Showing papers on "Spectrum of a matrix published in 1995"
TL;DR: The construction of an efficient iterative method which does not require from the user a prescription of several problem-dependent parameters to ensure the convergence, which can be used in the case when only a procedure for multiplying the coefficient matrix by a vector is available and which allows for an efficient parallel/vector implementation.
Abstract: The paper considers a possible approach to the construction of high-quality preconditionings for solving large sparse unsymmetric offdiagonally dominant, possibly indefinite linear systems. We are interested in the construction of an efficient iterative method which does not require from the user a prescription of several problem-dependent parameters to ensure the convergence, which can be used in the case when only a procedure for multiplying the coefficient matrix by a vector is available and which allows for an efficient parallel/vector implementation with only one additional assumption that the most of eigenvalues of the coefficient matrix are condensed in a vicinity of the point 1 of the complex plane. The suggested preconditioning strategy is based on consecutive translations of groups of spread eigenvalues into a vicinity of the point 1. Approximations to eigenvalues to be translated are computed by the Arnoldi procedure at several GMRES(k) iterations. We formulate the optimization problem to find optimal translations, present its suboptimal solution and prove the numerical stability of consecutive translations. The results of numerical experiments with the model CFD problem show the efficiency of the suggested preconditioning strategy.
83 citations
TL;DR: Two algorithms for determining the matrix numerically are proposed in this paper and, besides its easy implementation, offers a new proof of existence because of its global convergence property.
Abstract: Given two vectors $a,\lambda \in R^{n}$, the Schur--Horn theorem states that $a$ majorizes $\lambda$ if and only if there exists a Hermitian matrix $H$ with eigenvalues $\lambda$ and diagonal entries $a$. While the theory is regarded as classical by now, the known proof is not constructive. To construct a Hermitian matrix from its diagonal entries and eigenvalues therefore becomes an interesting and challenging inverse eigenvalue problem. Two algorithms for determining the matrix numerically are proposed in this paper. The lift and projection method is an iterative method that involves an interesting application of the Wielandt--Hoffman theorem. The projected gradient method is a continuous method that, besides its easy implementation, offers a new proof of existence because of its global convergence property.
47 citations
TL;DR: In this article, lower bounds for the toughness of a graph in terms of its eigenvalues were derived, and the best possible lower bounds were derived for each eigenvalue.
34 citations
TL;DR: In this paper, the majorization inequalities between the spectrum and the main diagonal of the Laplacian matrix of a simple graph on vertices V = 1, 1, n, n with vertices with eigenvalues λ 1≥…≥λn−0 were improved.
Abstract: Let G be a simple graph on vertices V={1,…,n}, with Laplacian matrix L=L(G). suppose L has eigenvalues λ1≥…≥λn−0, and that the degree sequence of G is d 1≥…λd n. In this paper we provide an improvement to the majorization inequalities between the spectrum and the main diagonal of L. The main result has as a corollary that
31 citations
TL;DR: In this article, it was shown that isolated eigenvalues in any gap of the essential spectrum of a self-adjoint operator are exactly the limits of eigen values of suitably chosen selfadjoint realizations An of τ on subintervals (an, bn) of (a, b) with an → a, b n → b.
Abstract: Let A be a self-adjoint operator defined by a general singular ordinary differential expression τ on an interval (a, b), − ∞ ≤ a < b ≤ ∞. We show that isolated eigenvalues in any gap of the essential spectrum of A are exactly the limits of eigenvalues of suitably chosen self-adjoint realizations An of τ on subintervals (an, bn) of (a, b) with an → a, bn → b. This means that eigenvalues of singular ordinary differential operators can be approximated by eigenvalues of regular operators.
30 citations
TL;DR: A simple formula is presented for computing an upper bound for departure from normality in the Frobenius norm that is cheaper to compute than the one derived by Henrici and sharp for Hermitian matrices, skew-HermitianMatrices and, in general, any matrix with eigenvalues that are horizontally or vertically aligned in the complex plane.
Abstract: The departure from normality of a matrix is a real scalar that is impractical to compute if the matrix is large and its eigenvalues are unknown. A simple formula is presented for computing an upper bound for departure from normality in the Frobenius norm. This new upper bound is cheaper to compute than the one derived by Henrici [Numer. Math., 4 (1962), pp. 24-40}]. Moreover, the new bound is sharp for Hermitian matrices, skew-Hermitian matrices and, in general, any matrix with eigenvalues that are horizontally or vertically aligned in the complex plane. In terms of applications, the new bound can be used in computing bounds for the spectral norm of matrix functions or bounds for the sensitivity of eigenvalues to matrix perturbations.
30 citations
TL;DR: The calculations exploit an analogy to the problem of finding a two-dimensional charge distribution on the interface of a semiconductor heterostructure under the influence of a split gate to find eigenvalues in a spectral interval of a large random matrix.
Abstract: We calculate the probability to find exactly $n$ eigenvalues in a spectral interval of a large random $N \times N$ matrix when this interval contains $s \ll N$ eigenvalues on average. The calculations exploit an analogy to the problem of finding a two-dimensional charge distribution on the interface of a semiconductor heterostructure under the influence of a split gate.
27 citations
TL;DR: In this article, a finite step algorithm is given such that for any two vectorsa, λ ∈ R n X n witha majorized by λ, it computes a symmetric matrixH ∈R n x n with the elements ofa and λ as its diagonal entries and eigenvalues, respectively.
Abstract: A finite step algorithm is given such that for any two vectorsa, λ ∈R
n
witha majorized by λ, it computes a symmetric matrixH ∈R
n x n
with the elements ofa and λ as its diagonal entries and eigenvalues, respectively.
20 citations
TL;DR: In this paper, tight perturbation bounds for the shifts in the eigenvalues and eigenvectors of a matrix were given for simple as well as multiple eigen values.
19 citations
01 Jan 1995
TL;DR: In this paper, the authors discuss the connection between the eigenvalues of linear operators and those of matrices, and discuss the spectrum of a linear operator corresponding to a given eigenvalue.
Abstract: This chapter discusses the concepts eigenvalues and eigenvectors. For any square matrix A over F , the polynomial det( A — xI ) is the characteristic polynomial of A . The equation det( A — xI ) = 0 is the characteristic equation of A , and the solutions of det( A — xI ) = 0 are called the eigenvalues of A . The set of all eigenvalues of A is called the spectrum of A . In the chapter, T denotes a field, V denotes a finite-dimensional vector space over T , and T denotes a linear operator on V . The chapter discusses the important connection between the eigenvalues of linear operators and those of matrices. If the n × n matrix A represents T relative to the basis A of V , then X is an eigenvector of A corresponding to λ only if X is the coordinate matrix relative to A of an eigenvector of T corresponding to the same eigenvalue.
18 citations
TL;DR: In this paper, the authors compared the Smith normal form (SNF) over the integers of an integral nonsingular matrix with its spectrum when its eigenvalues are integers and provided tight bounds on the size of the largest element of the SNF when the matrix is diagonalizable with nonzero integer eigen values.
TL;DR: In this article, Hess and Kato showed that there exists a unique λ > 0 such that the problem (0.1) has a solution, where m is a continuous bounded function on Ω.
Abstract: In a classical article, Hess and Kato [HK] study the problem
$$\left\{ {\begin{array}{*{20}{c}} {Au + \lambda mu = 0} \\ {0 \leqslant u \in D\left( A \right), u
e 0,} \end{array}} \right.$$
(0.1)
where A is a strongly elliptic operator on a bounded open set Ω of R n with Dirichlet boundary conditions and m is a continuous bounded function on Ω. They show that there exists a unique λ > 0 such that the problem (0.1) has a solution.
TL;DR: Asymptotic formulae for the angular spheroidal eigenvalues were found using the Bohr-Sommerfeld quantization rule and perturbation theory.
Abstract: Asymptotic formulae for the angular spheroidal eigenvalues are found using the Bohr-Sommerfeld quantization rule and perturbation theory. Approximate calculations for the separation between two consecutive eigenvalues are obtained.
TL;DR: In this paper, a transfer matrix method was used to calculate the asymptotic behavior of the nonlinear Schrodinger (NLS) equation in a self-defocusing medium for piecewise constant initial conditions.
Abstract: In this paper we use a transfer matrix method to calculate the asymptotic behavior of the nonlinear Schrodinger (NLS) equation in a self-defocusing medium for piecewise constant initial conditions. Treating initial conditions that consist of m repeated regions, we show that the eigenvalues associated with this problem appear in bands, and, as m tends to infinity, we obtain the eigenvalue density of states for these bands. Comparing results from the transfer matrix approach to the results for a Bloch function method, we show that the edges of a region with periodic initial conditions result in a finite number of additional eigenvalues that appear outside the bands.
TL;DR: In this paper, the eigenvalues of [X,A]=XA−AX, where A is an n by n fixed matrix and X runs over the set of the matrices of the same size, are characterized.
Abstract: We characterize the eigenvalues of [X,A]=XA−AX, where A is an n by n fixed matrix and X runs over the set of the matrices of the same size.
TL;DR: In this article, the authors propose and justify a new method of calculation of eigenvalues for discrete operators via the theory of regularized traces, which they call regularized trace theory.
Abstract: The authors propose and justify a new method of calculation of eigenvalues for discrete operators via the theory of regularized traces. Bibliography: 2 titles.
TL;DR: In this article, the authors established factorization theorems and properties of sets of eigenvectors for regular selfadjoint quatratic matrix polynomials whose leading coefficeint is indefinite or possibly singular, and for which all eigenvalues are real of definite type.
Abstract: Factorization theorems, and properties of sets of eigenvectors, are established for regular selfadjoint quatratic matrix polynomials L(λ) whose leading coefficeint is indefinite or possibly singular, and for which all eigenvalues are real of definite type. The two linear factors obtained have spectra which are just the eigenvalues of L(λ) of positive and negative types, respectively.
TL;DR: The universal connected correlations proposed recently between eigenvalues of unitary random matrices is examined numerically by the Monte Carlo sampling and shows a universal behavior after smoothing.
Abstract: The universal connected correlations proposed recently between eigenvalues of unitary random matrices is examined numerically. We perform an ensemble average by the Monte Carlo sampling. Although density of eigenvalues and a bare correlation of the eigenvalues are not universal, the connected correlation shows a universal behavior after smoothing.
TL;DR: In this article, the eigenvalues λ j of an n by n complex matrix A with its characteristic polynomial having real coefficients lie in the elliptic region defined by β 2 χ− tr A n 2 +α 2 y 2 ≤α 2 β 2, where α n−1 n ∑ n=1 n ( Re λ k ) 2 − ( Re(te A )) 2 n 1 2 and β= n− 1 n∑ k= 1 n ( Im λk ) 2 1 2
Abstract: We prove that the eigenvalues λ j of an n by n complex matrix A with its characteristic polynomial having real coefficients lie in the elliptic region defined by β 2 χ− tr A n 2 +α 2 y 2 ≤α 2 β 2 , where α n−1 n ∑ n=1 n ( Re λ k ) 2 − ( Re(te A )) 2 n 1 2 and β= n−1 n ∑ k=1 n ( Im λ k ) 2 1 2 This region is intersected with the strip |y|≤ 1 2 ∑ k=1 u Im λ k ) 2 1 2 to obtain an improved eigenvalue localization region. We also give bounds for the semiaxes, which can be computed without knowing the eigenvalues of A . When A has r n nonzero eigenvalues, we obtain a smaller elliptic region containing such nonzero eigenvalues.
TL;DR: In this article, a sufficient condition for a set of positive integers { g 1, g 2,..., g n } to be the geometric multiplicites of given eigenvalues for some strictly lower triangular completions of a partial matrix is given.
TL;DR: In this article, a method is given which improves the convergence of the linear variation method, where the matrix of the Hamiltonian is set up in an appropriate orthonormal basis set, the functions of which contain some parameters.
01 Jan 1995
TL;DR: A parallel algorithm for nding the singular values of a bidiagonal matrix B by computing the corresponding eigenvalues of the symmetric tridiagonal (ST) matrix B T B and taking the square roots of those eigen values.
Abstract: This paper describes a parallel algorithm for nding the singular values of a bidiagonal matrix B. The algorithm nds the largest singular values by nding the corresponding eigenvalues of the symmetric tridiagonal (ST) matrix B T B and taking the square roots of those eigenvalues. The smallest singular values are calculated by computing the corresponding eigenvalues of another ST matrix T, which contains zeroes in the main diagonal and entries of B in the oo-diagonals. Details of two implementations of the algorithm are described. One implementation uses the split-merge algorithm to nd the eigenvalues of ST matrices, and the other uses a bisection-based eigenvalue method. Performance results on an nCUBE-2 and a workstation cluster are presented.
TL;DR: In this article, the authors give explicit general solutions of the matrix equation AX−XB=C for both symmetric or skew-symmetric matrices A and B by using the notions of eigenprojections.
Abstract: Almost all of the existing results on the explicit solutions of the matrix equation AX−XB=C are obtained under the condition that A and B have no eigenvalues in common. For both symmetric or skewsymmetric matrices A and B, we shall give out the explicit general solutions of this equation by using the notions of eigenprojections. The results we obtained are applicable not only to any cases of eigenvalues regardless of their multiplicities, but also to the discussion of the general case of this equation.
TL;DR: In this article, an order-reduction method is proposed by means of which a part of the eigenvalues of a given large system in n-D space can be solved conveniently.
TL;DR: In this paper, the real and imaginary parts of the eigenvalues of an oriented-graph matrix have been studied, and it has been shown that each of the irreducible oriented graph matrices of order n ≥ 3 has at least three distinct eigen values.
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TL;DR: In this paper, the authors examined a purely geometric property of a point in the boundary of the numerical range of a Hilbert space operator that implies that such a point is a reducing essential eigenvalue of the given operator.
Abstract: We examine a purely geometric property of a point in the boundary of the numerical range of a (Hilbert space) operator that implies that such a point is a reducing essential eigenvalue of the given operator. Roughly speaking, such a property means that the boundary curve of the numerical range has infinite curvature at that point (we must exclude however linear verteces because they may be reducing eigenvalues without being reducing essential eigenvalues). This result allows us to give an elegant proof of a conjecture of Joel Anderson: {\it A compact perturbation of a scalar multiple of the identity operator can not have the closure of its numerical range equal to half a disk (neither equal to any acute circular sector).}