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.

Application of random matrix theory to study cross-correlations of stock prices

Abstract: We address the question of how to precisely identify correlated behavior between different firms in the economy by applying methods of random matrix theory (RMT) Specifically, we use methods of random matrix theory to analyze the cross-correlation matrix of price changes of the largest 1000 US stocks for the 2-year period 1994–1995 We find that the statistics of most of the eigenvalues in the spectrum of agree with the predictions of random matrix theory, but there are deviations for a few of the largest eigenvalues To prove that the rest of the eigenvalues are genuinely random, we test for universal properties such as eigenvalue spacings and eigenvalue correlations We demonstrate that shares universal properties with the Gaussian orthogonal ensemble of random matrices In addition, we quantify the number of significant participants, that is companies, of the eigenvectors using the inverse participation ratio, and find eigenvectors with large inverse participation ratios at both edges of the eigenvalue spectrum — a situation reminiscent of results in localization theory

On a method for computing eigenvalues and eigenfunctions of linear differential operators

Abstract: In this paper a simple method is presented to compute the eigenvalues and the eigenfunctions of second-order linear differential operators, with homogeneous boundary conditions, both in a finite interval and on the semi-line. Our technique overcomes the drawbacks of the method proposed by Calogero to compute the eigenvalues of Sturm-Liouville problems in a finite interval. An estimate of the convergence for the eigenvalues is given in the finite case and numerical tests are performed, exhibiting a very fast rate of convergence for the eigenvalues both for the finite interval and the semi-line cases. An excellent convergence for the eigenfunctions is also obtained in both cases.

Eigenvalues of Boundary Value Problems

Abstract: Here, we shall consider the n(≥ 2)th order difference equation $${\Delta ^n}y(k) + \lambda Q(k,y(k)), \cdots ,{\Delta ^{n - 2}}y(k))\, = \,\lambda P(k,y(k),\Delta y(k), \cdots ,{\Delta ^{n - 1}}y(k)),\,k \in N(0,J - 1)$$ (29.1) together with the boundary conditions (28.2) – (28.4), where the constants α, β, γ and δ satisfy the conditions (28.5) and (28.6). In (29.1), λ > 0, Q: N(0, J − 1) × ℝn−1 → ℝ, and P : N(0, J − 1) × ℝn → ℝ.

Exact eigenvalue spectrum of a class of fractal scale-free networks

Abstract: The eigenvalue spectrum of the transition matrix of a network encodes important information about its structural and dynamical properties. We study the transition matrix of a family of fractal scale-free networks and analytically determine all the eigenvalues and their degeneracies. We then use these eigenvalues to evaluate the closed-form solution to the eigentime for random walks on the networks under consideration. Through the connection between the spectrum of transition matrix and the number of spanning trees, we corroborate the obtained eigenvalues and their multiplicities.