- """
- 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 ]
-
- # 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)
-
- # Turn our matrices into long vectors...
- vectors = [ W(m.list()) for m in tensor_products ]
-
- # Vector space representation of Lyapunov-like matrices
- # (i.e. vec(L) where L is Luapunov-like).
- LL_vector = W.span(vectors).complement()
-
- # Now construct an ambient MatrixSpace in which to stick our
- # transformations.
- M = MatrixSpace(V.base_ring(), V.dimension())
-
- matrix_basis = [ M(v.list()) for v in LL_vector.basis() ]
-
- return matrix_basis
-
-
-
-def lyapunov_rank(K):
- r"""
- Compute the Lyapunov (or bilinearity) rank of this cone.
-
- The Lyapunov rank of a cone can be thought of in (mainly) two ways:
-
- 1. The dimension of the Lie algebra of the automorphism group of the
- cone.
-
- 2. The dimension of the linear space of all Lyapunov-like
- transformations on the cone.
-
- INPUT:
-
- A closed, convex polyhedral cone.
-
- OUTPUT:
-
- 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`
-
- ALGORITHM:
-
- The codimension formula from the second reference is used. We find
- all pairs `(x,s)` in the complementarity set of `K` such that `x`
- and `s` are rays of our cone. It is known that these vectors are
- sufficient to apply the codimension formula. Once we have all such
- pairs, we "brute force" the codimension formula by finding all
- linearly-independent `xs^{T}`.