Abstract
Let R n be a real n-dimensional space, let {A(x)∣x∈X} be a family of m=|X| linear operators in R n , and let K r be a sharp polyhedral cone formed by a set of rvectors, K r⊂ R n. Let K r be invariant under {A(x)∣x∈X}, i.e. K rA(x)=K r , for x∈X. We study a maximum set of non-collinear vectors derived from a vector h ∈K r 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?
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