An Alternative Framework for Irreducibility and Primitivity of Nonnegative Tensors

Jianhong Xu

An Alternative Framework for Irreducibility and Primitivity of Nonnegative Tensors

Rings and Algebras 2026-06-29 v1

Authors: Jianhong Xu

Abstract

Motivated by some recent studies on higher order Markov chains and well-known characterizations for irreducibility and primitivity of nonnegative matrices, we propose in this paper an alternative framework for irreducibility and primitivity of nonnegative tensors, giving rise to the concepts of s-irreducibility and s-primitivity. This framework includes the relevant results on matrices as its special cases, yet it expands existing results regarding irreducibility and primitivity for tensors. In addition to its tensor theoretic significance, such a framework has important implications for applied fields, especially when it comes to higher order Markov chains.

Keywords

nonnegative tensors irreducibility primitivity higher order Markov chains s-irreducibility s-primitivity

Cite

@article{arxiv.2606.30482,
  title  = {An Alternative Framework for Irreducibility and Primitivity of Nonnegative Tensors},
  author = {Jianhong Xu},
  journal= {arXiv preprint arXiv:2606.30482},
  year   = {2026}
}