1,750 research outputs found
The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations
In this paper, we investigate the asymptotic spectrum of complex or real
Deformed Wigner matrices defined by where
is an Hermitian (resp., symmetric) Wigner matrix whose
entries have a symmetric law satisfying a Poincar\'{e} inequality. The matrix
is Hermitian (resp., symmetric) and deterministic with all but finitely
many eigenvalues equal to zero. We first show that, as soon as the first
largest or last smallest eigenvalues of are sufficiently far from zero,
the corresponding eigenvalues of almost surely exit the limiting
semicircle compact support as the size becomes large. The corresponding
limits are universal in the sense that they only involve the variance of the
entries of . On the other hand, when is diagonal with a sole simple
nonnull eigenvalue large enough, we prove that the fluctuations of the largest
eigenvalue are not universal and vary with the particular distribution of the
entries of .Comment: Published in at http://dx.doi.org/10.1214/08-AOP394 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
An Overview of Polynomially Computable Characteristics of Special Interval Matrices
It is well known that many problems in interval computation are intractable,
which restricts our attempts to solve large problems in reasonable time. This
does not mean, however, that all problems are computationally hard. Identifying
polynomially solvable classes thus belongs to important current trends. The
purpose of this paper is to review some of such classes. In particular, we
focus on several special interval matrices and investigate their convenient
properties. We consider tridiagonal matrices, {M,H,P,B}-matrices, inverse
M-matrices, inverse nonnegative matrices, nonnegative matrices, totally
positive matrices and some others. We focus in particular on computing the
range of the determinant, eigenvalues, singular values, and selected norms.
Whenever possible, we state also formulae for determining the inverse matrix
and the hull of the solution set of an interval system of linear equations. We
survey not only the known facts, but we present some new views as well
Characterizing and approximating eigenvalue sets of symmetric interval matrices
We consider the eigenvalue problem for the case where the input matrix is
symmetric and its entries perturb in some given intervals. We present a
characterization of some of the exact boundary points, which allows us to
introduce an inner approximation algorithm, that in many case estimates exact
bounds. To our knowledge, this is the first algorithm that is able to guaran-
tee exactness. We illustrate our approach by several examples and numerical
experiments
Eigenvectors of random matrices: A survey
Eigenvectors of large matrices (and graphs) play an essential role in
combinatorics and theoretical computer science. The goal of this survey is to
provide an up-to-date account on properties of eigenvectors when the matrix (or
graph) is random.Comment: 64 pages, 1 figure; added Section 7 on localized eigenvector
Poisson Statistics for the Largest Eigenvalues in Random Matrix Ensemble
The paper studies the spectral properties of large Wigner, band and sample
covariance random matrices with heavy tails of the marginal distributions of
matrix entries.Comment: This is an extended version of my talk at the QMath 9 conference at
Giens, France on September 13-17, 200
- …