Spectrum of a matrix

About: Spectrum of a matrix is a research topic. Over the lifetime, 1064 publications have been published within this topic receiving 19841 citations. The topic is also known as: matrix spectrum.

Distribution of eigenvalues and eigenvectors of Wishart matrix when the population eigenvalues are infinitely dispersed

Abstract: We consider the asymptotic joint distribution of the eigenvalues and eigenvectors of Wishart matrix when the population eigenvalues become infinitely dispersed. We show that the normalized sample eigenvalues and the relevant elements of the sample eigenvectors are asymptotically all mutually independently distributed. The limiting distributions of the normalized sample eigenvalues are chi-squared distributions with varying degrees of freedom and the distribution of the relevant elements of the eigenvectors is the standard normal distribution. As an application of this result, we investigate tail minimaxity in the estimation of the population covariance matrix of Wishart distribution with respect to Stein’s loss function and the quadratic loss function. Under mild regularity conditions, we show that the behavior of a broad class of minimax estimators is identical when the sample eigenvalues become infinitely dispersed.

Spectra of weighted scale-free networks.

Abstract: Much information about the structure and dynamics of a network is encoded in the eigenvalues of its transition matrix. In this paper, we present a first study on the transition matrix of a family of weight driven networks, whose degree, strength, and edge weight obey power-law distributions, as observed in diverse real networks. We analytically obtain all the eigenvalues, as well as their multiplicities. We then apply the obtained eigenvalues to derive a closed-form expression for the random target access time for biased random walks occurring on the studied weighted networks. Moreover, using the connection between the eigenvalues of the transition matrix of a network and its weighted spanning trees, we validate the obtained eigenvalues and their multiplicities. We show that the power-law weight distribution has a strong effect on the behavior of random walks.

ASYMPTOTIC DISTRIBUTION OF EIGENVALUES FOR HYPOELLIPTIC SYSTEMS IN $R^n$

Abstract: General symmetric hypoelliptic systems of differential operators in with discrete spectrum are considered. Two-sided estimates, as , are found for , the number of eigenvalues in the interval . Under a regularity assumption on the behavior of the spectrum of the Weyl matrix symbol of the system, these estimates reduce to the asymptotics of with an estimate of the remainder term. In part the results are also new for the scalar case. Bibliography: 9 titles.

The symmetrized quantum thermal flux operator

Abstract: Analysis of the symmetrized thermal flux operator leads to explicit expressions for its eigenvalues and eigenfunctions. At any point in configuration space one finds two nonzero eigenvalues of opposite sign. The associated eigenfunctions are L2 integrable. The eigenfunctions and eigenvalues are expressed in terms of the thermal density matrix in the vicinity of the transition state. The positive eigenvalue of the thermal flux operator gives an upper bound to the rate and allows for a formulation of a quantum mechanical variational transition state theory. This new upper bound, though, is only a slight improvement over previous theories.