Investigate Invertibility of Sparse Symmetric Matrix

Abstract

In this paper, we investigate the invertibility of sparse symmetric matrices. We show that for an $n\times n$ sparse symmetric random matrix $A$ with $A_{ij}={\delta }_{ij}{\xi }_{ij}$ is invertible with high probability. Here, ${\delta }_{ij}$s, $i\ge j$ are i.i.d. Bernoulli random variables with $\mathbb{P}\left({\xi }_{ij}=1\right)=p\ge n^{-c}$, ${\xi }_{ij},i\ge j$ are i.i.d. random variables with mean 0, variance 1 and finite forth moment $M_4$, and $c$ is constant depending on $M_4$. More precisely, $$s_{\min }(A)>\epsilon \sqrt{\frac{p}{n}}.$$ with high probability.

arXiv admin note: substantial text overlap with arXiv:1102.0300, arXiv:1507.03525 by other authors

References in corpus (1)

• Sharp nonasymptotic bounds on the norm of random matrices with independent entries