Finiteness of a set of non-collinear vectors generated by a family of linear operators Original Research Article

Let image be a real n-dimensional space, let {A(x)midxset membership, variantX} be a family of m=X linear operators in image, and let Kr be a sharp polyhedral cone formed by a set of rvectors, image Let Kr be invariant under {A(x)midxset membership, variantX}, i.e. KrA(x)=Kr, for xset membership, variantX. We study a maximum set of non-collinear vectors derived from a vector hset membership, variantKr by the family {A(x)midxset membership, variantX} in this paper. It is shown that there is a function f(n,m,r) such that this set of non-collinear vectors is finite iff the cardinality of this set is not greater than f(n,m,r). This result can be used for solving the following problem: when does a channel simulated by a probabilistic automaton have a finite set of states?