On Eventual Non-negativity and Positivity for the Weighted Sum of Powers of Matrices

Abstract

The long run behaviour of linear dynamical systems is often studied by looking at eventual properties of matrices and recurrences that underlie the system. A basic problem in this setting is as follows: given a set of pairs of rational weights and matrices $$\{(w_1, A_1), \ldots , (w_m, A_m)\}$$ { ( w 1 , A 1 ) , … , ( w m , A m ) } , does there exist an integer N s.t for all $$n \ge N$$ n ≥ N , $$\sum _{i=1}^m w_i\cdot A_i^n \ge 0$$ ∑ i = 1 m w i · A i n ≥ 0 (resp. $$> 0$$ > 0 ). We study this problem, its applications and its connections to linear recurrence sequences. Our first result is that for $$m\ge 2$$ m ≥ 2 , the problem is as hard as the ultimate positivity of linear recurrences, a long standing open question (known to be $$\mathsf {coNP}$$ coNP -hard). Our second result is that for any $$m\ge 1$$ m ≥ 1 , the problem reduces to ultimate positivity of linear recurrences. This yields upper bounds for several subclasses of matrices by exploiting known results on linear recurrence sequences. Our third result is a general reduction technique for a large class of problems (including the above) from diagonalizable case to the case where the matrices are simple (have non-repeated eigenvalues). This immediately gives a decision procedure for our problem for diagonalizable matrices.