The identity is always Lyapunov-like in a nontrivial space::
sage: set_random_seed()
- sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 8)
+ sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
sage: L = identity_matrix(K.lattice_dim())
sage: is_lyapunov_like(L,K)
True
As is the "zero" transformation::
- sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 8)
+ sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
sage: R = K.lattice().vector_space().base_ring()
sage: L = zero_matrix(R, K.lattice_dim())
sage: is_lyapunov_like(L,K)
Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
on ``K``::
- sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6)
+ sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
sage: all([ is_lyapunov_like(L,K) for L in K.lyapunov_like_basis() ])
True
corollary in my paper::
sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 5)
+ sage: K = random_cone(max_ambient_dim=5)
sage: n = K.lattice_dim()
sage: m = K.dim()
sage: l = K.lineality()
The Z-property is possessed by every Z-transformation::
sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 6)
+ sage: K = random_cone(max_ambient_dim=6)
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
The lineality space of Z is LL::
sage: set_random_seed()
- sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6)
+ sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
sage: z_cone = Cone([ z.list() for z in Z_transformation_gens(K) ])
sage: z_cone.linear_subspace() == lls