [0 1] [ 0 0] [0 0] [ 0 0] [0 0]
+ [ 0 0], [0 0], [ 0 0], [0 0], [-1 0], [1 0], [ 0 -1], [0 1]
+ ]
+
+ A non-obvious application is to find the Z-transformations on the
+ right half-plane::
+
+ sage: K = Cone([(1,0),(0,1),(0,-1)])
+ sage: Z_transformation_gens(K)
+ [
+ [-1 0] [1 0] [ 0 0] [0 0] [ 0 0] [0 0]
+ [ 0 0], [0 0], [-1 0], [1 0], [ 0 -1], [0 1]
+ ]
+
+ Z-transformations on a subspace are Lyapunov-like and vice-versa::
+
+ sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
+ sage: K.is_full_space()
+ True
+ sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
+ sage: zs = span([ vector(z.list()) for z in Z_transformation_gens(K) ])
+ sage: zs == lls
+ True
+
+ TESTS:
+
+ The Z-property is possessed by every Z-transformation::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: Z_of_K = Z_transformation_gens(K)
+ sage: dcs = K.discrete_complementarity_set()
+ sage: all([(z*x).inner_product(s) <= 0 for z in Z_of_K
+ ....: for (x,s) in dcs])
+ True
+
+ The lineality space of the cone of Z-transformations is the space of
+ Lyapunov-like transformations::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: Z_cone = Cone([ z.list() for z in Z_transformation_gens(K) ],
+ ....: lattice=L,
+ ....: check=False)
+ sage: ll_basis = [ vector(l.list()) for l in K.lyapunov_like_basis() ]
+ sage: lls = L.vector_space().span(ll_basis)
+ sage: Z_cone.linear_subspace() == lls
+ True
+
+ The lineality of the Z-transformations on a cone is the Lyapunov
+ rank of that cone::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: Z_of_K = Z_transformation_gens(K)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: Z_cone = Cone([ z.list() for z in Z_of_K ],
+ ....: lattice=L,
+ ....: check=False)
+ sage: Z_cone.lineality() == K.lyapunov_rank()
+ True
+
+ The lineality spaces of the duals of the positive operator and
+ Z-transformation cones are equal. From this it follows that the
+ dimensions of the Z-transformation cone and positive operator cone
+ are equal::
+
+ sage: set_random_seed()
+ 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
+ """
+ # 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_solid() or K.is_strictly_convex():
+ # The lineality space of either ``K`` or ``K.dual()`` is
+ # trivial and it's easy to show that our generating set is
+ # minimal. I would love a proof that this works when ``K`` is
+ # neither pointed nor solid.
+ #
+ # Note that in that case we can get *duplicates*, since the
+ # tensor product of (x,s) is the same as that of (-x,-s).
+ 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)