Invariant subspaces for tightly clustered eigenvalues of tridiagonals

The invariant subspace of a real symmetric tridiagonal matrixT associated with a tight cluster ofm eigenvalues has a special structure. This structure is revealed by the envelope of the subspace (defined in Section 3) havingm high hills separated bym – 1 low valleys.This paper describes a long technical report that shows the existence ofm submatrices ofT each one having a single eigenvalue in the cluster interval whose normalized eigenvector has small entries in the first and last positions. These little eigenvectors determine a distinguished basis for an approximating subspace whose overlap matrix is tridiagonal and close to the identity. The only communication needed among the submatrices to make an orthogonal basis is between nearest neighbors.A variety of examples illustrate the theory.Supported by ONR, Contract N000014-90-J-1372.The ideas in this paper were presented at the ENUMATH meeting at CNRS, Paris, in September 1995 and at the ILAY workshop at CERFACS in October 1995.     

Asymptotic behavior of eigenvalues and random updating schemes

For a stochastic matrix (Q _ij ^T ) _i,j=1 ^M withQ _ij ^T exp(–U(ij)/T) at the off-diagonal positions, we develop an algorithm to evaluate the asymptotic convergence rate of all eigenvalues ofQ _ij ^T asT 0 using Ventcel's optimal graphs. As an application we can compare the convergence rates of some random updating schemes used in image processing.This research was partially supported by the National Science Council, Taiwan and Air Force Office of Scientific Research Contract No. F49620 S5C 0144, and was completed while Tzuu-Shuh Chiang was visiting the Center for Stochastic Processes, Department of Statistics, University of North Carolina, Chapel Hill, NC 27599-3260, USA.     

An a posteriori error estimation of the Rayleigh-Ritz eigenvalues

Summary LetA, B be essentially self-adjoint and positive definite differential operators defined inL ₂(G). Using Svirskij's construction of the base operator and some results from the analytic perturbation theory of linear operators a formula providing eigenvalue lower bounds of the problemAu=Bu is derived. In this formula a rough lower bound of some higher eigenvalue and the residual convergence of the Rayleigh-Ritz eigenfunction approximations are needed. Some numerical results are presented.     

A lower bound for eigenvalues of a clamped plate problem

In this paper, we study eigenvalues of a clamped plate problem. We obtain a lower bound for eigenvalues, which gives an important improvement of results due to Levine and Protter.     

Lower bound of the sum of positive eigenvalues of a graph

1.IntroductionA(general)graphGisagraphwithloopsormultiedges,whileasimplegraphhasneitherloopsnormultiedges.ForagivengraphG,theedgesetandtheadjacencymatrixofGaredenotedbyE(G)andA(G)respectively.Moreover,weuseS(G),P(G)andq(G)todenotethesumofpositiveeigenvalues,thepositiveandnegativeinertiaindicesofA(G).In[l],C.DelormepointedoutthatS(G)5#E(G)withequalityohlyformatchings.Hefurtherputforwardthequestion:WhatisthelowerboundofS(G)foragiven#E(G)?Isit~?Inthispaperwegiveanaffirmativeanswertoth…     

A lower bound for sums of eigenvalues of the Laplacian

Let be the th Dirichlet eigenvalue of a bounded domain in . According to Weyl's asymptotic formula we have

The optimal in view of this asymptotic relation lower estimate for the sums has been proven by P.Li and S.T.Yau (Comm. Math. Phys. 88 (1983), 309-318). Here we will improve this estimate by adding to its right-hand side a term of the order of that depends on the ratio of the volume to the moment of inertia of .

     

The error bound of the perturbation of the Drazin inverse

Let A and E be n×n matrices and B = A + E. Denote the Drazin inverse of A by A^D. In this paper we give an upper bound for the relative error ∥B^D ? A^D∥/∥A^D∥₂ and a lower bound for ∥B^D∥₂ under certain circumstances. The continuity properties and the derivative of the Drazin inverse are also considered.     

Upper bound for the non-maximal eigenvalues of irreducible nonnegative matrices

We present a lower and an upper bound for the second smallest eigenvalue of Laplacian matrices in terms of the averaged minimal cut of weighted graphs. This is used to obtain an upper bound for the real parts of the non-maximal eigenvalues of irreducible nonnegative matrices. The result can be applied to Markov chains.     

A refined bound for sum-free sets in groups of prime order

Improving upon earlier results of Freiman and the present authors,we show that if p is a sufficiently large prime and A is a sum-freesubset of the group of order p, such that n: = |A| > 0.318p,then A is contained in a dilation of the interval [n, p –n](mod p).     

An error bound of the Ritz method for a singular second-order differential equation

The paper presents an error bound of the Ritz method for the problem of minimizing the functional

High-dimensional Edgeworth expansion of a test statistic on independence and its error bound

In this paper, we calculate Edgeworth expansion of a test statistic on independence when some of the parameters are large, and simulate the goodness of fit of its approximation. We also calculate an error bound for Edgeworth expansion. Some tables of the error bound are given, which show that the derived bound is sufficiently small for practical use.     

Some results,concerning the variation of eigenvalues,when these depend on a real parameter

In this article we deal with a Hamiltonial of the form H(v) = Ho + A(v) where Ho is a self-adjoint bounded or unbounded operator on a Hilbert space and A(v) is a bounded self-adjoint perturbation depending on a real parameter v. In quantum mechanics a variety of results has been obtained by taking formally the derivative of the eigenvectors and eigenvalues of H(v).The differentiability of the eigenvectors and eigenvalues has been rigorously proved under several assumptions. Among these assumptions is the assumption that the eigenvalues are simple and the assumption that the perturbation A(v) is a uniformly bounded self-adjoint operator. A part of this article is dealing with examples, which show that these two assumptions are essential. The rest of this article is devoted to different applications concerning asymptotic relations of eigenvalues and a result for the solutions of the equation d_y/d_t= M(t)y in an abstract infinite dimensional Hilbert space, where iM(t)(1²=-1) is self-adjoint for every t in an interval. This result finds a succesful application to the theory of Toda and Langmuir lattices.     

The local eigenvalues of a bipartite distance-regular graph

We consider a bipartite distance-regular graph $\Gamma$ with vertex set $X$, diameter $D\ge 4$, and valency $k\ge 3$. For $0\le i\le D$, let ${\Gamma }_i\left(x\right)$ denote the set of vertices in $X$ that are distance $i$ from vertex $x$. We assume there exist scalars $r,s,t\in \mathbb{R}$, not all zero, such that $r|{\Gamma }_1\left(x\right)\cap {\Gamma }_1\left(y\right)\cap {\Gamma }_2\left(z\right)|+s|{\Gamma }_2\left(x\right)\cap {\Gamma }_2\left(y\right)\cap {\Gamma }_1\left(z\right)|+t=0$ for all $x,y,z\in X$ with path-length distances $\partial (x,y)=2,\partial (x,z)=3,\partial (y,z)=3$. Fix $x\in X$, and let ${\Gamma }_2^2$ denote the graph with vertex set $\tilde{X}=\{y\in X\mid \partial (x,y)=2\}$ and edge set $\tilde{R}=\{yz\mid y,z\in \tilde{X},\partial (y,z)=2\}$. We show that the adjacency matrix of the local graph ${\Gamma }_2^2$ has at most four distinct eigenvalues. We are motivated by the fact that our assumption above holds if $\Gamma$ is $Q$-polynomial.     

A semi-inverse variational method for generating the bound state energy eigenvalues in a quantum system: The Schrödinger equation

This paper deals with the semi-inverse variational method to extract the structures of bound states of the Schrödinger equation in a quantum environment. From realistic examples, some state configurations are presented to illustrate the effectiveness and the exactitude of the proposed method.     

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.