Related papers: On the singularity of random combinatorial matrice…

We prove two universality results for random tensors of arbitrary rank D. We first prove that a random tensor whose entries are N^D independent, identically distributed, complex random variables converges in distribution in the large N…

Let $M$ be a non-zero binary matrix with distinct rows where the rows are closed under certain logical operators. In this article, we investigate the existence of columns containing an equal or greater number of ones than zeros.…

This paper investigates the construction of deterministic matrices preserving the entropy of random vectors with a given probability distribution. In particular, it is shown that for random vectors having i.i.d. discrete components, this is…

Let $1\le k\le n$ and $M$ be a random $n\times n$ matrix with independent uniformly random $\{\pm 1\}$-entries. We show that there exists an absolute constant $c > 0$ such that \[\mathbf{P}[\operatorname{rank}(M)\le n-k]\le \exp(-c nk).\]

Consider an $n\times n$ matrix $P$ with the following properties. All entries in $P$ are positive or $0$, the sum of each row is 1 and for all $i$ and $j$ in $\{1,\dots,n\}$ there exists a natural number $k$ such that the $(i,j)$ entry of…

In this article one-sided (b, c)-inverses of arbitrary matrices as well as one-sided inverses along a (not necessarily square) matrix, will be studied. In adddition, the (b, c)-inverse and the inverse along an element will be also…

Let $M$ be a random $m \times n$ matrix with binary entries and i.i.d. rows. The weight (i.e., number of ones) of a row has a specified probability distribution, with the row chosen uniformly at random given its weight. Let $N(n,m)$ denote…

Let $A_n$ be a random symmetric matrix with Bernoulli $\{\pm 1\}$ entries. For any $\kappa>0$ and two real numbers $\lambda_1,\lambda_2$ with a separation $|\lambda_1-\lambda_2|\geq \kappa n^{1/2}$ and both lying in the bulk…

We study the gaps between consecutive singular values of random rectangular matrices. Specifically, if $M$ is an $n \times p$ random matrix with independent and identically distributed entries and $\Sigma$ is a $n \times n$ deterministic…

We consider random symmetric matrices with independent entries distributed according to the Haar measure on $\mathbb{Z}_p$ for odd primes $p$ and derive the distribution of their canonical form with respect to several equivalence relations.…

A well-known result in random matrix theory, proven by Kahn, Koml\'os and Szemer\'edi in 1995, states that a square random matrix with i.i.d. uniform $\{\pm 1\}$ entries is invertible with probability $1-\exp(-\Omega(n))$. As a natural…

We extend probability estimates on the smallest singular value of random matrices with independent entries to a class of sparse random matrices. We show that one can relax a previously used condition of uniform boundedness of the variances…

Let $n$ be a natural number and $\mathcal{M}$ a set of $n \times n$-matrices over the nonnegative integers such that the joint spectral radius of $\mathcal{M}$ is at most one. We show that if the zero matrix $0$ is a product of matrices in…

A connected graph can be associated with two distinct evolution algebras. In the first case, the structural matrix is the adjacency matrix of the graph itself. In the second case, the structural matrix is the transition probabilities matrix…

Let $\delta>1$ and $\beta>0$ be some real numbers. We prove that there are positive $u,v,N_0$ depending only on $\beta$ and $\delta$ with the following property: for any $N,n$ such that $N\ge \max(N_0,\delta n)$, any $N\times n$ random…

In this paper we consider Wigner random matrices -- symmetric n by n random matrices whose entries are independent identically distributed real random variables. We prove that the probability distribution of one or several eigenvalues close…

An exchangeable random matrix is a random matrix with distribution invariant under any permutation of the entries. For such random matrices, we show, as the dimension tends to infinity, that the empirical spectral distribution tends to the…

Random contractions (sub-unitary random matrices) appear naturally when considering quantized chaotic maps within a general theory of open linear stationary systems with discrete time. We analyze statistical properties of complex…

A concentration graph associated with a random vector is an undirected graph where each vertex corresponds to one random variable in the vector. The absence of an edge between any pair of vertices (or variables) is equivalent to full…

Let $\a$ be a real-valued random variable of mean zero and variance 1. Let $M_n(\a)$ denote the $n \times n$ random matrix whose entries are iid copies of $\a$ and $\sigma_n(M_n(\a))$ denote the least singular value of $M_n(\a)$.…