Canonical matrices with entries integers modulo p

TL;DR

This paper introduces a method to identify canonical matrices in the set of all matrices with entries modulo p, using lexicographic order and a unique integer representation, providing a clear criterion for canonicity.

Contribution

It establishes a necessary and sufficient condition for matrices to be canonical, and offers a unique integer-based representation for these matrices.

Findings

Characterization of canonical matrices with respect to lexicographic order

A necessary and sufficient condition for matrix canonicity

Unique integer representation of matrices

Abstract

The work considers an equivalence relation in the set of all $n\times m$ matrices with entries in the set $[p]=\{ 0,1,\ldots , p-1 \}$. In each element of the factor-set generated by this relation, we define the concept of canonical matrix, namely the minimal element with respect to the lexicographic order. We have found a necessary and sufficient condition for an arbitrary matrix with entries in the set $[p]$ to be canonical. For this purpose, the matrices are uniquely represented by ordered n-tuples of integers.