- sage: K = random_cone(max_ambient_dim=6)
- sage: dcs = discrete_complementarity_set(K)
- sage: sum([x.inner_product(s).abs() for (x,s) in dcs])
- 0
-
- """
- V = K.lattice().vector_space()
-
- # Convert rays to vectors so that we can compute inner products.
- xs = [V(x) for x in K.rays()]
-
- # We also convert the generators of the dual cone so that we
- # return pairs of vectors and not (vector, ray) pairs.
- ss = [V(s) for s in K.dual().rays()]
-
- return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0]
-
-
-def LL(K):
- r"""
- Compute a basis of Lyapunov-like transformations on this cone.
-
- OUTPUT:
-
- A list of matrices forming a basis for the space of all
- Lyapunov-like transformations on the given cone.
-
- EXAMPLES:
-
- The trivial cone has no Lyapunov-like transformations::
-
- sage: L = ToricLattice(0)
- sage: K = Cone([], lattice=L)
- sage: LL(K)
- []
-
- The Lyapunov-like transformations on the nonnegative orthant are
- simply diagonal matrices::
-
- sage: K = Cone([(1,)])
- sage: LL(K)
- [[1]]
-
- sage: K = Cone([(1,0),(0,1)])
- sage: LL(K)
- [
- [1 0] [0 0]
- [0 0], [0 1]
- ]
-
- sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
- sage: LL(K)
- [
- [1 0 0] [0 0 0] [0 0 0]
- [0 0 0] [0 1 0] [0 0 0]
- [0 0 0], [0 0 0], [0 0 1]
- ]
-
- Only the identity matrix is Lyapunov-like on the `L^{3}_{1}` and
- `L^{3}_{\infty}` cones [Rudolf et al.]_::
-
- sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
- sage: LL(L31)
- [
- [1 0 0]
- [0 1 0]
- [0 0 1]
- ]
-
- sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)])
- sage: LL(L3infty)
- [
- [1 0 0]
- [0 1 0]
- [0 0 1]
- ]
+ 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: pi_of_pK = positive_operator_gens(pK)
+ sage: actual = Cone([t.list() for t in pi_of_pK],
+ ....: lattice=L,
+ ....: check=False)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: expected = Cone([(p*t*p.inverse()).list() for t in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual.is_equivalent(expected)
+ True