On Primitivity of Sets of Matrices
Vincent D. Blondel, Raphael M. Jungers, and Alex Olshevsky

TL;DR
This paper studies the computational complexity of determining primitivity in sets of nonnegative matrices, showing it is generally hard but easier for matrices with no zero rows or columns, linking to automata theory.
Contribution
It proves that deciding primitivity is NP-hard in general, provides polynomial bounds for matrices in ${ m P}$, and connects matrix primitivity to automata synchronization conjectures.
Findings
Deciding primitivity is NP-hard unless P=NP.
Shortest positive product can be superpolynomial in general.
For matrices in ${ m P}$, shortest positive product length is polynomial, specifically O(n^3).
Abstract
A nonnegative matrix is called primitive if is positive for some integer . A generalization of this concept to finite sets of matrices is as follows: a set of matrices is primitive if is positive for some indices . The concept of primitive sets of matrices comes up in a number of problems within the study of discrete-time switched systems. In this paper, we analyze the computational complexity of deciding if a given set of matrices is primitive and we derive bounds on the length of the shortest positive product. We show that while primitivity is algorithmically decidable, unless it is not possible to decide primitivity of a matrix set in polynomial time. Moreover, we show that the length of the shortest positive sequence can be superpolynomial in the dimension of the…
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