-
- """
- beta = 0
-
- m = K.dim()
- n = K.lattice_dim()
- l = K.lineality()
-
- if m < n:
- # K is not solid, restrict to its span.
- K = _restrict_to_space(K, K.span())
-
- # Non-solid reduction lemma.
- beta += (n - m)*n
-
- if l > 0:
- # 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())
-
- # Non-pointed reduction lemma.
- beta += l * m
-
- beta += len(K.LL())
- 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)