We consider the real eigenvalues of an $(N \times N)$ elliptic Ginibre matrix whose entries are correlated through a non-Hermiticity parameter $\tau_N\in [0,1]$. In almost-Hermitian regime where $1-\tau_N=\Theta(N^{-1})$, we obtain large-$N$ expansion mean and variance number eigenvalues. Furthermore, derive limiting empirical distributions eigenvalues, which interpolate Wigner semicircle law u...

The paper studies the eigenvalue distribution of Schur complements of some special matrices, including nonstrictly diagonally dominant matrices and general H−matrices. Zhang, Xu, and Li [Theorem 4.1, The eigenvalue distribution on Schur complements of H-matrices. Linear Algebra Appl., 422:250–264, 2007] gave a condition for an n×n diagonally dominant matrix A to have |JR+(A)| eigenvalues with p...

Functional data analysis is a relatively new and rapidly growing area of statistics. This is partly due to technological advancements which have made it possible to generate new types of data that are in the form of curves. Because the data are functions, they lie in function spaces, which are of infinite dimension. To analyse functional data, one way, which is widely used, is to employ princip...

It has been shown that, if a model displays long-range (power-law) spatial correlations, its equal-time correlation matrix will also have a power law tail in the distribution of its high-lying eigenvalues. The purpose of this paper is to show that the converse is generally incorrect: a power-law tail in the high-lying eigenvalues of the correlation matrix may exist even in the absence of equal-...

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 ...

We investigate the statistical properties of the cross-correlation matrix between individual stocks traded in the Korean stock market using the random matrix theory (RMT) and observe how these affect the portfolio weights in the Markowitz portfolio theory. We find that the distribution of the cross-correlation matrix is positively skewed and changes over time. We find that the eigenvalue distri...

We give a short, operator-theoretic proof of the asymptotic independence of the minimal and maximal eigenvalue of the n× n Gaussian Unitary Ensemble in the large matrix limit n→ ∞. This is done by representing the joint probability distribution of those extreme eigenvalues as the Fredholm determinant of an operator matrix that asymptotically becomes diagonal. The method is amenable to explicitl...

Product graphs have been gainfully used in literature to generate mathematical models of complex networks which inherit properties of real networks. Realizing the duplication phenomena imbibed in the definition of corona product of two graphs, we define corona graphs. Given a small simple connected graph which we call basic graph, corona graphs are defined by taking corona product of the basic ...

Most pattern recognition applications require the eigenvalues and eigenvectors of the covariance matrix. It is well known that when the number of training samples is small, the eigenvalues of the covariance matrix contains bias, and the bias degrades recognition performance. There are some methods which ignore the small eigenvalues, or acquire better estimates of the covariance matrix by correc...

All information concerning the distribution of eigenvalues in any of the classical ensembles (GOE, GUE, Wishart) is encoded in the joint densities, for which we have obtained explicit formulas. Unfortunately, getting information out of the joint density requires integration, and many of the integrals that arise cannot be evaluated in any nice closed form. Nevertheless, it is possible to show th...