## Abstract

A nonnegative matrix A is called primitive if ^{Ak} is positive for some integer k>0. A generalization by Protasov and Voynov (2012) of this concept to finite sets of matrices is as follows: a set of matrices M={^{A1},^{A2},...,^{Am}} is primitive if A^{i1}A_{i2}...^{A}_{ik} is positive for some indices ^{i1},^{i2},.,^{ik}. 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 P=NP 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 matrices. On the other hand, defining P to be the set of matrices with no zero rows or columns, we give a combinatorial proof of the Protasov-Voynov characterization (2012) of primitivity for matrices in P which can be tested in polynomial time. This latter observation is related to the well-known 1964 conjecture of Černý on synchronizing automata; in fact, any bound on the minimal length of a synchronizing word for synchronizing automata immediately translates into a bound on the length of the shortest positive product of a primitive set of matrices in P. In particular, any primitive set of n×n matrices in P has a positive product of length O(^{n3}).

Original language | English (US) |
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Pages (from-to) | 80-88 |

Number of pages | 9 |

Journal | Automatica |

Volume | 61 |

DOIs | |

State | Published - Nov 2015 |

## Keywords

- Complexity theory
- Control algorithms
- Nonnegative matrices
- State trajectories
- Switched networks

## ASJC Scopus subject areas

- Control and Systems Engineering
- Electrical and Electronic Engineering