Factorisability of Low Dimensional Non-Negative Integer Matrices

2026-09-22

AI Summary

研究了二维非负整数矩阵是否可分解为两个同类矩阵的乘积的问题，并提出了一种有效算法来判断矩阵的素性及寻找复合矩阵的分解。

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 $2 \times 2$ 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.

Problem

• non-negative integer matrices
• factorisation
• primality
• computational group theory
• incidence matrices

Innovation

• efficient algorithm
• matrix factorisation
• non-negative integer matrices
• primality testing