scholarly journals On minimal singular values of random matrices with correlated entries

2015 ◽  
Vol 04 (02) ◽  
pp. 1550006 ◽  
Author(s):  
F. Götze ◽  
A. Naumov ◽  
A. Tikhomirov
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Singular Values ◽  
Singular Value ◽  
The Matrix ◽  
Class Of Matrices ◽  
Least Singular Value ◽  
Mutually Independent

Let X be a random matrix whose pairs of entries Xjk and Xkj are correlated and vectors (Xjk, Xkj), for 1 ≤ j < k ≤ n, are mutually independent. Assume that the diagonal entries are independent from off-diagonal entries as well. We assume that [Formula: see text], for any j, k = 1, …, n and 𝔼 XjkXkj = ρ for 1 ≤ j < k ≤ n. Let Mn be a non-random n × n matrix with ‖Mn‖ ≤ KnQ, for some positive constants K > 0 and Q ≥ 0. Let sn(X + Mn) denote the least singular value of the matrix X + Mn. It is shown that there exist positive constants A and B depending on K, Q, ρ only such that [Formula: see text] As an application of this result we prove the elliptic law for this class of matrices with non-identically distributed correlated entries.

scholarly journals Complex random matrices have no real eigenvalues

2018 ◽  
Vol 07 (01) ◽  
pp. 1750014 ◽  
Author(s):  
Kyle Luh
Keyword(s):  
Random Matrices ◽  
Random Variables ◽  
Principal Component ◽  
Random Variable ◽  
Singular Values ◽  
Singular Value ◽  
Small Ball ◽  
Circular Law ◽  
Least Singular Value ◽  
Tail Bound

Let [Formula: see text] where [Formula: see text] are iid copies of a mean zero, variance one, subgaussian random variable. Let [Formula: see text] be an [Formula: see text] random matrix with entries that are iid copies of [Formula: see text]. We prove that there exists a [Formula: see text] such that the probability that [Formula: see text] has any real eigenvalues is less than [Formula: see text] where [Formula: see text] only depends on the subgaussian moment of [Formula: see text]. The bound is optimal up to the value of the constant [Formula: see text]. The principal component of the proof is an optimal tail bound on the least singular value of matrices of the form [Formula: see text] where [Formula: see text] is a deterministic complex matrix with the condition that [Formula: see text] for some constant [Formula: see text] depending on the subgaussian moment of [Formula: see text]. For this class of random variables, this result improves on the results of Pan–Zhou [Circular law, extreme singular values and potential theory, J. Multivariate Anal. 101(3) (2010) 645–656] and Rudelson–Vershynin [The Littlewood–Offord problem and invertibility of random matrices, Adv. Math. 218(2) (2008) 600–633]. In the proof of the tail bound, we develop an optimal small-ball probability bound for complex random variables that generalizes the Littlewood–Offord theory developed by Tao–Vu [From the Littlewood–Offord problem to the circular law: Universality of the spectral distribution of random matrices, Bull. Amer. Math. Soc.[Formula: see text]N.S.[Formula: see text] 46(3) (2009) 377–396; Inverse Littlewood–Offord theorems and the condition number of random discrete matrices, Ann. of Math.[Formula: see text] 169(2) (2009) 595–632] and Rudelson–Vershynin [The Littlewood–Offord problem and invertibility of random matrices, Adv. Math. 218(2) (2008) 600–633; Smallest singular value of a random rectangular matrix, Comm. Pure Appl. Math. 62(12) (2009) 1707–1739].

scholarly journals On the limiting spectral density of random matrices filled with stochastic processes

2019 ◽  
Vol 27 (2) ◽  
pp. 89-105 ◽  
Author(s):  
Matthias Löwe ◽  
Kristina Schubert
Keyword(s):  
Stochastic Process ◽  
Spectral Density ◽  
Random Matrices ◽  
Markov Processes ◽  
Random Matrix ◽  
Matrix Theory ◽  
Gibbs Measures ◽  
State Spaces ◽  
The Matrix ◽  
Finite State

Abstract We discuss the limiting spectral density of real symmetric random matrices. In contrast to standard random matrix theory, the upper diagonal entries are not assumed to be independent, but we will fill them with the entries of a stochastic process. Under assumptions on this process which are satisfied, e.g., by stationary Markov chains on finite sets, by stationary Gibbs measures on finite state spaces, or by Gaussian Markov processes, we show that the limiting spectral distribution depends on the way the matrix is filled with the stochastic process. If the filling is in a certain way compatible with the symmetry condition on the matrix, the limiting law of the empirical eigenvalue distribution is the well-known semi-circle law. For other fillings we show that the semi-circle law cannot be the limiting spectral density.

scholarly journals The Expected Norm of Random Matrices

2000 ◽  
Vol 9 (2) ◽  
pp. 149-166 ◽  
Author(s):  
YOAV SEGINER
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Operator Norm ◽  
Euclidean Norm ◽  
Constant Independent ◽  
Multiplicative Constant ◽  
The Matrix

We compare the Euclidean operator norm of a random matrix with the Euclidean norm of its rows and columns. In the first part of this paper, we show that if A is a random matrix with i.i.d. zero mean entries, then E∥A∥h [les ] Kh (E maxi ∥ai[bull ] ∥h + E maxj ∥aj[bull ] ∥h), where K is a constant which does not depend on the dimensions or distribution of A (h, however, does depend on the dimensions). In the second part we drop the assumption that the entries of A are i.i.d. We therefore consider the Euclidean operator norm of a random matrix, A, obtained from a (non-random) matrix by randomizing the signs of the matrix's entries. We show that in this case, the best inequality possible (up to a multiplicative constant) is E∥A∥h [les ] (c log1/4 min {m, n})h (E maxi ∥ai[bull ] ∥h + E maxj ∥aj[bull ] ∥h) (m, n the dimensions of the matrix and c a constant independent of m, n).

The Empirical Distribution of the Singular Values of a Random Hankel Matrix

2015 ◽  
Vol 14 (03) ◽  
pp. 1550027 ◽  
Author(s):  
Mansi Ghodsi ◽  
Nader Alharbi ◽  
Hossein Hassani
Keyword(s):  
Normal Distribution ◽  
Random Matrix ◽  
Empirical Distribution ◽  
Hankel Matrix ◽  
Singular Values ◽  
Shape Parameters ◽  
The Matrix ◽  
Non Parametric

The empirical distribution of the eigenvalues of the matrix HHT divided by its trace is considered, where H is a Hankel random matrix. The normal distribution with different parameters are considered and the effect of scale and shape parameters are evaluated. The correlation among eigenvalues are assessed using parametric and non-parametric association criteria.

scholarly journals RANDOM MATRICES: THE CIRCULAR LAW

2008 ◽  
Vol 10 (02) ◽  
pp. 261-307 ◽  
Author(s):  
TERENCE TAO ◽  
VAN VU
Keyword(s):  
Strong Convergence ◽  
Random Matrices ◽  
Random Variable ◽  
Singular Value ◽  
Additive Combinatorics ◽  
Sparse Random Matrices ◽  
Circular Law ◽  
Bounded Variance ◽  
Complex Random Variable ◽  
Least Singular Value

Let x be a complex random variable with mean zero and bounded variance σ2. Let Nn be a random matrix of order n with entries being i.i.d. copies of x. Let λ1, …, λn be the eigenvalues of [Formula: see text]. Define the empirical spectral distributionμn of Nn by the formula [Formula: see text] The following well-known conjecture has been open since the 1950's: Circular Law Conjecture: μn converges to the uniform distribution μ∞ over the unit disk as n tends to infinity. We prove this conjecture, with strong convergence, under the slightly stronger assumption that the (2 + η)th-moment of x is bounded, for any η > 0. Our method builds and improves upon earlier work of Girko, Bai, Götze–Tikhomirov, and Pan–Zhou, and also applies for sparse random matrices. The new key ingredient in the paper is a general result about the least singular value of random matrices, which was obtained using tools and ideas from additive combinatorics.

scholarly journals RANDOM MATRICES: UNIVERSAL PROPERTIES OF EIGENVECTORS

2012 ◽  
Vol 01 (01) ◽  
pp. 1150001 ◽  
Author(s):  
TERENCE TAO ◽  
VAN VU
Keyword(s):  
Random Matrices ◽  
Real Axis ◽  
Central Limit ◽  
Random Matrix ◽  
Joint Distribution ◽  
Limit Theorems ◽  
The Matrix ◽  
Universal Properties ◽  
The Mean ◽  
First Four Moments

The four moment theorem asserts, roughly speaking, that the joint distribution of a small number of eigenvalues of a Wigner random matrix (when measured at the scale of the mean eigenvalue spacing) depends only on the first four moments of the entries of the matrix. In this paper, we extend the four moment theorem to also cover the coefficients of the eigenvectors of a Wigner random matrix. A similar result (with different hypotheses) has been proved recently by Knowles and Yin, using a different method. As an application, we prove some central limit theorems for these eigenvectors. In another application, we prove a universality result for the resolvent, up to the real axis. This implies universality of the inverse matrix.

scholarly journals Dynamical correlation functions for products of random matrices

2015 ◽  
Vol 04 (04) ◽  
pp. 1550020 ◽  
Author(s):  
Eugene Strahov
Keyword(s):  
Random Matrices ◽  
Discrete Time ◽  
Random Matrix ◽  
Correlation Functions ◽  
Singular Values ◽  
Matrix Product ◽  
Dynamical Correlation ◽  
Products Of Random Matrices ◽  
Special Cases ◽  
Correlation Kernels

We introduce and study a family of random processes with a discrete time related to products of random matrices. Such processes are formed by singular values of random matrix products, and the number of factors in a random matrix product plays a role of a discrete time. We consider in detail the case when the (squared) singular values of the initial random matrix form a polynomial ensemble, and the initial random matrix is multiplied by standard complex Gaussian matrices. In this case, we show that the random process is a discrete-time determinantal point process. For three special cases (the case when the initial random matrix is a standard complex Gaussian matrix, the case when it is a truncated unitary matrix, or the case when it is a standard complex Gaussian matrix with a source) we compute the dynamical correlation functions explicitly, and find the hard edge scaling limits of the correlation kernels. The proofs rely on the Eynard–Mehta theorem, and on contour integral representations for the correlation kernels suitable for an asymptotic analysis.

scholarly journals THE BETA-MANOVA ENSEMBLE WITH GENERAL COVARIANCE

2014 ◽  
Vol 03 (01) ◽  
pp. 1450002 ◽  
Author(s):  
ALEXANDER DUBBS ◽  
ALAN EDELMAN
Keyword(s):  
Matrix Model ◽  
Random Matrix ◽  
Singular Values ◽  
Positive Definite ◽  
Singular Value ◽  
General Covariance ◽  
Generalized Singular Value Decomposition ◽  
Random Matrix Model ◽  
Value Decomposition ◽  
Order Continuous

We consider adding arbitrary covariance to the β-Jacobi random matrix model. We recall that for β = 1 the Jacobi random matrix model may be thought of as the eigenvalues, λi, of YtY(XTX + YtY)-1 where X and Y are matrices whose elements are i.i.d. standard normals. Equivalently we can take the generalized cosine singular values of (Y, X), ci, and use [Formula: see text]. When β = 1 we add covariance by considering YtY(YtY + ΩXtXΩ)-1, for a positive definite diagonal matrix Ω. Equivalently, and preferably, we consider the generalized singular value decomposition (gsvd) of (Y, XΩ). We refer to Ω = I as the Jacobi case and the general Ω case as the MANOVA case. In this paper, we provide a matrix model for the general β-MANOVA ensemble. In particular, we provide an algorithm for the numerical sampling of eigenvalues or generalized cosine singular values. The β-MANOVA algorithm uses the β-Wishart algorithm of Forrester and Dubbs–Edelman–Koev–Venkataramana as a subroutine, perhaps making β-MANOVA the first "second-order" continuous-β random matrix algorithm. Our proofs make use of a conjecture of MacDonald (proven by Baker and Forrester), a theorem of Kaneko, and many identities from Forrester's Log-Gases and Random Matrices. We supply numerical evidence that our theorems are correct.

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