## Abstract

Let Q_{n,d} denote the random combinatorial matrix whose rows are independent of one another and such that each row is sampled uniformly at random from the subset of vectors in {0,1}^{n} having precisely d entries equal to 1. We present a short proof of the fact that [Formula presented], whenever ω(n^{1/2}log^{3/2}n)=d≤n/2. In particular, our proof accommodates sparse random combinatorial matrices in the sense that d=o(n) is allowed. We also consider the singularity of deterministic integer matrices A randomly perturbed by a sparse combinatorial matrix. In particular, we prove that [Formula presented], again, whenever ω(n^{1/2}log^{3/2}n)=d≤n/2 and A has the property that (1,−d) is not an eigenpair of A.

Original language | English |
---|---|

Article number | 113017 |

Journal | Discrete Mathematics |

Volume | 345 |

Issue number | 11 |

DOIs | |

State | Published - Nov 2022 |

## Keywords

- Random matrices
- Random perturbation
- Singularity

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics