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

Abstract

Let ℝn be a real n-dimensional space, let {A(x) | x ∈ X} be a family of m = |X| linear operators in ℝn, and let Kr be a sharp polyhedral cone formed by a set of rvectors, Kr ⊂ ℝn. Let Kr be invariant under {A(x) | x ∈ X}, i.e. KrA(x) = Kr, for x ∈ X. We study a maximum set of non-collinear vectors derived from a vector h ∈ Kr by the family {A(x) | x ∈ X} 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? © 1999 Elsevier Science Inc. All rights reserved.