Related papers: Surjectivity of near square random matrices
Low-rank approximation of a matrix by means of random sampling has been consistently efficient in its empirical studies by many scientists who applied it with various sparse and structured multipliers, but adequate formal support for this…
We extend Latimer and MacDuffee's theorem to a general commutative domain and apply this result to study similarity of matrices over integral rings of number fields. We also conjecture similarity over discrete valuation rings can be descent…
Random matrices arise in many mathematical contexts, and it is natural to ask about the properties that such matrices satisfy. If we choose a matrix with integer entries at random, for example, what is the probability that it will have a…
Given a set C in R^d, let p(C) be the probability that a random d-dimensional unimodular lattice, chosen according to Haar measure on SL(d,Z)\SL(d,R), is disjoint from C\{0}. For special convex sets C we prove bounds on p(C) which are sharp…
Let $A$ be an $n \times n$ random matrix with iid entries over a finite field of order $q$. Suppose that the entries do not take values in any additive coset of the field with probability greater than $1 - \alpha$ for some fixed $0 < \alpha…
We study the singularity probability of n*n random matrices with i.i.d. entries from highly biased discrete distributions. We obtain sharp non-asymptotic bounds for this probability and derive estimates on the least singular values. Our…
Let $M_n$ be a class of symmetric sparse random matrices, with independent entries $M_{ij} = \delta_{ij} \xi_{ij}$ for $i \leq j$. $\delta_{ij}$ are i.i.d. Bernoulli random variables taking the value $1$ with probability $p \geq…
Let $K$ be a number field, let $A$ be a finite-dimensional $K$-algebra, let $\mathrm{J}(A)$ denote the Jacobson radical of $A$, and let $\Lambda$ be an $\mathcal{O}_{K}$-order in $A$. Suppose that each simple component of the semisimple…
Chiral random matrix theory makes very detailed predictions for the spectral correlations of the QCD Dirac operator, both in the bulk of the spectrum and near zero virtuality. These predictions have been successfully tested in lattice QCD…
Random matrices are used in fields as different as the study of multi-orthogonal polynomials or the enumeration of discrete surfaces. Both of them are based on the study of a matrix integral. However, this term can be confusing since the…
We determine the rank of a random matrix over an arbitrary field with prescribed numbers of non-zero entries in each row and column. As an application we obtain a formula for the rate of low-density parity check codes. This formula…
We determine the probability that a random n x n symmetric matrix over {1, 2, ... , m} has determinant divisible by m.
We investigate invertible matrices over finite additively idempotent semirings. The main result provides a criterion for the invertibility of such matrices. We also give a construction of the inverse matrix and a formula for the number of…
We present two sharp, closed-form empirical Bernstein inequalities for symmetric random matrices with bounded eigenvalues. By sharp, we mean that both inequalities adapt to the unknown variance in a tight manner: the deviation captured by…
We determine the rank of a random matrix A over a finite field with prescribed numbers of non-zero entries in each row and column. As an application we obtain a formula for the rate of low-density parity check codes. This formula verifies a…
Consider a symmetric (finite) matrix ensemble, with a certain probability distribution. What is the probability that the spectrum belongs to a certain interval or union of intervals on the real line? In this paper, we show that, upon…
This paper gives the relation between quasi completely inverse semirings and strongly additively quasi completely inverse semirings and generates the necessary and sufficient conditions for a strongly additively quasi completely inverse…
The probability that a tuple of matrices together with all scalars generates a finite incidence ring is calculated. It is proved that all real and complex finite-dimensional incidence algebras are generated by two randomly chosen matrices.
Let $\Sigma_d^{++}$ be the set of positive definite matrices with determinant 1 in dimension $d\ge 2$. Identifying any two $SL_d(\mathbb{Z})$-congruent elements in $\Sigma_d^{++}$ gives rise to the space of reduced quadratic forms of…
We derive exact matrix integral representations for different sums over partitions. The characteristic feature of all obtained matrix models is the presence of logarithmic (or, vice versa, exponential) terms in the potential. Our derivation…