The First Order Asymptotics of the Extreme Eigenvectors of Certain Hermitian Toeplitz Matrices

The paper is concerned with Hermitian Toeplitz matrices generated by a class of unbounded symbols that emerge in several applications. The main result gives the third order asymptotics of the extreme eigenvalues and the first order asymptotics of the extreme eigenvectors of the matrices as their dimension increases to infinity. This work was partially supported by CONACYT projects 60160 and 80504, Mexico.     

Pseudodifferential operators on<Emphasis Type="Italic">SU</Emphasis>(2)

We construct an algebra of left-invariant pseudodifferential operators on SU(2). We require only that the symbols be homogeneous and C². For Fourier-bandlimited symbols, we derive the expected formulae for composition and commutators and construct an orthonormal basis of common approximate eigenvectors that could be used to study spectral theory. Some remarks on applications to matrices of operators are made.     

Pseudodifferential Operators on SU(2)

We construct an algebra of left-invariant pseudodifferential operators on SU(2). We require only that the symbols be homogeneous and C ² . For Fourier-bandlimited symbols, we derive the expected formulae for composition and commutators and construct an orthonormal basis of common approximate eigenvectors that could be used to study spectral theory. Some remarks on applications to matrices of operators are made.     

On the characteristic polynomial,eigenvectors and determinant of heptadiagonal matrices

In this paper, the characteristic polynomial of general heptadiagonal matrices is derived as well as eigenvectors associated to a prescribed eigenvalue. A symbolic algorithm to compute the determinant of heptadiagonal matrices is also presented allowing a suite implementation through computational software programs.     

A computational method for eigenvalues and eigenvectors of a matrix with real eigenvalues

This paper describes a new computational procedure for calculating eigenvalues and eigenvectors of a square matrix. The method is based on a matrix function, the sign of a matrix. Eigenvalues and eigenvectors of matrices with distinct eigenvalues and nondefective matrices with repeated roots can be determined in a straightforward manner. Defective matrices require additional calculations.     

Using principal eigenvectors of adjacency matrices with added diagonal weights to compose centrality measures and identify bowtie structures for a digraph

Principal eigenvectors of adjacency matrices are often adopted as measures of centrality for a graph or digraph. However, previous principal-eigenvector-like measures for a digraph usually consider only the strongly connected component whose adjacency submatrix has the largest eigenvalue. In this paper, for each and every strongly connected component in a digraph, we add weights to diagonal elements of its member nodes in the adjacency matrix such that the modified matrix will have the new unique largest eigenvalue and corresponding principal eigenvectors. Consequently, we use the new principal eigenvectors of the modified matrices, based on different strongly connected components, not only to compose centrality measures but also to identify bowtie structures for a digraph.     

Variational equations for the eigenvalues and eigenvectors of nonsymmetric matrices

The tracking of eigenvalues and eigenvectors for parameterized matrices is of major importance in optimization and stability problems. In the present paper, we consider a one-parameter family of matrices with distinct eigenvalues. A complete system of differential equations is developed for both the eigenvalues and the right and left eigenvectors. The computational feasibility of the differential system is demonstrated by means of a numerical example.The work of R. Kalaba and L. Tesfatsion was partially supported by the National Science Foundation under Grant No. ENG-77-28432 and by the National Institutes of Health under Grant No. GM-23732-03.     

RQI DYNAMICS FOR NON-NORMAL MATRICES WITHREAL EIGENVALUES

1.IntroductionAswellknown,RQI(RayleighQuotientiteration)isapracticalalgorithmforeigen-valueproblemsofsymmetricmatrices.In1974,B.N.ParlettprovedthatthesequencegeneratedbyRQIalwaysconvergestoaneigenvectorforalmostallofinitialvectorsifthematrixinquestionisanormalone.NamelythesetofvectorsinR",forwhichRQIdiverges,hajszeromeasure.Nevertheless,healsopointedouttheconvergelitspeedbeingcubicone[1].In1989,S.BarttsonandJ.SmillieconsideredRQIforsymmetricmatrixagain.Theydiscoveredthatthedynamicsof…     

Approximate eigensolution of Laplacian matrices for locally modified graph products

Laplacian matrices and their spectrum are of great importance in algebraic graph theory. There exist efficient formulations for eigensolutions of the Laplacian matrices associated with a special class of graphs called product graphs. In this paper, the problem of determining a few approximate smallest eigenvalues and eigenvectors of large scale product graphs modified through the addition or deletion of some nodes and/or members, is investigated. The eigenproblem associated with a modified graph model is reduced using the set of master eigenvectors and linear approximated slave eigenvectors from the original model. Implicitly restarted Lanczos method is employed to obtain the required eigenpairs of the reduced problem. Examples of large scale models are included to demonstrate the efficiency of the proposed method compared to the direct application of the IRL method.     

Approximate eigensolution of Laplacian matrices for locally modified graph products

Laplacian matrices and their spectrum are of great importance in algebraic graph theory. There exist efficient formulations for eigensolutions of the Laplacian matrices associated with a special class of graphs called product graphs. In this paper, the problem of determining a few approximate smallest eigenvalues and eigenvectors of large scale product graphs modified through the addition or deletion of some nodes and/or members, is investigated. The eigenproblem associated with a modified graph model is reduced using the set of master eigenvectors and linear approximated slave eigenvectors from the original model. Implicitly restarted Lanczos method is employed to obtain the required eigenpairs of the reduced problem. Examples of large scale models are included to demonstrate the efficiency of the proposed method compared to the direct application of the IRL method.     

Eigenvalues of Hermitian Toeplitz matrices with smooth simple-loop symbols

The paper presents higher-order asymptotic formulas for the eigenvalues of large Hermitian Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line. The formulas are established not only for the extreme eigenvalues, but also for the inner eigenvalues. The results extend and make more precise existing results, which so far pertain to banded matrices or to matrices with infinitely differentiable symbols. Also given is a fixed-point equation for the eigenvalues which may be solved numerically by an iteration method.     

A matrix analysis of Arnoldi and Lanczos methods

Summary. In this paper we propose a matrix analysis of the Arnoldi and Lanczos procedures when used for approximating the eigenpairs of a non-normal matrix. By means of a new relation between the respective representation matrices, we relate the corresponding eigenvalues and eigenvectors. Moreover, backward error analysis is used to theoretically justify some unexpected experimental behaviors of non-normal matrices and in particular of banded Toeplitz matrices. Received June 19, 1996 / Revised version received November 3, 1997     

On some tridiagonal k-Toeplitz matrices: Algebraic and analytical aspects. Applications

In this paper, we use the analytic theory for 2 and 3-Toeplitz matrices to obtain the explicit expressions for the eigenvalues, eigenvectors and the spectral measure associated to the corresponding infinite matrices. As an application we consider two solvable models related with the so-called chain model. Some numerical experiments are also included.     

Effective construction of eigenvectors for a class of singular sparse matrices

Fundamental results and an efficient algorithm for constructing eigenvectors corresponding to non-zero eigenvalues of matrices with zero rows and/or columns are developed. The formulation is based on the relation between eigenvectors of such matrices and the eigenvectors of their submatrices after removing all zero rows and columns. While being easily implemented, the algorithm decreases the computation time needed for numerical eigenanalysis, and resolves potential numerical eigensolver instabilities.     

Asymptotic Formulas for Determinants of a Sum of Finite Toeplitz and Hankel Matrices

The purpose of this paper is to describe asymptotic formulas for determinants of a sum of finite Toeplitz and Hankel matrices with singular generating functions. The formulas are similar to those of the analogous problem for finite Toeplitz matrices for a certain class of symbols. However, the appearance of the Hankel matrices changes the nature of the asymptotics in some instances depending on the location of the singularities. Several concrete examples are also described in the paper.     

Computation of roots of real and complex matrices

An algorithm is presented in this paper by which the rth root of real or complex matrices can be found without the computation of the eigenvalues and eigenvectors of the matrix. All required computations are in the real domain. The method is based on the Newton-Raphson algorithm and is capable of finding roots even when the matrix is defective. Computing the root of a matrix from eigenvalues and eigenvectors would be the preferred method if these data were available.     

Pseudoeigenvector bases and deflated GMRES for highly nonnormal matrices

Pseudoeigenvalues have been extensively studied for highly nonnormal matrices. This paper focuses on the corresponding pseudoeigenvectors. The properties and uses of pseudoeigenvector bases are investigated. It is shown that pseudoeigenvector bases can be much better conditioned than eigenvector bases. We look at the stability and the varying quality of pseudoeigenvector bases. Then applications are considered including the exponential of a matrix. Several aspects of GMRES convergence are looked at, including why using approximate eigenvectors to deflate eigenvalues can be effective even when there is not a basis of eigenvectors.     

Eigenvector sensitivity under general and structured perturbations of tridiagonal Toeplitz‐type matrices

The sensitivity of eigenvalues of structured matrices under general or structured perturbations of the matrix entries has been thoroughly studied in the literature. Error bounds are available, and the pseudospectrum can be computed to gain insight. Few investigations have focused on analyzing the sensitivity of eigenvectors under general or structured perturbations. This paper discusses this sensitivity for tridiagonal Toeplitz and Toeplitz‐type matrices.     

On Matrices with Perron–Frobenius Properties and Some Negative Entries

We consider those n-by-n matrices with a strictly dominant positive eigenvalue of multiplicity 1 and associated positive left and right eigenvectors. Such matrices may have negative entries and generalize the primitive matrices in important ways. Several ways of constructing such matrices, including a very geometric one, are discussed. This paper grew out of a recent survey talk about nonnegative matrices by the first author and a joint paper, with others, by the second author about the symmetric case [Tarazaga et al. (2001) Linear Algebra Appl. 328: 57].     

Block diagonalization and eigenvalues

This paper deals with block diagonalization of partitioned (not necessarily square) matrices. The process is shown to be analogous to calculating eigenvalues and eigenvectors. Computer techniques and examples are provided. Several various types of applications are discussed including application to liver disease.