X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=b4d2be0d73e10ed9f58d1189c84ed4bec3d0ad94;hb=db016cc07b5f0cb45a96022eb468e3080786078b;hp=421fb3c8f04abb23fd420271ed10e0a7e65815d4;hpb=f8e779ee533992a940e1014d00960fc58c3c4b79;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 421fb3c..b4d2be0 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -8,77 +8,235 @@ addsitedir(abspath('../../')) from sage.all import * -def project_span(K): +def _basically_the_same(K1, K2): r""" - Project ``K`` into its own span. + Test whether or not ``K1`` and ``K2`` are "basically the same." - EXAMPLES:: + This is a hack to get around the fact that it's difficult to tell + when two cones are linearly isomorphic. We have a proposition that + equates two cones, but represented over `\mathbb{Q}`, they are + merely linearly isomorphic (not equal). So rather than test for + equality, we test a list of properties that should be preserved + under an invertible linear transformation. + + OUTPUT: + + ``True`` if ``K1`` and ``K2`` are basically the same, and ``False`` + otherwise. + + EXAMPLES: + + Any proper cone with three generators in `\mathbb{R}^{3}` is + basically the same as the nonnegative orthant:: + + sage: K1 = Cone([(1,0,0), (0,1,0), (0,0,1)]) + sage: K2 = Cone([(1,2,3), (3, 18, 4), (66, 51, 0)]) + sage: _basically_the_same(K1, K2) + True + + Negating a cone gives you another cone that is basically the same:: + + sage: K = Cone([(0,2,-5), (-6, 2, 4), (0, 51, 0)]) + sage: _basically_the_same(K, -K) + True + + TESTS: + + Any cone is basically the same as itself:: + + sage: K = random_cone(max_ambient_dim = 8) + sage: _basically_the_same(K, K) + True + + After applying an invertible matrix to the rows of a cone, the + result should be basically the same as the cone we started with:: + + sage: K1 = random_cone(max_ambient_dim = 8) + sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') + sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) + sage: _basically_the_same(K1, K2) + True + + """ + if K1.lattice_dim() != K2.lattice_dim(): + return False + + if K1.nrays() != K2.nrays(): + return False + + if K1.dim() != K2.dim(): + return False + + if K1.lineality() != K2.lineality(): + return False + + if K1.is_solid() != K2.is_solid(): + return False + + if K1.is_strictly_convex() != K2.is_strictly_convex(): + return False + + if len(LL(K1)) != len(LL(K2)): + return False + + C_of_K1 = discrete_complementarity_set(K1) + C_of_K2 = discrete_complementarity_set(K2) + if len(C_of_K1) != len(C_of_K2): + return False + + if len(K1.facets()) != len(K2.facets()): + return False + + return True + + + +def _restrict_to_space(K, W): + r""" + Restrict this cone a subspace of its ambient space. + + INPUT: + + - ``W`` -- The subspace into which this cone will be restricted. + + OUTPUT: + + A new cone in a sublattice corresponding to ``W``. + + EXAMPLES: + + When this cone is solid, restricting it into its own span should do + nothing:: sage: K = Cone([(1,)]) - sage: project_span(K) == K + sage: _restrict_to_space(K, K.span()) == K True + A single ray restricted into its own span gives the same output + regardless of the ambient space:: + sage: K2 = Cone([(1,0)]) - sage: project_span(K2).rays() + sage: K2_S = _restrict_to_space(K2, K2.span()).rays() + sage: K2_S N(1) in 1-d lattice N sage: K3 = Cone([(1,0,0)]) - sage: project_span(K3).rays() + sage: K3_S = _restrict_to_space(K3, K3.span()).rays() + sage: K3_S N(1) in 1-d lattice N - sage: project_span(K2) == project_span(K3) + sage: K2_S == K3_S True TESTS: The projected cone should always be solid:: - sage: K = random_cone(max_dim = 10) - sage: K_S = project_span(K) - sage: K_S.is_solid() + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim = 8) + sage: _restrict_to_space(K, K.span()).is_solid() + True + + And the resulting cone should live in a space having the same + dimension as the space we restricted it to:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim = 8) + sage: K_P = _restrict_to_space(K, K.dual().span()) + sage: K_P.lattice_dim() == K.dual().dim() + True + + This function should not affect the dimension of a cone:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim = 8) + sage: K.dim() == _restrict_to_space(K,K.span()).dim() + True + + Nor should it affect the lineality of a cone:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim = 8) + sage: K.lineality() == _restrict_to_space(K, K.span()).lineality() + True + + No matter which space we restrict to, the lineality should not + increase:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim = 8) + sage: S = K.span(); P = K.dual().span() + sage: K.lineality() >= _restrict_to_space(K,S).lineality() + True + sage: K.lineality() >= _restrict_to_space(K,P).lineality() True If we do this according to our paper, then the result is proper:: - sage: K = random_cone(max_dim = 10) - sage: K_S = project_span(K) - sage: P = project_span(K_S.dual()).dual() - sage: P.is_proper() + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim = 8) + sage: K_S = _restrict_to_space(K, K.span()) + sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual() + sage: K_SP.is_proper() + True + sage: K_SP = _restrict_to_space(K_S, K_S.dual().span()) + sage: K_SP.is_proper() + True + + Test the proposition in our paper concerning the duals and + restrictions. Generate a random cone, then create a subcone of + it. The operation of dual-taking should then commute with + _restrict_to_space:: + + sage: set_random_seed() + sage: J = random_cone(max_ambient_dim = 8) + sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) + sage: K_W_star = _restrict_to_space(K, J.span()).dual() + sage: K_star_W = _restrict_to_space(K.dual(), J.span()) + sage: _basically_the_same(K_W_star, K_star_W) True """ - L = K.lattice() - F = L.base_field() - Q = L.quotient(K.sublattice_complement()) - vecs = [ vector(F, reversed(list(Q(r)))) for r in K.rays() ] + # First we want to intersect ``K`` with ``W``. The easiest way to + # do this is via cone intersection, so we turn the subspace ``W`` + # into a cone. + W_cone = Cone(W.basis() + [-b for b in W.basis()], lattice=K.lattice()) + K = K.intersection(W_cone) - newL = None - if len(vecs) == 0: - newL = ToricLattice(0) + # We've already intersected K with the span of K2, so every + # generator of K should belong to W now. + K_W_rays = [ W.coordinate_vector(r) for r in K.rays() ] - return Cone(vecs, lattice=newL) + L = ToricLattice(W.dimension()) + return Cone(K_W_rays, lattice=L) def discrete_complementarity_set(K): r""" - Compute the discrete complementarity set of this cone. + Compute a discrete complementarity set of this cone. - The complementarity set of this cone is the set of all orthogonal - pairs `(x,s)` such that `x` is in this cone, and `s` is in its - dual. The discrete complementarity set restricts `x` and `s` to be - generators of their respective cones. + A discrete complementarity set of `K` is the set of all orthogonal + pairs `(x,s)` such that `x \in G_{1}` and `s \in G_{2}` for some + generating sets `G_{1}` of `K` and `G_{2}` of its dual. Polyhedral + convex cones are input in terms of their generators, so "the" (this + particular) discrete complementarity set corresponds to ``G1 + == K.rays()`` and ``G2 == K.dual().rays()``. OUTPUT: A list of pairs `(x,s)` such that, - * `x` is in this cone. - * `x` is a generator of this cone. - * `s` is in this cone's dual. - * `s` is a generator of this cone's dual. + * Both `x` and `s` are vectors (not rays). + * `x` is one of ``K.rays()``. + * `s` is one of ``K.dual().rays()``. * `x` and `s` are orthogonal. + REFERENCES: + + .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an + Improper Cone. Work in-progress. + EXAMPLES: The discrete complementarity set of the nonnegative orthant consists @@ -109,24 +267,43 @@ def discrete_complementarity_set(K): sage: discrete_complementarity_set(K) [] + Likewise when this cone is trivial (its dual is the entire space):: + + sage: L = ToricLattice(0) + sage: K = Cone([], ToricLattice(0)) + sage: discrete_complementarity_set(K) + [] + TESTS: The complementarity set of the dual can be obtained by switching the components of the complementarity set of the original cone:: - sage: K1 = random_cone(max_dim=10, max_rays=10) + sage: set_random_seed() + sage: K1 = random_cone(max_ambient_dim=6) sage: K2 = K1.dual() sage: expected = [(x,s) for (s,x) in discrete_complementarity_set(K2)] sage: actual = discrete_complementarity_set(K1) - sage: actual == expected + sage: sorted(actual) == sorted(expected) True + The pairs in the discrete complementarity set are in fact + complementary:: + + sage: set_random_seed() + 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 the rays to vectors so that we can compute inner - # products. + # 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] @@ -134,8 +311,7 @@ def discrete_complementarity_set(K): def LL(K): r""" - Compute the space `\mathbf{LL}` of all Lyapunov-like transformations - on this cone. + Compute a basis of Lyapunov-like transformations on this cone. OUTPUT: @@ -192,24 +368,47 @@ def LL(K): [0 0 1] ] + If our cone is the entire space, then every transformation on it is + Lyapunov-like:: + + sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)]) + sage: M = MatrixSpace(QQ,2) + sage: M.basis() == LL(K) + True + TESTS: The inner product `\left< L\left(x\right), s \right>` is zero for every pair `\left( x,s \right)` in the discrete complementarity set of the cone:: - sage: K = random_cone(max_dim=8, max_rays=10) + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=8) sage: C_of_K = discrete_complementarity_set(K) sage: l = [ (L*x).inner_product(s) for (x,s) in C_of_K for L in LL(K) ] sage: sum(map(abs, l)) 0 + The Lyapunov-like transformations on a cone and its dual are related + by transposition, but we're not guaranteed to compute transposed + elements of `LL\left( K \right)` as our basis for `LL\left( K^{*} + \right)` + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=8) + sage: LL2 = [ L.transpose() for L in LL(K.dual()) ] + sage: V = VectorSpace( K.lattice().base_field(), K.lattice_dim()^2) + sage: LL1_vecs = [ V(m.list()) for m in LL(K) ] + sage: LL2_vecs = [ V(m.list()) for m in LL2 ] + sage: V.span(LL1_vecs) == V.span(LL2_vecs) + True + """ V = K.lattice().vector_space() C_of_K = discrete_complementarity_set(K) - tensor_products = [s.tensor_product(x) for (x,s) in C_of_K] + tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ] # Sage doesn't think matrices are vectors, so we have to convert # our matrices to vectors explicitly before we can figure out how @@ -240,7 +439,7 @@ def LL(K): def lyapunov_rank(K): r""" - Compute the Lyapunov (or bilinearity) rank of this cone. + Compute the Lyapunov rank (or bilinearity rank) of this cone. The Lyapunov rank of a cone can be thought of in (mainly) two ways: @@ -259,16 +458,7 @@ def lyapunov_rank(K): An integer representing the Lyapunov rank of the cone. If the dimension of the ambient vector space is `n`, then the Lyapunov rank will be between `1` and `n` inclusive; however a rank of `n-1` is - not possible (see the first reference). - - .. note:: - - In the references, the cones are always assumed to be proper. We - do not impose this restriction. - - .. seealso:: - - :meth:`is_proper` + not possible (see [Orlitzky/Gowda]_). ALGORITHM: @@ -311,8 +501,8 @@ def lyapunov_rank(K): [Orlitzky/Gowda]_:: sage: R5 = VectorSpace(QQ, 5) - sage: gens = R5.basis() + [ -r for r in R5.basis() ] - sage: K = Cone(gens) + sage: gs = R5.basis() + [ -r for r in R5.basis() ] + sage: K = Cone(gs) sage: lyapunov_rank(K) 25 @@ -345,11 +535,11 @@ def lyapunov_rank(K): sage: neg_e1 = (-1,0,0,0,0) sage: e2 = (0,1,0,0,0) sage: neg_e2 = (0,-1,0,0,0) - sage: zero = (0,0,0,0,0) - sage: K = Cone([e1, neg_e1, e2, neg_e2, zero, zero, zero]) + sage: z = (0,0,0,0,0) + sage: K = Cone([e1, neg_e1, e2, neg_e2, z, z, z]) sage: lyapunov_rank(K) 19 - sage: K.lattice_dim()**2 - K.dim()*(K.lattice_dim() - K.dim()) + sage: K.lattice_dim()**2 - K.dim()*K.codim() 19 The Lyapunov rank should be additive on a product of proper cones @@ -381,16 +571,31 @@ def lyapunov_rank(K): The Lyapunov rank should be additive on a product of proper cones [Rudolf et al.]_:: - sage: K1 = random_cone(max_dim=10, strictly_convex=True, solid=True) - sage: K2 = random_cone(max_dim=10, strictly_convex=True, solid=True) + sage: set_random_seed() + sage: K1 = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=True) + sage: K2 = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=True) sage: K = K1.cartesian_product(K2) sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2) True + The Lyapunov rank is invariant under a linear isomorphism + [Orlitzky/Gowda]_:: + + sage: K1 = random_cone(max_ambient_dim = 8) + sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') + sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) + sage: lyapunov_rank(K1) == lyapunov_rank(K2) + True + The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` itself [Rudolf et al.]_:: - sage: K = random_cone(max_dim=10, max_rays=10) + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=8) sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) True @@ -400,7 +605,10 @@ def lyapunov_rank(K): trivial cone in a trivial space as well. However, in zero dimensions, the Lyapunov rank of the trivial cone will be zero:: - sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True) + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=True) sage: b = lyapunov_rank(K) sage: n = K.lattice_dim() sage: (n == 0 or 1 <= b) and b <= n @@ -411,7 +619,8 @@ def lyapunov_rank(K): In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have Lyapunov rank `n-1` in `n` dimensions:: - sage: K = random_cone(max_dim=10) + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=8) sage: b = lyapunov_rank(K) sage: n = K.lattice_dim() sage: b == n-1 @@ -420,42 +629,274 @@ def lyapunov_rank(K): The calculation of the Lyapunov rank of an improper cone can be reduced to that of a proper cone [Orlitzky/Gowda]_:: - sage: K = random_cone(max_dim=10) + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=8) sage: actual = lyapunov_rank(K) - sage: K_S = project_span(K) - sage: P = project_span(K_S.dual()).dual() - sage: l = K.linear_subspace().dimension() - sage: codim = K.lattice_dim() - K.dim() - sage: expected = lyapunov_rank(P) + K.dim()*(l + codim) + codim**2 + sage: K_S = _restrict_to_space(K, K.span()) + sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual() + sage: l = K.lineality() + sage: c = K.codim() + sage: expected = lyapunov_rank(K_SP) + K.dim()*(l + c) + c**2 sage: actual == expected True - The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``:: + The Lyapunov rank of any cone is just the dimension of ``LL(K)``:: - sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True) + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=8) sage: lyapunov_rank(K) == len(LL(K)) True + We can make an imperfect cone perfect by adding a slack variable + (a Theorem in [Orlitzky/Gowda]_):: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=True) + sage: L = ToricLattice(K.lattice_dim() + 1) + sage: K = Cone([ r.list() + [0] for r in K.rays() ], lattice=L) + sage: lyapunov_rank(K) >= K.lattice_dim() + True + """ beta = 0 m = K.dim() n = K.lattice_dim() - l = K.linear_subspace().dimension() + l = K.lineality() if m < n: - # K is not solid, project onto its span. - K = project_span(K) + # K is not solid, restrict to its span. + K = _restrict_to_space(K, K.span()) - # Lemma 2 - beta += m*(n - m) + (n - m)**2 + # Non-solid reduction lemma. + beta += (n - m)*n if l > 0: - # K is not pointed, project its dual onto its span. - K = project_span(K.dual()).dual() + # K is not pointed, restrict to the span of its dual. Uses a + # proposition from our paper, i.e. this is equivalent to K = + # _rho(K.dual()).dual(). + K = _restrict_to_space(K, K.dual().span()) - # Lemma 3 - beta += m * l + # Non-pointed reduction lemma. + beta += l * m beta += len(LL(K)) return beta + + + +def is_lyapunov_like(L,K): + r""" + Determine whether or not ``L`` is Lyapunov-like on ``K``. + + We say that ``L`` is Lyapunov-like on ``K`` if `\left\langle + L\left\lparenx\right\rparen,s\right\rangle = 0` for all pairs + `\left\langle x,s \right\rangle` in the complementarity set of + ``K``. It is known [Orlitzky]_ that this property need only be + checked for generators of ``K`` and its dual. + + INPUT: + + - ``L`` -- A linear transformation or matrix. + + - ``K`` -- A polyhedral closed convex cone. + + OUTPUT: + + ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``, + and ``False`` otherwise. + + .. WARNING:: + + If this function returns ``True``, then ``L`` is Lyapunov-like + on ``K``. However, if ``False`` is returned, that could mean one + of two things. The first is that ``L`` is definitely not + Lyapunov-like on ``K``. The second is more of an "I don't know" + answer, returned (for example) if we cannot prove that an inner + product is zero. + + REFERENCES: + + .. [Orlitzky] M. Orlitzky. The Lyapunov rank of an + improper cone (preprint). + + EXAMPLES: + + The identity is always Lyapunov-like in a nontrivial space:: + + sage: set_random_seed() + sage: K = random_cone(min_ambient_dim = 1, max_rays = 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_rays = 5) + sage: R = K.lattice().vector_space().base_ring() + sage: L = zero_matrix(R, K.lattice_dim()) + sage: is_lyapunov_like(L,K) + True + + Everything in ``LL(K)`` should be Lyapunov-like on ``K``:: + + sage: K = random_cone(min_ambient_dim = 1, max_rays = 5) + sage: all([is_lyapunov_like(L,K) for L in LL(K)]) + True + + """ + return all([(L*x).inner_product(s) == 0 + for (x,s) in discrete_complementarity_set(K)]) + + +def random_element(K): + 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. + + EXAMPLES: + + A random element of the trivial cone is zero:: + + 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) + + TESTS: + + Any cone should contain an element of itself:: + + sage: set_random_seed() + sage: K = random_cone(max_rays = 8) + sage: K.contains(random_element(K)) + True + + """ + 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 + + +def positive_operators(K): + r""" + Compute generators of the cone of positive operators on this cone. + + 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. + + EXAMPLES: + + The trivial cone in a trivial space has no positive operators:: + + sage: K = Cone([], ToricLattice(0)) + sage: positive_operators(K) + [] + + Positive operators on the nonnegative orthant are nonnegative matrices:: + + sage: K = Cone([(1,)]) + sage: positive_operators(K) + [[1]] + + sage: K = Cone([(1,0),(0,1)]) + sage: positive_operators(K) + [ + [1 0] [0 1] [0 0] [0 0] + [0 0], [0 0], [1 0], [0 1] + ] + + Every operator is positive on the ambient vector space:: + + sage: K = Cone([(1,),(-1,)]) + sage: K.is_full_space() + True + sage: positive_operators(K) + [[1], [-1]] + + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() + True + sage: positive_operators(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] + ] + + TESTS: + + A positive operator on a cone should send its generators into the cone:: + + sage: K = random_cone(max_ambient_dim = 6) + sage: pi_of_k = positive_operators(K) + sage: all([K.contains(p*x) for p in pi_of_k for x in K.rays()]) + True + + """ + V = K.lattice().vector_space() + + # Sage doesn't think matrices are vectors, so we have to convert + # our matrices to vectors explicitly before we can figure out how + # many are linearly-indepenedent. + # + # The space W has the same base ring as V, but dimension + # dim(V)^2. So it has the same dimension as the space of linear + # transformations on V. In other words, it's just the right size + # to create an isomorphism between it and our matrices. + W = VectorSpace(V.base_ring(), V.dimension()**2) + + G1 = [ V(x) for x in K.rays() ] + G2 = [ V(s) for s in K.dual().rays() ] + + tensor_products = [ s.tensor_product(x) for x in G1 for s in G2 ] + + # Turn our matrices into long vectors... + vectors = [ W(m.list()) for m in tensor_products ] + + # Create the *dual* cone of the positive operators, expressed as + # long vectors.. + L = ToricLattice(W.dimension()) + pi_dual = Cone(vectors, lattice=L) + + # Now compute the desired cone from its dual... + pi_cone = pi_dual.dual() + + # And finally convert its rays back to matrix representations. + M = MatrixSpace(V.base_ring(), V.dimension()) + + return [ M(v.list()) for v in pi_cone.rays() ]