Related papers: On the singularity of random combinatorial matrice…

Let $A$ be a $m\times m$ complex matrix with zero trace and let $\e>0$. Then there are $m\times m$ matrices $B$ and $C$ such that $A=[B,C]$ and $\|B\|\|C\|\le K_\e m^\e\|A\|$ where $K_\e$ depends only on $\e$. Moreover, the matrix $B$ can…

Given some binary matrix $M$, suppose we are presented with the collection of its rows and columns in independent arbitrary orderings. From this information, are we able to recover the unique original orderings and matrix? We present an…

The universality phenomenon asserts that the distribution of the eigenvalues of random matrix with i.i.d. zero mean, unit variance entries does not depend on the underlying structure of the random entries. For example, a plot of the…

An elementary proof of the two-sidedness of the matrix-inverse is given using only linear independence and the reduced row-echelon form of a matrix. In addition, it is shown that a matrix is invertible if and only if it is row-equivalent to…

A 0/1-simplex is the convex hull of n+1 affinely independent vertices of the unit n-cube I^n. It is nonobtuse if none its dihedral angles is obtuse, and acute if additionally none of them is right. Acute 0/1-simplices in I^n can be…

The number of $n \times n$ matrices whose entries are either -1, 0, or 1, whose row- and column- sums are all 1, and such that in every row and every column the non-zero entries alternate in sign, is proved to be $[1!4! >...…

Let $R=\mathcal{O}_{\Q(\sqrt{d})}$ for $d<0$, squarefree, $d\neq -1,-3$. We prove Lehmer's conjecture for associated reciprocal polynomials of $R$-matrices; that is, any noncyclotomic $R$-matrix has Mahler measure at least…

Let $X_1,..., X_N\in\R^n$ be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability at least $1 - 3 \exp(-c\sqrt{n}\r)$ one has $ \sup_{x\in…

We consider a square random matrix made by i.i.d. rows with any distribution and prove that, for any given dimension, the probability for the least singular value to be in [0; $\epsilon$) is at least of order $\epsilon$. This allows us to…

Let A be an n by n matrix, whose entries are independent copies of a centered random variable satisfying the subgaussian tail estimate. We prove that the operator norm of A^{-1} does not exceed Cn^{3/2} with probability close to 1.

In this paper, we investigate the following question: How often is a random matrix normal? We consider a random $n\times n$ matrix, $M_n$, whose entries are i.i.d. Rademacher random variables (taking values $\{ \pm1 \}$ with probability…

Let $\epsilon_1, \dotsc, \epsilon_n$ be i.i.d. Rademacher random variables taking values $\pm 1$ with probability $1/2$ each. Given an integer vector $\boldsymbol{a} = (a_1, \dotsc, a_n)$, its concentration probability is the quantity…

We study the conjugation action of orthogonal matrices on symmetric random matrices. Given a fixed orthogonal matrix over an algebraic number field and a random matrix with entries sufficiently uniform in the ring of integers, we wonder…

We study the distribution of the least singular value associated to an ensemble of sparse random matrices. Our motivating example is the ensemble of $N\times N$ matrices whose entries are chosen independently from a Bernoulli distribution…

Let $M$ be a matroid on a finite ground set $E$, and suppose that the automorphism group of $M$ acts transitively on $E$. We show the following: if $X_1,\ldots,X_K$ are sampled independently from a distribution $p$ on $E$, then the…

We prove that the (non-symmetric) adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices is asymptotically almost surely invertible, assuming $\min(d,n-d)\ge C\log^2n$ for a sufficiently large constant $C>0$. The…

We consider unimodular matrices $M$ such that neither $M$ nor $M^{-1}$ contain zero entries. Matrices typically exhibit a trade-off: small $M$ imply large $M^{-1}$. We investigate rare cases where both remain small, classify these matrices…

Permanents of random matrices with independent and identically distributed (i.i.d.) entries have extensively studied in literature and convergence and concentration properties are known under varying assumptions on the distributions. In…

Using standard techniques from combinatorics, model theory, and algebraic geometry, we prove generalized versions of several basic results in the theory of spectrally arbitrary matrix patterns. Also, we point out a counterexample to a…

A known result in random matrix theory states the following: Given a random Wigner matrix $X$ which belongs to the Gaussian Orthogonal Ensemble (GOE), then such matrix $X$ has an invariant distribution under orthogonal conjugations. The…