LIMIT THEOREMS FOR β-LAGUERRE AND β-JACOBI ENSEMBLES

We use tridiagonal models to study the limiting behavior of β-Laguerre and β-Jacobi ensembles,focusing on the limiting behavior of the extremal eigenvalues and the central limit theorem for the two ensembles.For the central limit theorem of β-Laguerre ensembles,we follow the idea in[1]while giving a modified version for the generalized case.Then we use the total variation distance between the two sorts of ensembles to obtain the limiting behavior of β-Jacobi ensembles.

A SMALLEST SINGULAR VALUE METHOD FOR SOLVING INVERSE EIGENVALUE PROBLEMS

Utilizing the properties of the smallest singular value of a matrix, we propose a new, efficient and reliable algorithm for solving nonsymmetric matrix inverse eigenvalue problems, and compare it with a known method. We also present numerical experiments which illustrate our results.

On the Second Smallest and the Largest Normalized Laplacian Eigenvalues of a Graph

Let G be a simple connected graph with order n.Let L(G)and Q(G)be the normalized Laplacian and normalized signless Laplacian matrices of G,respectively.Letλk(G)be the k-th smallest normalized Laplacian eigenvalue of G.Denote byρ(A)the spectral radius of the matrix A.In this paper,we study the behaviors ofλ2(G)andρ(L(G))when the graph is perturbed by three operations.We also study the properties ofρ(L(G))and X for the connected bipartite graphs,where X is a unit eigenvector of L(G)corresponding toρ(L(G)).Meanwhile we characterize all the simple connected graphs withρ(L(G))=ρ(Q(G)).

An Inequality for the Perron Pair of an Irreducible and Symmetric Nonnegative Tensor with Application

The main purpose of this paper is to consider the Perron pair of an irreducible and symmetric nonnegative tensor and the smallest eigenvalue of an irreducible and symmetric nonsingular M-tensor.We analyze the analytical property of an algebraic simple eigenvalue of symmetric tensors.We also derive an inequality about the Perron pair of nonnegative tensors based on plane stochastic tensors.We finally consider the perturbation of the smallest eigenvalue of nonsingular M-tensors and design a strategy to compute its smallest eigenvalue.We verify our results via random numerical examples.

A DYNAMICS APPROACH TO THE COMPUTATION OF EIGENVECTORS OF MATRICES

We construct a family of dynamical systems whose evolution converges to the eigenvectors of a general square matrix, not necessarily symmetric. We analyze the convergence of those systems and perform numerical tests. Some examples and comparisons with the power methods are presented.