``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.
for (x,s) in K.discrete_complementarity_set()])
-def positive_operator_gens(K):
+def positive_operator_gens(K1, K2 = None):
r"""
- Compute generators of the cone of positive operators on this cone.
+ 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.
+
+ INPUT:
+
+ - ``K2`` -- (default: ``K1``) the codomain cone; the image of this
+ cone under the returned operators is a subset of ``K2``.
OUTPUT:
- 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.
+ 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.
+
+ REFERENCES:
+
+ .. [Orlitzky-Pi-Z]
+ M. Orlitzky.
+ Positive and Z-operators on closed convex cones.
+
+ .. [Tam]
+ B.-S. Tam.
+ Some results of polyhedral cones and simplicial cones.
+ Linear and Multilinear Algebra, 4:4 (1977) 281--284.
EXAMPLES:
TESTS:
- Each positive operator generator should send the generators of the
- cone into the cone::
+ Each positive operator generator should send the generators of one
+ cone into the other cone::
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 ])
+ 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 the
- cone into the cone::
+ Each positive operator generator should send a random element of one
+ cone into the other cone::
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*K.random_element(QQ)) for P in pi_of_K ])
+ 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 the cone into the cone::
+ generators of one cone into the other cone::
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([ g.list() for g in pi_of_K ],
+ 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(K.lattice_dim(), pi_cone.random_element(QQ).list())
- sage: all([ K.contains(P*x) for x in K ])
+ 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 the cone into the cone::
+ element of one cone into the other cone::
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([ g.list() for g in pi_of_K ],
+ 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(K.lattice_dim(), pi_cone.random_element(QQ).list())
- sage: K.contains(P*K.random_element(ring=QQ))
+ 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
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 ]
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).
+ if K1.is_proper() and K2.is_proper():
+ # All of the generators involved are extreme vectors and
+ # therefore minimal [Tam]_. 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
pi_cone = pi_dual.dual()
# And finally convert its rays back to matrix representations.
- M = MatrixSpace(F, n)
+ M = MatrixSpace(F, m, n)
return [ M(v.list()) for v in pi_cone ]
-def Z_transformation_gens(K):
+def Z_operator_gens(K):
r"""
- Compute generators of the cone of Z-transformations on this cone.
+ Compute generators of the cone of Z-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.
+
+ REFERENCES:
+
+ M. Orlitzky.
+ Positive and Z-operators on closed convex cones.
EXAMPLES:
- Z-transformations on the nonnegative orthant are just Z-matrices.
+ Z-operators on the nonnegative orthant are just Z-matrices.
That is, matrices whose off-diagonal elements are nonnegative::
sage: K = Cone([(1,0),(0,1)])
- sage: Z_transformation_gens(K)
+ sage: Z_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]
]
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)
+ sage: all([ z[i][j] <= 0 for z in Z_operator_gens(K)
....: for i in range(z.nrows())
....: for j in range(z.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 Z-operators::
sage: K = Cone([], ToricLattice(0))
- sage: Z_transformation_gens(K)
+ sage: Z_operator_gens(K)
[]
- Every operator is a Z-transformation on the ambient vector space::
+ Every operator is a Z-operator on the ambient vector space::
sage: K = Cone([(1,),(-1,)])
sage: K.is_full_space()
True
- sage: Z_transformation_gens(K)
+ sage: Z_operator_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)
+ sage: Z_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 Z-transformations on the
+ A non-obvious application is to find the Z-operators on the
right half-plane::
sage: K = Cone([(1,0),(0,1),(0,-1)])
- sage: Z_transformation_gens(K)
+ sage: Z_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]
]
- Z-transformations on a subspace are Lyapunov-like and vice-versa::
+ Z-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 = span([ vector(z.list()) for z in Z_operator_gens(K) ])
sage: zs == lls
True
TESTS:
- The Z-property is possessed by every Z-transformation::
+ 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_transformation_gens(K)
+ 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
- The lineality space of the cone of Z-transformations is the space of
- Lyapunov-like transformations::
+ The lineality space of the cone of Z-operators is the space of
+ Lyapunov-like operators::
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) ],
+ sage: Z_cone = Cone([ z.list() for z in Z_operator_gens(K) ],
....: lattice=L,
....: check=False)
sage: ll_basis = [ vector(l.list()) for l in K.lyapunov_like_basis() ]
sage: Z_cone.linear_subspace() == lls
True
- The lineality of the Z-transformations on a cone is the Lyapunov
+ The lineality of the Z-operators 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: Z_of_K = Z_operator_gens(K)
sage: L = ToricLattice(K.lattice_dim()**2)
sage: Z_cone = Cone([ z.list() for z in Z_of_K ],
....: lattice=L,
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::
+ The lineality spaces of the duals of the positive and Z-operator
+ cones are equal. From this it follows that the dimensions of the
+ Z-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: Z_of_K = Z_transformation_gens(K)
+ sage: Z_of_K = Z_operator_gens(K)
sage: L = ToricLattice(K.lattice_dim()**2)
sage: pi_cone = Cone([p.list() for p in pi_of_K],
....: lattice=L,
sage: K.is_trivial()
True
sage: L = ToricLattice(n^2)
- sage: Z_of_K = Z_transformation_gens(K)
+ sage: Z_of_K = Z_operator_gens(K)
sage: Z_cone = Cone([z.list() for z in Z_of_K],
....: lattice=L,
....: check=False)
sage: K = K.dual()
sage: K.is_full_space()
True
- sage: Z_of_K = Z_transformation_gens(K)
+ sage: Z_of_K = Z_operator_gens(K)
sage: Z_cone = Cone([z.list() for z in Z_of_K],
....: lattice=L,
....: check=False)
sage: actual == n^2
True
sage: K = Cone([(1,0),(0,1),(0,-1)])
- sage: Z_of_K = Z_transformation_gens(K)
+ sage: Z_of_K = Z_operator_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::
+ The Z-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: Z_of_pK = Z_transformation_gens(pK)
+ sage: Z_of_pK = Z_operator_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: Z_of_K = Z_operator_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::
+ An operator is a Z-operator on a cone if and only if its
+ adjoint is a Z-operator on the dual of that cone::
sage: set_random_seed()
sage: K = random_cone(max_ambient_dim=4)
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_of_K = Z_operator_gens(K)
+ sage: Z_of_K_star = Z_operator_gens(K.dual())
sage: Z_cone = Cone([p.list() for p in Z_of_K],
....: lattice=L,
....: check=False)
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() ]
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).
+ 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,
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
+ # But first, make them negative, so we get Z-operators and
# not cross-positive ones.
M = MatrixSpace(F, n)
return [ -M(v.list()) for v in Sigma_cone ]
+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 Z_cone(K):
- gens = Z_transformation_gens(K)
+ gens = Z_operator_gens(K)
L = ToricLattice(K.lattice_dim()**2)
return Cone([ g.list() for g in gens ], lattice=L, check=False)
projects / sage.d.git / blobdiff