On eigenvalue distributions of large autocovariance matrices

On singular value distribution of large-dimensional autocovariance matrices

On the nonnegative inverse eigenvalue problem of traditional matrices

In this paper, at first for a given set of real or complex numbers $sigma$ with nonnegative summation, we introduce some special conditions that with them there is no nonnegative tridiagonal matrix in which $sigma$ is its spectrum. In continue we present some conditions for existence such nonnegative tridiagonal matrices.

Large Random Matrices: Eigenvalue Distribution

A recursive method is derived to calculate all eigenvalue correlation functions of a random hermitian matrix in the large size limit, and after smoothing of the short scale oscillations. The property that the two-point function is universal, is recovered and the three and four-point functions are given explicitly. One observes that higher order correlation functions are linear combinations of u...

Asymptotic eigenvalue distribution of large Toeplitz matrices

We study the asymptotic eigenvalue distribution of Toeplitz matrices generated by a singular symbol. It has been conjectured by Widom that, for a generic symbol, the eigenvalues converge to the image of the symbol. In this paper we ask how the eigenvalues converge to the image. For a given Toeplitz matrix Tn(a) of size n, we take the standard approach of looking at det(ζ − Tn(a)), of which the ...

Numerical Methods for Eigenvalue Distributions of Random Matrices

We present efficient numerical techniques for calculation of eigenvalue distributions of random matrices in the beta-ensembles. We compute histograms using direct simulations on very large matrices, by using tridiagonal matrices with appropriate simplifications. The distributions are also obtained by numerical solution of the Painlevé II and V equations with high accuracy. For the spacings we s...