On the number of eigenvalues of a matrix operator

We consider a matrix operator H in the Fock space. We prove the finiteness of the number of negative eigenvalues of H if the corresponding generalized Friedrichs model has the zero eigenvalue (0 = min σ _ess(H)). We also prove that H has infinitely many negative eigenvalues accumulating near zero (the Efimov effect) if the generalized Friedrichs model has zero energy resonance. We obtain asymptotics for the number of negative eigenvalues of H below z as z → −0.     

The sign matrix and the separation of matrix eigenvalues

The sign matrices uniquely associated with the matrices (M ? ζ_jI)², where ζ_j are the corners of a rectangle oriented at π/4 to the axes of a Cartesian coordinate system, may be used to compute the number of eigenvalues of the arbitrarily chosen matrix M which lie within the rectangle, and to determine the left and right invariant subspaces of M associated with these eigenvalues. This paper is concerned with the proof of this statement, and with the details of the computation of the required sign matrices.     

Distribution of eigenvalues and eigenvectors of Wishart matrix when the population eigenvalues are infinitely dispersed and its application to minimax estimation of covariance matrix

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 tail minimax estimators is identical when the sample eigenvalues become infinitely dispersed.     

The no-response approach and its relation to non-iterative methods for the inverse scattering

This paper addresses the inverse obstacle scattering problem. In the recent years several non-iterative methods have been proposed to reconstruct obstacles (penetrable or impenetrable) from near or far field measurements. In the chronological order, we cite among others the linear sampling method, the factorization method, the probe method and the singular sources method. These methods use differently the measurements to detect the unknown obstacle and they require the use of many incident fields (i.e. the full or a part of the far field map). More recently, two other approaches have been added. They are the no-response test and the range test. Both of them use few incident fields to detect some informations about the scatterer. All the mentioned methods are based on building functions depending on some parameter. These functions share the property that their behaviors with respect to the parameter change drastically. The surface of the obstacle is located at most in the interface where these functions become large. The goal of this work is to investigate the relation between some of the non-iterative reconstruction schemes regarding the convergence issue. A given method is said to be convergent if it reconstructs a part or the entire obstacle by using few or many incident fields respectively. For simplicity we consider the obstacle reconstruction problem from far field data for the Helmholtz equation. Gen Nakamura is partially supported by Grant-in-Aid for Scientific research (B)(2)(N.14340038) of Japan Society for Promotion of Science. Mourad Sini is supported by Japan Society for Promotion of Science.     

The determinant of the sum of two normal matrices with prescribed eigenvalues

Given complex numbers α₁,...,α_n, β₁,...,β_n, what can we say about the determinant of A+B, where A (B) is an n×n normal matrix with eigenvalues α₁,...,α_n (β₁,...,β_n)? Some partial answers are offered to this question.     

The number of Hecke eigenvalues of same signs

We give the best possible lower bounds in order of magnitude for the number of positive and negative Hecke eigenvalues. This improves upon a recent work of Kohnen, Lau and Shparlinski. Also, we study an analogous problem for short intervals.     

A converse to the Jensen inequality,its matrix extensions and inequalities for minors and eigenvalues

We obtain a scalar inequality, converse to the Jensen inequality. We also derive an operator converse to the Jensen inequality. As special cases, we obtain inequalities, similar to the Kantorovich one as well as some operator generalizations of them. Using some exterior algebra, we prove a generalization of the Sylvester determinant theorem. We also deduce some determinant analogs of the additive and multiplicative Kantorovich inequalities.     

Powers of a matrix and a formula for the moduli of its eigenvalues

The spectral radius of a complex square matrix A is given by ρ(A) = lim sup_{k → ∞} (TrA^k)^1/k. A more general result is proved which gives information about the moduli of all eigenvalues of A.     

The relation between the number of leaves of a tree and its diameter

Czechoslovak Mathematical Journal - Let L(n, d) denote the minimum possible number of leaves in a tree of order n and diameter d. Lesniak (1975) gave the lower bound B(n,d) = ⌈2(n −...     

The eigenvalues of a tridiagonal matrix in biogeography

We derive the eigenvalues of a tridiagonal matrix with a special structure. A conjecture about the eigenvalues was presented in a previous paper, and here we prove the conjecture. The matrix structure that we consider has applications in biogeography theory.     

星匹配数与(无符号)拉普拉斯特征值

设G是一个简单无向图,s 3是一个正整数.文章中,若K1,s-匹配数为m(G)的n阶连通图G满足n(s+1)m(G),则G的第m(G)大L-特征值μm(G)s+1,然后证明了类似结论对于Q-谱也成立.最后给出了几个判断图的哈密顿性的Q-特征值条件.     

Computational aspect to the nearest matrix with two prescribed eigenvalues

Given a complex square matrix A and two complex numbers λ₁ and λ₂, we present a method to calculate the distance from A to the set of matrices X that have λ₁ and λ₂ as some of their eigenvalues. We also find the nearest matrix X.     

The scattering matrix with respect to an Hermitian matrix of a graph

Recently, Gnutzmann and Smilansky [5] presented a formula for the bond scattering matrix of a graph with respect to an Hermitian matrix. We present another proof for this formula by a technique use in the zeta function of a graph. Furthermore, we generalize Gnutzmann and Smilansky's formula to a regular covering of a graph. Finally, we define an L-function of a graph, and present a determinant expression. As a corollary, we express the generalization of Gnutzmann and Smilansky's formula to a regular covering of a graph by using its L-functions.     

Normal and seminormal eigenvalues of matrix functions

LetP() be ann×n analytic matrix function andW(P) be its numerical range. In this paper classical results on the normality of matrix eigenvalues on W(P) are generalized to the context of such matrix functions. Special attention is paid to corners of W(P) and to the special case of matrix polynomialsP().