Related papers: The number of 2x2 integer matrices having a prescr…
This paper considers random (non-Hermitian) circulant matrices, and proves several results analogous to recent theorems on non-Hermitian random matrices with independent entries. In particular, the limiting spectral distribution of a random…
In this brief paper the probability density of a random real, complex and quaternion determinant is rederived using singular values. The behaviour of suitably rescaled random determinants is studied in the limit of infinite order of the…
Number theorists have studied extensively the connections between the distribution of zeros of the Riemann $\zeta$-function, and of some generalizations, with the statistics of the eigenvalues of large random matrices. It is interesting to…
We describe two algorithms for the eigenvalue, eigenvector problem which, on input a Gaussian matrix with complex entries, finish with probability 1 and in average polynomial time.
Let M be an arbitrary Hermitian matrix of order n, and k be a positive integer less than or equal to n. We show that if k is large, the distribution of eigenvalues on the real line is almost the same for almost all principal submatrices of…
An interval matrix is a matrix whose entries are intervals in the set of real numbers. We generalize this concept, which has been broadly studied, to other fields. Precisely we define a rational interval matrix to be a matrix whose entries…
We characterize the idempotent stable range one $2\times 2$ matrices over commutative rings and in particular, the integral matrices with this property. Several special cases and examples complete the subject.
Motivated by the question of whether a random polynomial with integer coefficients is likely to be irreducible, we study the probability that a monic polynomial with integer coefficients has a low-degree factor over the integers, which is…
The goal of this article is to study how much the eigenvalues of large Hermitian random matrices deviate from certain deterministic locations -- or in other words, to investigate optimal rigidity estimates for the eigenvalues. We do this in…
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 develop an efficient algorithm for sampling the eigenvalues of random matrices distributed according to the Haar measure over the orthogonal or unitary group. Our technique samples directly a factorization of the Hessenberg form of such…
We provide a self-contained introduction to random matrices. While some applications are mentioned, our main emphasis is on three different approaches to random matrix models: the Coulomb gas method and its interpretation in terms of…
The choice of the parameter value for regularized inverse problems is critical to the results and remains a topic of interest. This article explores a criterion for selecting a good parameter value by maximizing the probability of the data,…
We use the martingale-theoretic approach of game-theoretic probability to incorporate imprecision into the study of randomness. In particular, we define several notions of randomness associated with interval, rather than precise,…
Let $n$ be a positive integer and $X = [x_{ij}]_{1 \leq i, j \leq n}$ be an $n \times n$\linebreak \noindent sized matrix of independent random variables having joint uniform distribution $$\hbox{Pr} {x_{ij} = k \hbox{for} 1 \leq k \leq n}…
We apply the operation of random independent thinning on the eigenvalues of $n\times n$ Haar distributed unitary random matrices. We study gap probabilities for the thinned eigenvalues, and we study the statistics of the eigenvalues of…
Given a random matrix A with eigenvalues between -1 and 1, we analyze the number of iterations needed to solve the linear equation (I-A)x=b with the Neumann series iteration. We give sufficient conditions for convergence of an upper bound…
We discuss the solution of eigenvalue problems associated with partial differential equations that can be written in the generalized form $\m{A}x=\lambda\m{B}x$, where the matrices $\m{A}$ and/or $\m{B}$ may depend on a scalar parameter.…
We consider random hermitian matrices in which distant above-diagonal entries are independent but nearby entries may be correlated. We find the limit of the empirical distribution of eigenvalues by combinatorial methods. We also prove that…
The probability that an interval $I$ is free of eigenvalues in a matrix ensemble with unitary symmetry is given by a Fredholm determinant. When the weight function in the matrix ensemble is a classical weight function, and the interval $I$…