``K``. It is known [Orlitzky]_ that this property need only be
checked for generators of ``K`` and its dual.
+ There are faster ways of checking this property. For example, we
+ could compute a `lyapunov_like_basis` of the cone, and then test
+ whether or not the given matrix is contained in the span of that
+ basis. The value of this function is that it works on symbolic
+ matrices.
+
INPUT:
- ``L`` -- A linear transformation or matrix.
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
for (x,s) in K.discrete_complementarity_set()])
-def random_element(K):
+def positive_operator_gens(K1, K2 = None):
r"""
- Return a random element of ``K`` from its ambient vector space.
-
- ALGORITHM:
-
- The cone ``K`` is specified in terms of its generators, so that
- ``K`` is equal to the convex conic combination of those generators.
- To choose a random element of ``K``, we assign random nonnegative
- coefficients to each generator of ``K`` and construct a new vector
- from the scaled rays.
-
- A vector, rather than a ray, is returned so that the element may
- have non-integer coordinates. Thus the element may have an
- arbitrarily small norm.
+ Compute generators of the cone of positive operators on this cone. A
+ linear operator on a cone is positive if the image of the cone under
+ the operator is a subset of the cone. This concept can be extended
+ to two cones, where the image of the first cone under a positive
+ operator is a subset of the second cone.
- EXAMPLES:
+ INPUT:
- A random element of the trivial cone is zero::
+ - ``K2`` -- (default: ``K1``) the codomain cone; the image of this
+ cone under the returned operators is a subset of ``K2``.
- sage: set_random_seed()
- sage: K = Cone([], ToricLattice(0))
- sage: random_element(K)
- ()
- sage: K = Cone([(0,)])
- sage: random_element(K)
- (0)
- sage: K = Cone([(0,0)])
- sage: random_element(K)
- (0, 0)
- sage: K = Cone([(0,0,0)])
- sage: random_element(K)
- (0, 0, 0)
+ OUTPUT:
- TESTS:
+ A list of `m`-by-``n`` matrices where ``m == K2.lattice_dim()`` and
+ ``n == K1.lattice_dim()``. Each matrix ``P`` in the list should have
+ the property that ``P*x`` is an element of ``K2`` whenever ``x`` is
+ an element of ``K1``. Moreover, any nonnegative linear combination of
+ these matrices shares the same property.
- Any cone should contain an element of itself::
+ .. SEEALSO::
- sage: set_random_seed()
- sage: K = random_cone(max_rays = 8)
- sage: K.contains(random_element(K))
- True
+ :meth:`cross_positive_operator_gens`, :meth:`Z_operator_gens`,
- """
- V = K.lattice().vector_space()
- F = V.base_ring()
- coefficients = [ F.random_element().abs() for i in range(K.nrays()) ]
- vector_gens = map(V, K.rays())
- scaled_gens = [ coefficients[i]*vector_gens[i]
- for i in range(len(vector_gens)) ]
-
- # Make sure we return a vector. Without the coercion, we might
- # return ``0`` when ``K`` has no rays.
- v = V(sum(scaled_gens))
- return v
+ REFERENCES:
+ .. [Tam]
+ B.-S. Tam.
+ Some results of polyhedral cones and simplicial cones.
+ Linear and Multilinear Algebra, 4:4 (1977) 281--284.
-def positive_operator_gens(K):
- r"""
- Compute generators of the cone of positive operators on this cone.
+ EXAMPLES:
- OUTPUT:
+ Positive operators on the nonnegative orthant are nonnegative matrices::
- A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
- Each matrix ``P`` in the list should have the property that ``P*x``
- is an element of ``K`` whenever ``x`` is an element of
- ``K``. Moreover, any nonnegative linear combination of these
- matrices shares the same property.
+ sage: K = Cone([(1,)])
+ sage: positive_operator_gens(K)
+ [[1]]
- EXAMPLES:
+ sage: K = Cone([(1,0),(0,1)])
+ sage: positive_operator_gens(K)
+ [
+ [1 0] [0 1] [0 0] [0 0]
+ [0 0], [0 0], [1 0], [0 1]
+ ]
The trivial cone in a trivial space has no positive operators::
sage: positive_operator_gens(K)
[]
- Positive operators on the nonnegative orthant are nonnegative matrices::
+ Every operator is positive on the trivial cone::
- sage: K = Cone([(1,)])
+ sage: K = Cone([(0,)])
sage: positive_operator_gens(K)
- [[1]]
+ [[1], [-1]]
- sage: K = Cone([(1,0),(0,1)])
+ sage: K = Cone([(0,0)])
+ sage: K.is_trivial()
+ True
sage: positive_operator_gens(K)
[
- [1 0] [0 1] [0 0] [0 0]
- [0 0], [0 0], [1 0], [0 1]
+ [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]
]
Every operator is positive on the ambient vector space::
[0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1]
]
+ A non-obvious application is to find the positive operators on the
+ right half-plane::
+
+ sage: K = Cone([(1,0),(0,1),(0,-1)])
+ sage: positive_operator_gens(K)
+ [
+ [1 0] [0 0] [ 0 0] [0 0] [ 0 0]
+ [0 0], [1 0], [-1 0], [0 1], [ 0 -1]
+ ]
+
TESTS:
- A positive operator on a cone should send its generators into the cone::
+ Each positive operator generator should send the generators of one
+ cone into the other cone::
+
+ sage: set_random_seed()
+ sage: K1 = random_cone(max_ambient_dim=4)
+ sage: K2 = random_cone(max_ambient_dim=4)
+ sage: pi_K1_K2 = positive_operator_gens(K1,K2)
+ sage: all([ K2.contains(P*x) for P in pi_K1_K2 for x in K1 ])
+ True
+
+ Each positive operator generator should send a random element of one
+ cone into the other cone::
- sage: K = random_cone(max_ambient_dim = 6)
+ sage: set_random_seed()
+ sage: K1 = random_cone(max_ambient_dim=4)
+ sage: K2 = random_cone(max_ambient_dim=4)
+ sage: pi_K1_K2 = positive_operator_gens(K1,K2)
+ sage: all([ K2.contains(P*K1.random_element(QQ)) for P in pi_K1_K2 ])
+ True
+
+ A random element of the positive operator cone should send the
+ generators of one cone into the other cone::
+
+ sage: set_random_seed()
+ sage: K1 = random_cone(max_ambient_dim=4)
+ sage: K2 = random_cone(max_ambient_dim=4)
+ sage: pi_K1_K2 = positive_operator_gens(K1,K2)
+ sage: L = ToricLattice(K1.lattice_dim() * K2.lattice_dim())
+ sage: pi_cone = Cone([ g.list() for g in pi_K1_K2 ],
+ ....: lattice=L,
+ ....: check=False)
+ sage: P = matrix(K2.lattice_dim(),
+ ....: K1.lattice_dim(),
+ ....: pi_cone.random_element(QQ).list())
+ sage: all([ K2.contains(P*x) for x in K1 ])
+ True
+
+ A random element of the positive operator cone should send a random
+ element of one cone into the other cone::
+
+ sage: set_random_seed()
+ sage: K1 = random_cone(max_ambient_dim=4)
+ sage: K2 = random_cone(max_ambient_dim=4)
+ sage: pi_K1_K2 = positive_operator_gens(K1,K2)
+ sage: L = ToricLattice(K1.lattice_dim() * K2.lattice_dim())
+ sage: pi_cone = Cone([ g.list() for g in pi_K1_K2 ],
+ ....: lattice=L,
+ ....: check=False)
+ sage: P = matrix(K2.lattice_dim(),
+ ....: K1.lattice_dim(),
+ ....: pi_cone.random_element(QQ).list())
+ sage: K2.contains(P*K1.random_element(ring=QQ))
+ True
+
+ The lineality space of the dual of the cone of positive operators
+ can be computed from the lineality spaces of the cone and its dual::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
sage: pi_of_K = positive_operator_gens(K)
- sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()])
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: pi_cone = Cone([ g.list() for g in pi_of_K ],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = pi_cone.dual().linear_subspace()
+ sage: U1 = [ vector((s.tensor_product(x)).list())
+ ....: for x in K.lines()
+ ....: for s in K.dual() ]
+ sage: U2 = [ vector((s.tensor_product(x)).list())
+ ....: for x in K
+ ....: for s in K.dual().lines() ]
+ sage: expected = pi_cone.lattice().vector_space().span(U1 + U2)
+ sage: actual == expected
+ True
+
+ The lineality of the dual of the cone of positive operators
+ is known from its lineality space::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: n = K.lattice_dim()
+ sage: m = K.dim()
+ sage: l = K.lineality()
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: L = ToricLattice(n**2)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = pi_cone.dual().lineality()
+ sage: expected = l*(m - l) + m*(n - m)
+ sage: actual == expected
True
The dimension of the cone of positive operators is given by the
corollary in my paper::
- sage: K = random_cone(max_ambient_dim = 6)
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
sage: n = K.lattice_dim()
sage: m = K.dim()
sage: l = K.lineality()
sage: pi_of_K = positive_operator_gens(K)
- sage: actual = Cone([p.list() for p in pi_of_K]).dim()
- sage: expected = n**2 - l*(n - l) - (n - m)*m
+ sage: L = ToricLattice(n**2)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = pi_cone.dim()
+ sage: expected = n**2 - l*(m - l) - (n - m)*m
+ sage: actual == expected
+ 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: pi_of_K = positive_operator_gens(K)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = pi_cone.dim()
+ sage: actual == n^2
+ True
+ sage: K = K.dual()
+ sage: K.is_full_space()
+ True
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = pi_cone.dim()
+ sage: actual == n^2
+ True
+ sage: K = Cone([(1,0),(0,1),(0,-1)])
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: actual = Cone([p.list() for p in pi_of_K], check=False).dim()
+ sage: actual == 3
+ True
+
+ The lineality of the cone of positive operators follows from the
+ description of its generators::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: n = K.lattice_dim()
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: L = ToricLattice(n**2)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = pi_cone.lineality()
+ sage: expected = n**2 - K.dim()*K.dual().dim()
+ sage: actual == expected
+ True
+
+ The trivial cone, full space, and half-plane all give rise to the
+ expected linealities::
+
+ sage: n = ZZ.random_element().abs()
+ sage: K = Cone([[0] * n], ToricLattice(n))
+ sage: K.is_trivial()
+ True
+ sage: L = ToricLattice(n^2)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = pi_cone.lineality()
+ sage: actual == n^2
+ True
+ sage: K = K.dual()
+ sage: K.is_full_space()
+ True
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K], lattice=L)
+ sage: pi_cone.lineality() == n^2
+ True
+ sage: K = Cone([(1,0),(0,1),(0,-1)])
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: pi_cone = Cone([p.list() for p in pi_of_K], check=False)
+ sage: actual = pi_cone.lineality()
+ sage: actual == 2
+ True
+
+ A cone is proper if and only if its cone of positive operators
+ is proper::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: pi_of_K = positive_operator_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: K.is_proper() == pi_cone.is_proper()
+ True
+
+ The positive operators 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: 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
+
+ A transformation is positive on a cone if and only if its adjoint is
+ positive 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: pi_of_K = positive_operator_gens(K)
+ sage: pi_of_K_star = positive_operator_gens(K.dual())
+ sage: pi_cone = Cone([p.list() for p in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: pi_star = Cone([p.list() for p in pi_of_K_star],
+ ....: lattice=L,
+ ....: check=False)
+ sage: M = MatrixSpace(F, n)
+ sage: L = M(pi_cone.random_element(ring=QQ).list())
+ sage: pi_star.contains(W(L.transpose().list()))
+ True
+
+ sage: L = W.random_element()
+ sage: L_star = W(M(L.list()).transpose().list())
+ sage: pi_cone.contains(L) == pi_star.contains(L_star)
+ True
+
+ The Lyapunov rank of the positive operator cone is the product of
+ the Lyapunov ranks of the associated cones if they're all proper::
+
+ sage: K1 = random_cone(max_ambient_dim=4,
+ ....: strictly_convex=True,
+ ....: solid=True)
+ sage: K2 = random_cone(max_ambient_dim=4,
+ ....: strictly_convex=True,
+ ....: solid=True)
+ sage: pi_K1_K2 = positive_operator_gens(K1,K2)
+ sage: L = ToricLattice(K1.lattice_dim() * K2.lattice_dim())
+ sage: pi_cone = Cone([ g.list() for g in pi_K1_K2 ],
+ ....: lattice=L,
+ ....: check=False)
+ sage: beta1 = K1.lyapunov_rank()
+ sage: beta2 = K2.lyapunov_rank()
+ sage: pi_cone.lyapunov_rank() == beta1*beta2
+ True
+
+ The Lyapunov-like operators on a proper polyhedral positive operator
+ cone can be computed from the Lyapunov-like operators on the cones
+ with respect to which the operators are positive::
+
+ sage: K1 = random_cone(max_ambient_dim=4,
+ ....: strictly_convex=True,
+ ....: solid=True)
+ sage: K2 = random_cone(max_ambient_dim=4,
+ ....: strictly_convex=True,
+ ....: solid=True)
+ sage: pi_K1_K2 = positive_operator_gens(K1,K2)
+ sage: F = K1.lattice().base_field()
+ sage: m = K1.lattice_dim()
+ sage: n = K2.lattice_dim()
+ sage: L = ToricLattice(m*n)
+ sage: M1 = MatrixSpace(F, m, m)
+ sage: M2 = MatrixSpace(F, n, n)
+ sage: LL_K1 = [ M1(x.list()) for x in K1.dual().lyapunov_like_basis() ]
+ sage: LL_K2 = [ M2(x.list()) for x in K2.lyapunov_like_basis() ]
+ sage: tps = [ s.tensor_product(x) for x in LL_K1 for s in LL_K2 ]
+ sage: W = VectorSpace(F, (m**2)*(n**2))
+ sage: expected = span(F, [ W(x.list()) for x in tps ])
+ sage: pi_cone = Cone([p.list() for p in pi_K1_K2],
+ ....: lattice=L,
+ ....: check=False)
+ sage: LL_pi = pi_cone.lyapunov_like_basis()
+ sage: actual = span(F, [ W(x.list()) for x in LL_pi ])
sage: actual == expected
True
"""
+ if K2 is None:
+ K2 = K1
+
# 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()
+ F = K1.lattice().base_field()
+ n = K1.lattice_dim()
+ m = K2.lattice_dim()
- tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ]
+ tensor_products = [ s.tensor_product(x) for x in K1 for s in K2.dual() ]
# Convert those tensor products to long vectors.
- W = VectorSpace(F, n**2)
+ W = VectorSpace(F, n*m)
vectors = [ W(tp.list()) for tp in tensor_products ]
- # Create the *dual* cone of the positive operators, expressed as
- # long vectors..
- pi_dual = Cone(vectors, ToricLattice(W.dimension()))
+ check = True
+ if K1.is_proper() and K2.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 positive operators, expressed as
+ # long vectors.
+ pi_dual = Cone(vectors, ToricLattice(W.dimension()), check=check)
# Now compute the desired cone from its dual...
pi_cone = pi_dual.dual()
# And finally convert its rays back to matrix representations.
- M = MatrixSpace(F, n)
- return [ M(v.list()) for v in pi_cone.rays() ]
+ M = MatrixSpace(F, m, n)
+ return [ M(v.list()) for v in pi_cone ]
-def Z_transformation_gens(K):
+def cross_positive_operator_gens(K):
r"""
- Compute generators of the cone of Z-transformations on this cone.
+ Compute generators of the cone of cross-positive operators on this
+ cone.
OUTPUT:
A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
Each matrix ``L`` in the list should have the property that
- ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element the
- discrete complementarity set of ``K``. Moreover, any nonnegative
- linear combination of these matrices shares the same property.
+ ``(L*x).inner_product(s) >= 0`` whenever ``(x,s)`` is an element of
+ this cone's :meth:`discrete_complementarity_set`. Moreover, any
+ conic (nonnegative linear) combination of these matrices shares the
+ same property.
+
+ .. SEEALSO::
+
+ :meth:`positive_operator_gens`, :meth:`Z_operator_gens`,
EXAMPLES:
- Z-transformations on the nonnegative orthant are just Z-matrices.
- That is, matrices whose off-diagonal elements are nonnegative::
+ Cross-positive operators on the nonnegative orthant are negations
+ of Z-matrices; that is, matrices whose off-diagonal elements are
+ nonnegative::
sage: K = Cone([(1,0),(0,1)])
- sage: Z_transformation_gens(K)
+ sage: cross_positive_operator_gens(K)
[
- [ 0 -1] [ 0 0] [-1 0] [1 0] [ 0 0] [0 0]
- [ 0 0], [-1 0], [ 0 0], [0 0], [ 0 -1], [0 1]
+ [0 1] [0 0] [1 0] [-1 0] [0 0] [ 0 0]
+ [0 0], [1 0], [0 0], [ 0 0], [0 1], [ 0 -1]
]
sage: K = Cone([(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)])
- sage: all([ z[i][j] <= 0 for z in Z_transformation_gens(K)
- ....: for i in range(z.nrows())
- ....: for j in range(z.ncols())
+ sage: all([ c[i][j] >= 0 for c in cross_positive_operator_gens(K)
+ ....: for i in range(c.nrows())
+ ....: for j in range(c.ncols())
....: if i != j ])
True
- The trivial cone in a trivial space has no Z-transformations::
+ The trivial cone in a trivial space has no cross-positive operators::
sage: K = Cone([], ToricLattice(0))
- sage: Z_transformation_gens(K)
+ sage: cross_positive_operator_gens(K)
[]
- Z-transformations on a subspace are Lyapunov-like and vice-versa::
+ Every operator is a cross-positive operator on the ambient vector
+ space::
+
+ sage: K = Cone([(1,),(-1,)])
+ sage: K.is_full_space()
+ True
+ sage: cross_positive_operator_gens(K)
+ [[1], [-1]]
+
+ sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
+ sage: K.is_full_space()
+ True
+ sage: cross_positive_operator_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 cross-positive operators
+ on the right half-plane::
+
+ sage: K = Cone([(1,0),(0,1),(0,-1)])
+ sage: cross_positive_operator_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]
+ ]
+
+ Cross-positive operators 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
+ sage: cs = span([ vector(c.list()) for c in cross_positive_operator_gens(K) ])
+ sage: cs == lls
True
TESTS:
- The Z-property is possessed by every Z-transformation::
+ The cross-positive property is possessed by every cross-positive
+ operator::
sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim = 6)
- sage: Z_of_K = Z_transformation_gens(K)
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: Sigma_of_K = cross_positive_operator_gens(K)
sage: dcs = K.discrete_complementarity_set()
- sage: all([(z*x).inner_product(s) <= 0 for z in Z_of_K
+ sage: all([(c*x).inner_product(s) >= 0 for c in Sigma_of_K
....: for (x,s) in dcs])
True
- The lineality space of Z is LL::
+ The lineality space of the cone of cross-positive operators is the
+ space of Lyapunov-like operators::
sage: set_random_seed()
- 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
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: Sigma_cone = Cone([ c.list() for c in cross_positive_operator_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: Sigma_cone.linear_subspace() == lls
+ True
+
+ The lineality of the cross-positive operators on a cone is the
+ Lyapunov rank of that cone::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: Sigma_of_K = cross_positive_operator_gens(K)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: Sigma_cone = Cone([ c.list() for c in Sigma_of_K ],
+ ....: lattice=L,
+ ....: check=False)
+ sage: Sigma_cone.lineality() == K.lyapunov_rank()
+ True
+
+ The lineality spaces of the duals of the positive and cross-positive
+ operator cones are equal. From this it follows that the dimensions of
+ the cross-positive operator 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: Sigma_of_K = cross_positive_operator_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: Sigma_cone = Cone([ c.list() for c in Sigma_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: pi_cone.dim() == Sigma_cone.dim()
True
+ sage: pi_star = pi_cone.dual()
+ sage: sigma_star = Sigma_cone.dual()
+ sage: pi_star.linear_subspace() == sigma_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: Sigma_of_K = cross_positive_operator_gens(K)
+ sage: Sigma_cone = Cone([c.list() for c in Sigma_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = Sigma_cone.dim()
+ sage: actual == n^2
+ True
+ sage: K = K.dual()
+ sage: K.is_full_space()
+ True
+ sage: Sigma_of_K = cross_positive_operator_gens(K)
+ sage: Sigma_cone = Cone([ c.list() for c in Sigma_of_K ],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = Sigma_cone.dim()
+ sage: actual == n^2
+ True
+ sage: K = Cone([(1,0),(0,1),(0,-1)])
+ sage: Sigma_of_K = cross_positive_operator_gens(K)
+ sage: Sigma_cone = Cone([ c.list() for c in Sigma_of_K ], check=False)
+ sage: Sigma_cone.dim() == 3
+ True
+
+ The cross-positive operators 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: Sigma_of_pK = cross_positive_operator_gens(pK)
+ sage: actual = Cone([t.list() for t in Sigma_of_pK],
+ ....: lattice=L,
+ ....: check=False)
+ sage: Sigma_of_K = cross_positive_operator_gens(K)
+ sage: expected = Cone([ (p*t*p.inverse()).list() for t in Sigma_of_K ],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual.is_equivalent(expected)
+ True
+
+ An operator is cross-positive on a cone if and only if its
+ adjoint is cross-positive 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: Sigma_of_K = cross_positive_operator_gens(K)
+ sage: Sigma_of_K_star = cross_positive_operator_gens(K.dual())
+ sage: Sigma_cone = Cone([ p.list() for p in Sigma_of_K ],
+ ....: lattice=L,
+ ....: check=False)
+ sage: Sigma_star = Cone([ p.list() for p in Sigma_of_K_star ],
+ ....: lattice=L,
+ ....: check=False)
+ sage: M = MatrixSpace(F, n)
+ sage: L = M(Sigma_cone.random_element(ring=QQ).list())
+ sage: Sigma_star.contains(W(L.transpose().list()))
+ True
+
+ sage: L = W.random_element()
+ sage: L_star = W(M(L.list()).transpose().list())
+ sage: Sigma_cone.contains(L) == Sigma_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
n = K.lattice_dim()
# These tensor products contain generators for the dual cone of
- # the cross-positive transformations.
+ # the cross-positive operators.
tensor_products = [ s.tensor_product(x)
for (x,s) in K.discrete_complementarity_set() ]
W = VectorSpace(F, n**2)
vectors = [ W(m.list()) for m in tensor_products ]
- # Create the *dual* cone of the cross-positive operators,
- # expressed as long vectors..
- Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension()))
+ 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.rays() ]
+ return [ M(v.list()) for v in Sigma_cone ]
+
+
+def Z_operator_gens(K):
+ r"""
+ Compute generators of the cone of Z-operators on this cone.
+
+ The Z-operators on a cone generalize the Z-matrices over the
+ nonnegative orthant. They are simply negations of the
+ :meth:`cross_positive_operators`.
+
+ OUTPUT:
+
+ A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
+ Each matrix ``L`` in the list should have the property that
+ ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element of
+ this cone's :meth:`discrete_complementarity_set`. Moreover, any
+ conic (nonnegative linear) combination of these matrices shares the
+ same property.
+
+ .. SEEALSO::
+
+ :meth:`positive_operator_gens`, :meth:`cross_positive_operator_gens`,
+
+ TESTS:
+
+ The Z-property is possessed by every Z-operator::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: Z_of_K = Z_operator_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
+ """
+ return [ -cp for cp in cross_positive_operator_gens(K) ]
+
+
+def LL_cone(K):
+ gens = K.lyapunov_like_basis()
+ L = ToricLattice(K.lattice_dim()**2)
+ return Cone([ g.list() for g in gens ], lattice=L, check=False)
+
+def Sigma_cone(K):
+ gens = cross_positive_operator_gens(K)
+ L = ToricLattice(K.lattice_dim()**2)
+ return Cone([ g.list() for g in gens ], lattice=L, check=False)
+
+def Z_cone(K):
+ gens = Z_operator_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)
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