The full rank condition for sparse random matrices

TL;DR

This paper establishes a sufficient and nearly necessary condition for sparse random matrices to have full row rank, applicable over finite fields and rationals, advancing understanding of their algebraic properties.

Contribution

It introduces a comprehensive condition for the full rank of sparse random matrices, covering both finite field and rational cases, which was previously not well-understood.

Findings

Derived a sufficient condition for full row rank.

Condition is generally necessary as well.

Applicable to matrices over finite fields and rationals.

Abstract

We derive a sufficient condition for a sparse random matrix with given numbers of non-zero entries in the rows and columns having full row rank. The result covers both matrices over finite fields with independent non-zero entries and $\{0,1\}$-matrices over the rationals. The sufficient condition is generally necessary as well.