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PreprintCC BY 4.0arXiv

Factorisability of Low Dimensional Non-Negative Integer Matrices

  • Paul C. Bell
  • Eva Foster
  • Daniel Reidenbach
  • Pavel Semukhin
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Abstract

We consider the problem of determining if a given two-dimensional nonnegative integer matrix M is the product of two such matrices, excluding trivial units. A matrix M with no such factorisation is called prime and therefore belongs to the minimal (infinite rank) generator of 2 × 2 matrices over the natural numbers, otherwise it is called composite. We also consider the problem of finding a (non-unique) factorisation of a composite matrix. Our results have applications in computational group theory and the theory of codes, where such matrices are called incidence matrices. We analyse the complexity of primality and finding a factorisation for a composite matrix, providing a first efficient algorithm.

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Cite

BibTeXRIS
BibTeX
@misc{bell2026factorisability,
  title = {{Factorisability of Low Dimensional Non-Negative Integer Matrices}},
  author = {Paul C. Bell and Eva Foster and Daniel Reidenbach and Pavel Semukhin},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26033},
  eprint = {2609.26033},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26033},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Factorisability of Low Dimensional Non-Negative Integer Matrices
AU  - Paul C. Bell
AU  - Eva Foster
AU  - Daniel Reidenbach
AU  - Pavel Semukhin
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26033
UR  - https://arxiv.org/abs/2609.26033
AB  - We consider the problem of determining if a given two-dimensional nonnegative integer matrix M is the product of two such matrices, excluding trivial units. A matrix M with no such factorisation is called prime and therefore belongs to the minimal (infinite rank) generator of 2 × 2 matrices over the natural numbers, otherwise it is called composite. We also consider the problem of finding a (non-unique) factorisation of a composite matrix. Our results have applications in computational group theory and the theory of codes, where such matrices are called incidence matrices. We analyse the complexity of primality and finding a factorisation for a composite matrix, providing a first efficient algorithm.
ER  -