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).

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 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 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.

Sparse signal recovery using a new class of random matrices

2017 ◽  
Vol 8 (2) ◽  
Author(s):  
Enrico Au-Yeung
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Sparse Matrices ◽  
Random Variables ◽  
Signal Recovery ◽  
Original Matrix ◽  
Bernoulli Random Variables ◽  
New Class ◽  
The Matrix ◽  
New Type

Abstract We propose a new class of random matrices that enables the recovery of signals with sparse representation in a known basis with overwhelmingly high probability. To construct a matrix in this class, we begin with a fixed non-random matrix that satisfies two very general conditions. Then we decompose the matrix into pieces of sparse matrices. A random sum (involving Bernoulli random variables) of these pieces of sparse matrices is used to construct the final matrix. We say that the random matrix is the randomized Bernoulli transform of the original matrix. The random matrix is not created by filling all its entries with random variables, as in the case of Gaussian or Bernoulli matrices. Therefore, as a benefit, far fewer number of random variables are needed to generate this new type of random matrices. We prove that the number of samples needed to recover a random signal is proportional to the sparsity of the signal, up to a logarithmic factor, and hence this number is nearly optimal.

scholarly journals Octonions in random matrix theory

2017 ◽  
Vol 473 (2200) ◽  
pp. 20160800 ◽  
Author(s):  
Peter J. Forrester
Keyword(s):  
Random Matrices ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Quantum Physics ◽  
Matrix Theory ◽  
Analytic Theory ◽  
Division Algebras ◽  
Number Systems ◽  
The Matrix ◽  
Real Complex

The octonions are one of the four normed division algebras, together with the real, complex and quaternion number systems. The latter three hold a primary place in random matrix theory, where in applications to quantum physics they are determined as the entries of ensembles of Hermitian random matrices by symmetry considerations. Only for N =2 is there an existing analytic theory of Hermitian random matrices with octonion entries. We use a Jordan algebra viewpoint to provide an analytic theory for N =3. We then proceed to consider the matrix structure X † X , when X has random octonion entries. Analytic results are obtained from N =2, but are observed to break down in the 3×3 case.

VICTORIA transform, RESPECT and REFORM methods for the proof of the G-Elliptic Law under G-Lindeberg condition and twice stochastic condition for the variances and covariances of the entries of some random matrices

2020 ◽  
Vol 28 (2) ◽  
pp. 131-162
Author(s):  
Vyacheslav L. Girko
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Lindeberg Condition

AbstractThe G-Elliptic law under the G-Lindeberg condition for the independent pairs of the entries of a random matrix is proven.

scholarly journals An asymptotic property of large matrices with identically distributed Boolean independent entries

2019 ◽  
Vol 22 (04) ◽  
pp. 1950024
Author(s):  
Mihai Popa ◽  
Zhiwei Hao
Keyword(s):  
Recent Result ◽  
Recent Work ◽  
Random Matrices ◽  
Asymptotic Property ◽  
Limit Distribution ◽  
Asymptotic Independence ◽  
Moment Conditions ◽  
The Matrix ◽  
Combinatorial Condition ◽  
Independence Relations

Motivated by the recent work on asymptotic independence relations for random matrices with non-commutative entries, we investigate the limit distribution and independence relations for large matrices with identically distributed and Boolean independent entries. More precisely, we show that, under some moment conditions, such random matrices are asymptotically [Formula: see text]-diagonal and Boolean independent from each other. This paper also gives a combinatorial condition under which such matrices are asymptotically Boolean independent from the matrix obtained by permuting the entries (thus extending a recent result in Boolean probability). In particular, we show that the random matrices considered are asymptotically Boolean independent from some of their partial transposes. The main results of the paper are based on combinatorial techniques.

From random matrices to random tensors

2021 ◽  
pp. 166-177
Author(s):  
Adrian Tanasa
Keyword(s):  
Matrix Models ◽  
Random Matrices ◽  
Mathematical Physics ◽  
Natural Generalization ◽  
Random Surface ◽  
The Third ◽  
Planar Surfaces ◽  
Tensor Models ◽  
Feynman Graphs ◽  
The Matrix

After a brief presentation of random matrices as a random surface QFT approach to 2D quantum gravity, we focus on two crucial mathematical physics results: the implementation of the large N limit (N being here the size of the matrix) and of the double-scaling mechanism for matrix models. It is worth emphasizing that, in the large N limit, it is the planar surfaces which dominate. In the third section of the chapter we introduce tensor models, seen as a natural generalization, in dimension higher than two, of matrix models. The last section of the chapter presents a potential generalisation of the Bollobás–Riordan polynomial for tensor graphs (which are the Feynman graphs of the perturbative expansion of QFT tensor models).

scholarly journals Edge universality for non-Hermitian random matrices

2020 ◽  
Author(s):  
Giorgio Cipolloni ◽  
László Erdős ◽  
Dominik Schröder
Keyword(s):  
Unit Circle ◽  
Random Matrices ◽  
Matrix Elements ◽  
Spectral Edge ◽  
Ginibre Ensemble ◽  
Eigenvalue Statistics ◽  
The Matrix ◽  
Independent Identically Distributed

Abstract We consider large non-Hermitian real or complex random matrices $$X$$ X with independent, identically distributed centred entries. We prove that their local eigenvalue statistics near the spectral edge, the unit circle, coincide with those of the Ginibre ensemble, i.e. when the matrix elements of $$X$$ X are Gaussian. This result is the non-Hermitian counterpart of the universality of the Tracy–Widom distribution at the spectral edges of the Wigner ensemble.

scholarly journals Bernstein-type inequality for a class of dependent random matrices

2016 ◽  
Vol 05 (02) ◽  
pp. 1650006 ◽  
Author(s):  
Marwa Banna ◽  
Florence Merlevède ◽  
Pierre Youssef
Keyword(s):  
Random Matrices ◽  
Type Inequality ◽  
Bernstein Inequality ◽  
Mathematical Statistics ◽  
Strong Mixing ◽  
Mixing Conditions ◽  
Bernstein Type ◽  
Bernstein Type Inequality ◽  
The Matrix ◽  
The Laplace Transform

In this paper, we obtain a Bernstein-type inequality for the sum of self-adjoint centered and geometrically absolutely regular random matrices with bounded largest eigenvalue. This inequality can be viewed as an extension to the matrix setting of the Bernstein-type inequality obtained by Merlevède et al. [Bernstein inequality and moderate deviations under strong mixing conditions, in High Dimensional Probability V: The Luminy Volume, Institute of Mathematical Statistics Collection, Vol. 5 (Institute of Mathematical Statistics, Beachwood, OH, 2009), pp. 273–292.] in the context of real-valued bounded random variables that are geometrically absolutely regular. The proofs rely on decoupling the Laplace transform of a sum on a Cantor-like set of random matrices.

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