Level crossing in random matrices. II. Random perturbation of a random matrix

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.

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The Expected Norm of Random Matrices

Combinatorics Probability Computing ◽

10.1017/s096354830000420x ◽

2000 ◽

Vol 9 (2) ◽

pp. 149-166 ◽

Cited By ~ 30

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).

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FLUCTUATIONS OF MATRIX ENTRIES OF ANALYTIC FUNCTIONS OF NON-HERMITIAN RANDOM MATRICES

Random Matrices Theory and Application ◽

10.1142/s2010326312500086 ◽

2012 ◽

Vol 01 (03) ◽

pp. 1250008

Author(s):

SEAN O'ROURKE

Keyword(s):

Random Matrices ◽

Random Matrix ◽

Analytic Functions ◽

Random Variables

Consider an n × n non-Hermitian random matrix Mn whose entries are independent real random variables. Under suitable conditions on the entries, we study the fluctuations of the entries of f(Mn) as n tends to infinity, where f is analytic on an appropriate domain. This extends the results in [19, 20, 23] from symmetric random matrices to the non-Hermitian case.

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A Random Matrix Approach to Manipulator Jacobian

Volume 3: Nonlinear Estimation and Control; Optimization and Optimal Control; Piezoelectric Actuation and Nanoscale Control; Robotics and Manipulators; Sensing; System Identification (Estimation for Automotive Applications, Modeling, Therapeutic Control in Bio-Systems); Variable Structure/Sliding-Mode Control; Vehicles and Human Robotics; Vehicle Dynamics and Control; Vehicle Path Planning and Collision Avoidance; Vibrational and Mechanical Systems; Wind Energy Systems and Control ◽

10.1115/dscc2013-3950 ◽

2013 ◽

Cited By ~ 2

Author(s):

Javad Sovizi ◽

Aliakbar Alamdari ◽

Venkat N. Krovi

Keyword(s):

Random Matrix ◽

Jacobian Matrix ◽

Random Perturbation ◽

Random Variable ◽

System Response ◽

Kinematic Modeling ◽

Perturbation Matrix ◽

Uncertainty Model ◽

System Information

Traditional kinematic analysis of manipulators, built upon a deterministic articulated kinematic modeling often proves inadequate to capture uncertainties affecting the performance of the real robotic systems. While a probabilistic framework is necessary to characterize the system response variability, the random variable/vector based approaches are unable to effectively and efficiently characterize the system response uncertainties. Hence in this paper, we propose a random matrix formulation for the Jacobian matrix of a robotic system. It facilitates characterization of the uncertainty model using limited system information in addition to taking into account the structural inter-dependencies and kinematic complexity of the manipulator. The random Jacobian matrix is modeled such that it adopts a symmetric positive definite random perturbation matrix. The maximum entropy principle permits characterization of this perturbation matrix in the form of a Wishart distribution with specific parameters. Comparing to the random variable/vector based schemes, the benefits now include: incorporating the kinematic configuration and complexity in the probabilistic formulation, achieving the uncertainty model using limited system information (mean and dispersion parameter), and realizing a faster simulation process. A case study of a 6R serial manipulator (PUMA 560) is presented to highlight the critical aspects of the process. A Monte Carlo analysis is performed to capture the deviations of distal path from the desired trajectory and the statistical analysis on the realizations of the end effector position and orientation shows how the uncertainty propagates throughout the system.

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A Block Compressive Sensing Based Scalable Encryption Framework for Protecting Significant Image Regions

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127416501911 ◽

2016 ◽

Vol 26 (11) ◽

pp. 1650191 ◽

Cited By ~ 17

Author(s):

Yushu Zhang ◽

Jiantao Zhou ◽

Fei Chen ◽

Leo Yu Zhang ◽

Di Xiao ◽

...

Keyword(s):

Compressive Sensing ◽

Random Matrices ◽

Random Matrix ◽

Chaotic Maps ◽

Sampling Rate ◽

Equal Size ◽

Edge Detector ◽

Scalable Encryption ◽

Significant Block ◽

Block Encryption

The existing Block Compressive Sensing (BCS) based image ciphers adopted the same sampling rate for all the blocks, which may lead to the desirable result that after subsampling, significant blocks lose some more-useful information while insignificant blocks still retain some less-useful information. Motivated by this observation, we propose a scalable encryption framework (SEF) based on BCS together with a Sobel Edge Detector and Cascade Chaotic Maps. Our work is firstly dedicated to the design of two new fusion techniques, chaos-based structurally random matrices and chaos-based random convolution and subsampling. The basic idea is to divide an image into some blocks with an equal size and then diagnose their respective significance with the help of the Sobel Edge Detector. For significant block encryption, chaos-based structurally random matrix is applied to significant blocks whereas chaos-based random convolution and subsampling are responsible for the remaining insignificant ones. In comparison with the BCS based image ciphers, the SEF takes lightweight subsampling and severe sensitivity encryption for the significant blocks and severe subsampling and lightweight robustness encryption for the insignificant ones in parallel, thus better protecting significant image regions.

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Eigenvectors of a matrix under random perturbation

Random Matrices Theory and Application ◽

10.1142/s2010326321500234 ◽

2020 ◽

pp. 2150023

Author(s):

Florent Benaych-Georges ◽

Nathanaël Enriquez ◽

Alkéos Michaïl

Keyword(s):

Random Matrix ◽

Hermitian Matrix ◽

Random Perturbation ◽

Perturbative Expansion ◽

The State ◽

Operator Norm ◽

Spectral Measures ◽

Large Size ◽

Variance Profile

In this text, based on elementary computations, we provide a perturbative expansion of the coordinates of the eigenvectors of a Hermitian matrix of large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first one are independent, centered, with a variance profile. This is done through a perturbative expansion of spectral measures associated to the state defined by a given vector.

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Identifiability of parametric random matrix models

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025719500188 ◽

2019 ◽

Vol 22 (03) ◽

pp. 1950018

Author(s):

Tomohiro Hayase

Keyword(s):

Probability Theory ◽

Matrix Models ◽

Random Matrices ◽

Random Matrix ◽

Free Probability ◽

Parameter Identifiability ◽

Free Probability Theory ◽

Noise Type ◽

Random Matrix Models ◽

Spectral Distributions

We investigate parameter identifiability of spectral distributions of random matrices. In particular, we treat compound Wishart type and signal-plus-noise type. We show that each model is identifiable up to some kind of rotation of parameter space. Our method is based on free probability theory.

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Permutations Without Long Decreasing Subsequences and Random Matrices

The Electronic Journal of Combinatorics ◽

10.37236/929 ◽

2007 ◽

Vol 14 (1) ◽

Cited By ~ 1

Author(s):

Piotr Šniady

Keyword(s):

Random Matrices ◽

Young Diagram ◽

Random Matrix ◽

Random Permutation ◽

Fixed Number ◽

Plancherel Measure ◽

Young Diagrams ◽

Gaussian Unitary Ensemble ◽

Longest Increasing Subsequence ◽

Unitary Ensemble

We study the shape of the Young diagram $\lambda$ associated via the Robinson–Schensted–Knuth algorithm to a random permutation in $S_n$ such that the length of the longest decreasing subsequence is not bigger than a fixed number $d$; in other words we study the restriction of the Plancherel measure to Young diagrams with at most $d$ rows. We prove that in the limit $n\to\infty$ the rows of $\lambda$ behave like the eigenvalues of a certain random matrix (namely the traceless Gaussian Unitary Ensemble random matrix) with $d$ rows and columns. In particular, the length of the longest increasing subsequence of such a random permutation behaves asymptotically like the largest eigenvalue of the corresponding random matrix.

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