The smallest singular value of signed random combinatorial matrices

TL;DR

This paper proves that the smallest singular value of a specific class of signed random combinatorial matrices is typically bounded away from zero, with the probability of singularity decreasing exponentially with matrix size.

Contribution

It extends the analysis of singular values to constrained signed matrices using the CLCD method, showing exponential decay in singularity probability.

Findings

Probability of singularity is exponentially small.

Smallest singular value is typically at least proportional to n^{-1/2}.

Method adapts CLCD to constrained matrix setting.

Abstract

Let $M_n$ be an $n\times n$ signed random combinatorial matrix whose rows are independent and uniformly distributed over the set of $\{-1,0,1\}$-vectors with exactly $n/2$ zero coordinates. Despite the dependence induced by the row constraints, we prove that there exist constants $C,c > 0$ such that for any $\varepsilon\ge0$, \begin{align*} \textbf{P}\left(s_{n}(M_n)\le {\varepsilon}{n^{-1/2}}\right)\le C\varepsilon+e^{-cn}. \end{align*} In particular, the probability that $M_n$ is singular is exponentially small. Our approach builds on the Combinatorial Least Common Denominator (CLCD) introduced by Tran and develops the method in the present constrained setting.