We consider a class of matrices of the form Cn = (1/N)A 1/2 n XnBnX ∗ nA 1/2 n , where Xn is an n × N matrix consisting of i.i.d. standardized complex entries, A n is a non-negative definite Hermitian square-root of the non-negative definite matrix An, and Bn is diagonal with nonnegative diagonal entries. Under the assumption that the distribution of the eigenvalues of An and Bn converge to pro...

When discussing the convergence properties of the Lanczos iteration method for the real symmetric eigenvalue problem, Trefethen and Bau noted that the Lanczos method tends to find eigenvalues in regions that have too little charge when compared to an equilibrium distribution. In this paper a quantitative version of this rule of thumb is presented. We describe, in an asymptotic sense, the region...

When discussing the convergence properties of the Lanczos iteration method for the real symmetric eigenvalue problem, Trefethen and Bau noted that the Lanczos method tends to find eigenvalues in regions that have too little charge when compared to an equilibrium distribution. In this paper a quantitative version of this rule of thumb is presented. We describe, in an asymptotic sense, the region...

For the understandings of the magnetohydrodynamic (MHD) characteristics in plasma, the spectrum of the resistive MHD modes are investigated in detail by solving the eigenvalue problem of the reduced MHD equations in cylindrical tokamak plasmas. The eigenvalues and eigenfunctions of the resistive MHD modes are clarified for small toroidal and poloidal mode numbers, and the discrete eigenvalues w...

In this paper, we investigate the spectral properties of the adjacency and the Laplacian matrices of random graphs. We prove that: (i) the law of large numbers for the spectral norms and the largest eigenvalues of the adjacency and the Laplacian matrices; (ii) under some further independent conditions, the normalized largest eigenvalues of the Laplacian matrices are dense in a compact interval ...

In this paper, we investigate the spectral properties of the adjacency and the Laplacian matrices of random graphs. We prove that (i) the law of large numbers for the spectral norms and the largest eigenvalues of the adjacency and the Laplacian matrices; (ii) under some further independent conditions, the normalized largest eigenvalues of the Laplacian matrices are dense in a compact interval a...

We discuss the existence of large isolated (non-unit) eigenvalues of the Perron– Frobenius operator for expanding interval maps. Corresponding to these eigenvalues (or ‘resonances’) are distributions which approach the invariant density (or equilibrium distribution) at a rate slower than that prescribed by the minimal expansion rate. We consider the transitional behaviour of the eigenfunctions ...

the stability analysis of islanded inverter-based microgrids (ibmgs) is increasingly an important and challenging topic due to the nonlinearity of ibmgs. in this paper, a new linear model for such microgrids as well as an iterative method to correct the linear model is proposed. using the linear model makes it easy to analyze the eigenvalues and stability of ibmgs due to the fact that it derive...

Abstract We consider random, complex sample covariance matrices 1 N X ∗X , where X is a p×N random matrix with i.i.d. entries of distribution μ. It has been conjectured that both the distribution of the distance between nearest neighbor eigenvalues in the bulk and that of the smallest eigenvalues become, in the limit N → ∞, p N → 1, the same as that identified for a complex Gaussian distributio...

This work is concerned with an asymptotical distribution of eigenvalues of sparse random matrices. It is shown that the semicircle law which is known for random matrices is also valid for the sparse random matrices with sparsity nIN=o(1), where n is the matrix size and 2N the number of non-zero elements of the matrix. The degree of degeneration is also estimated for the matrices with 2N—cn (c>0...