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On the eigenvalues of some tridiagonal matrices

Fonseca, C. M. da
Fonte: Universidade de Coimbra Publicador: Universidade de Coimbra
Tipo: Artigo de Revista Científica Formato: aplication/PDF
ENG
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A solution is given for a problem on eigenvalues of some symmetric tridiagonal matrices suggested by William Trench. The method presented can be generalizable to other problems.; http://www.sciencedirect.com/science/article/B6TYH-4J9N0SV-1/1/f15fb954c01649af39599f88edf73d60

On the location of the eigenvalues of Jacobi matrices

Fonseca, C. M. da
Fonte: Universidade de Coimbra Publicador: Universidade de Coimbra
Tipo: Artigo de Revista Científica Formato: aplication/PDF
ENG
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Using some well known concepts on orthogonal polynomials, some recent results on the location of eigenvalues of tridiagonal matrices of very large order are extended. A significant number of important papers are unified.; http://www.sciencedirect.com/science/article/B6TY9-4JB9MTY-1/1/7ae6d931eb463437bae718429bb02246

A note on the eigenvalues of a special class of matrices

CUMINATO, J. A.; MCKEE, S.
Fonte: ELSEVIER SCIENCE BV Publicador: ELSEVIER SCIENCE BV
Tipo: Artigo de Revista Científica
ENG
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In the analysis of stability of a variant of the Crank-Nicolson (C-N) method for the heat equation on a staggered grid a class of non-symmetric matrices appear that have an interesting property: their eigenvalues are all real and lie within the unit circle. In this note we shall show how this class of matrices is derived from the C-N method and prove that their eigenvalues are inside [-1, 1] for all values of m (the order of the matrix) and all values of a positive parameter a, the stability parameter sigma. As the order of the matrix is general, and the parameter sigma lies on the positive real line this class of matrices turns out to be quite general and could be of interest as a test set for eigenvalue solvers, especially as examples of very large matrices. (C) 2010 Elsevier B.V. All rights reserved.

Decaimento dos autovalores de operadores integrais gerados por núcleos positivos definidos; Decay rates for eigenvalues of integral operators generated by positive definite kernels

Ferreira, Jose Claudinei
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Dissertação de Mestrado Formato: application/pdf
Publicado em 11/02/2008 PT
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Inicialmente, estudamos alguns resultados clássicos da teoria dos núcleos positivos definidos e alguns resultados pertinentes. Estudamos em seguida, o Teorema de Mercer e algumas de suas generalizações e conseqüências, incluindo a caracterização da transformada de Fourier de um núcleo positivo definido com domínio Rm£Rm, m ¸ 1. O trabalho traz um enfoque especial nos núcleos cujo domínio é um subconjunto não-compacto de Rm £ Rm, uma vez que os demais casos são considerados de maneira extensiva na literatura. Aplicamos esses estudos na análise do decaimento dos autovalores de operadores integrais gerados por núcleos positivos definidos; Firstly, we study some classical results from the theory of positive definite kernels along with some related results. Secondly, we focus on generalizations of Mercer's theorem and some of their implications. Special attention is given to the cases where the domain of the kernel is not compact, once the other cases are considered consistently in the literature. We include a characterization for the Fourier transform of a positive definite kernel on Rm£Rm, m ¸ 1. Finally, we apply the previous study in the analysis of decay rates for eigenvalues of integral operators generated by positive definite kernels

Perturbation splitting for more accurate eigenvalues

Ralha, Rui
Fonte: Society for Industrial and Applied Mathematics (SIAM) Publicador: Society for Industrial and Applied Mathematics (SIAM)
Tipo: Artigo de Revista Científica
Publicado em /02/2009 ENG
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Let $T$ be a symmetric tridiagonal matrix with entries and eigenvalues of different magnitudes. For some $T$, small entrywise relative perturbations induce small errors in the eigenvalues, independently of the size of the entries of the matrix; this is certainly true when the perturbed matrix can be written as $widetilde{T}=X^{T}TX$ with small $||X^{T}X-I||$. Even if it is not possible to express in this way the perturbations in every entry of $T$, much can be gained by doing so for as many as possible entries of larger magnitude. We propose a technique which consists of splitting multiplicative and additive perturbations to produce new error bounds which, for some matrices, are much sharper than the usual ones. Such bounds may be useful in the development of improved software for the tridiagonal eigenvalue problem, and we describe their role in the context of a mixed precision bisection-like procedure. Using the very same idea of splitting perturbations (multiplicative and additive), we show that when $T$ defines well its eigenvalues, the numerical values of the pivots in the usual decomposition $T-lambda I=LDL^{T}$ may be used to compute approximations with high relative precision.; Fundação para a Ciência e Tecnologia (FCT) - POCI 2010

Reliable eigenvalues of symmetric tridiagonals

Ralha, Rui
Fonte: SIAM Publicador: SIAM
Tipo: Artigo de Revista Científica
Publicado em /12/2011 ENG
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For the eigenvalues of a symmetric tridiagonal matrix T, the most accurate algorithms deliver approximations which are the exact eigenvalues of a matrix whose entries differ from the corresponding entries of T by small relative perturbations. However, for matrices with eigenvalues of different magnitudes, the number of correct digits in the computed approximations for eigenvalues of size smaller than ‖T‖₂ depends on how well such eigenvalues are defined by the data. Some classes of matrices are known to define their eigenvalues to high relative accuracy but, in general, there is no simple way to estimate well the number of correct digits in the approximations. To remedy this, we propose a method that provides sharp bounds for the eigenvalues of T. We present some numerical examples to illustrate the usefulness of our method.; FEDER (Programa Operacional Factores de Competitividade); FCT (Projecto PEst-C/MAT/UI0013/2011

On the eigenvalues of Jordan products

Martins, E.A.; Silva, F.C.
Fonte: Elsevier Publicador: Elsevier
Tipo: Artigo de Revista Científica
ENG
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This paper studies the possible eigenvalues of the Jordan product XA+AX, when A is fixed and X varies. © 2002 Elsevier Science Inc. All rights reserved.; Centro de Estruturas Lineares e Combinatórias; FCT

Main eigenvalues and (κ, τ)-regular sets

Cardoso, D.M.; Sciriha, I.; Zerafa, C.
Fonte: Elsevier Publicador: Elsevier
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
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A (κ, τ)-regular set is a subset of the vertices of a graph G, inducing a κ-regular subgraph such that every vertex not in the subset has τ neighbors in it. A main eigenvalue of the adjacency matrix A of a graph G has an eigenvector not orthogonal to the all-one vector j. For graphs with a (κ, τ)-regular set a necessary and sufficient condition for an eigenvalue be non-main is deduced and the main eigenvalues are characterized. These results are applied to the construction of infinite families of bidegreed graphs with two main eigenvalues and the same spectral radius (index) and some relations with strongly regular graphs are obtained. Finally, the determination of (κ, τ)-regular sets is analyzed. © 2009 Elsevier Inc. All rights reserved.; CEOC; FCT; FEDER/POCI 2010; University of Malta

Eigenvalues of Matrix commutators

Martins, Enide Andrade; Silva, Fernando
Fonte: Taylor & Francis Publicador: Taylor & Francis
Tipo: Artigo de Revista Científica
ENG
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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.

Eigenvalues of a H-generalized join graph operation constrained by vertex subsets

Cardoso, Domingos M.; Martins, Enide A.; Robbiano, Maria; Rojo, Oscar
Fonte: Elsevier Publicador: Elsevier
Tipo: Artigo de Revista Científica
ENG
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36.6%
A generalized H-join operation of a family of graphs G1, . . . , Gp, where H has order p, constrained by a family of vertex subsets Si ⊆V(Gi), i = 1, . . . , p, is introduced. When each vertex subset Si is (ki, τi)-regular, it is deduced that all non-main adjacency eigenvalues of Gi , different from ki−τi , remain as eigenvalues of the graph G obtained by this operation. If each Gi is ki-regular and all the vertex subsets are such that Si = V(Gi), the H-generalized join constrained by these vertex subsets coincides with the H-join operation. Furthermore, some applications on the spread of graphs are presented.

On the eigenvalues of Euclidean distance matrices

Alfakih,A.Y.
Fonte: Sociedade Brasileira de Matemática Aplicada e Computacional Publicador: Sociedade Brasileira de Matemática Aplicada e Computacional
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/01/2008 EN
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In this paper, the notion of equitable partitions (EP) is used to study the eigenvalues of Euclidean distance matrices (EDMs). In particular, EP is used to obtain the characteristic polynomials of regular EDMs and non-spherical centrally symmetric EDMs. The paper also presents methods for constructing cospectral EDMs and EDMs with exactly three distinct eigenvalues.

Error bound for a perturbed minimization problem related with the sum of smallest eigenvalues

Travaglia,Marcos Vinicio
Fonte: Sociedade Brasileira de Matemática Aplicada e Computacional Publicador: Sociedade Brasileira de Matemática Aplicada e Computacional
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/06/2010 EN
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Let C be a n×n symmetric matrix. For each integer 1 < k < n we consider the minimization problem m(ε): = minX{ Tr{CX} + εƒ(X)}. Here the variable X is an n×n symmetric matrix, whose eigenvalues satisfy the number ε is a positive (perturbation) parameter and ƒ is a Lipchitz-continuous function (in general nonlinear). It is well known that when ε = 0 the minimum value, m(0), is the sum of the smallest k eigenvalues of C. Assuming that the eigenvalues of C satisfy λ1(C) < ... < λk(C) < λk+1(C) < ∙∙∙ < λn(C), we establish the following upper and lower bounds for the minimum value m(ε): where is the minimum value of ƒ over the solution set of unperturbed problem and L is the Lipschitz-constant of ƒ. The above inequality shows that the error by replacing the upper bound (or the lower bound) by the exact value is at least quadratic in the perturbation parameter. We also treat the case that λk+1(C) = λk(C). We compare the exact solution with the upper and lower bounds for some examples. Mathematical subject classification: 15A42...

Transformações lineares, autovalores e autovetores; Linear transformations, eigenvalues and eigenvectors

Ramos, Marco Aurélio David
Fonte: Universidade Federal de Goiás; Brasil; UFG; Programa de Pós-graduação em PROFMAT (RG); Instituto de Matemática e Estatística - IME (RG) Publicador: Universidade Federal de Goiás; Brasil; UFG; Programa de Pós-graduação em PROFMAT (RG); Instituto de Matemática e Estatística - IME (RG)
Tipo: Dissertação Formato: application/pdf
POR
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In this thesis we study linear transformations, eigenvalues and eigenvectors with the objective of solve a system of linear ordinary differential equations with constant coefficients.; Nesta dissertação estudamos transformações lineares, autovalores e autovetores com o intuito de resolvermos um sistema de equações diferenciais ordinárias lineares com coeficientes constantes.

On eigenvalues, case deletion and extremes in regression

Velilla, Santiago
Fonte: Universidade Carlos III de Madrid Publicador: Universidade Carlos III de Madrid
Tipo: info:eu-repo/semantics/workingPaper; info:eu-repo/semantics/workingPaper Formato: application/pdf
Publicado em /10/1990 ENG
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This paper presents an approximation for assessing the effect of deleting an observation in the eigenvalues of the correlation matrix of a multiple linear regression modelo Applications in connection with the detection of collinearityinfluential observations are explored.

Deflation for block eigenvalues of block partitioned matrices with an application to matrix polynomials of commuting matrices

Pereira, E.; Vitória, J.
Fonte: Universidade de Coimbra Publicador: Universidade de Coimbra
Tipo: Artigo de Revista Científica Formato: aplication/PDF
ENG
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A method for computing a complete set of block eigenvalues for a block partitioned matrix using a generalized form of Wielandt's deflation is presented. An application of this process is given to compute a complete set of solvents of matrix polynomials where the coefficients and the variable are commuting matrices.; http://www.sciencedirect.com/science/article/B6TYJ-444G6J6-H/1/c3a6cac92904c18a24a2d28ebf602b0f

A soma dos maiores autovalores da matriz laplaciana sem sinal em famílias de grafos; The sum of the largest eigenvalues of singless Laplacian matrix on graphs families

Bruno Dias Amaro
Fonte: Biblioteca Digital da Unicamp Publicador: Biblioteca Digital da Unicamp
Tipo: Tese de Doutorado Formato: application/pdf
Publicado em 05/12/2014 PT
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A Teoria Espectral de Grafos é um ramo da Matemática Discreta que se preocupa com a relação entre as propriedades algébricas do espectro de certas matrizes associadas a grafos, como a matriz de adjacência, laplaciana ou laplaciana sem sinal e a topologia dos mesmos. Os autovalores e autovetores das matrizes associadas a um grafo são os invariantes que formam o autoespaço de grafos. Em Teoria Espectral de Grafos a conjectura proposta por Brouwer e Haemers, que associa a soma dos k maiores autovalores da matriz Laplaciana de um grafo G com seu número de arestas mais um fator combinatório (que depende do valor k adotado) é uma das questões interessantes e que está em aberto na literatura. Essa mostra diversos trabalhos que tentam provar tal conjectura. Em 2013, Ashraf et al. estenderam essa conjectura para a matriz laplaciana sem sinal e provaram que ela é válida para a soma dos 2 maiores autovalores e que também é válida para todo k, caso o grafo seja regular. Nosso trabalho aborda a versão dessa conjectura para a matriz laplaciana sem sinal. Conseguimos obter uma família de grafos que satisfaz a conjectura para a soma dos 3 maiores autovalores da matriz laplaciana sem sinal e a família de grafos split completo mais uma aresta satisfaz a conjectura para todos os autovalores. Ainda...

Carrier frequency effect on the MIMO eigenvalues in an indoor environment; Efecto de la frecuencia portadora sobre los valores propios de los sistemas MIMO en interiores

Llano, Gonzalo; García Ariza, Alexis Paolo; Rubio, Lorenzo; Reig, Juan
Fonte: Universidad Icesi; Facultad de Ingeniería Publicador: Universidad Icesi; Facultad de Ingeniería
Tipo: article; Artículo Formato: PDF; p.33-41; Electrónico
SPA
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El efecto de la frecuencia portadora sobre los valores propios de los sistemas MIMO (multiple-input multiple-output) es investigado experimentalmente en un entorno indoor, considerando condiciones de línea de vista (LOS: line-of-sight) y sin línea de vista (NLOS: non-line-of-sight). Los resultados muestran una reducción en la potencia media de los valores propios del sistema MIMO, lo cual es debido a un incremento en la correlación espacial entre los sub-canales cuando la frecuencia portadora se incrementa. Este efecto causa una reducción en la capacidad del sistema MIMO.; The effect of the carrier frequency on the multiple-input multiple-output (MIMO) system eigenvalues is investigated experimentally in an indoor environment considering lineof- sight (LOS) and non-line-of-sight (NLOS) conditions. The results show a reduction of the mean power gain in the MIMO system eigenvalues due to a correlation increment between the spatial subchannels when the carrier frequency increases. This effect causes a slightly reduction of the MIMO capacity.

Acyclic Digraphs and Eigenvalues of O,1 Matrices

McKay, Brendan; Oggier, Frederique; Royle, Gordon; Sloane, N J A; Wanless, Ian; Wilf, Herbert
Fonte: University of Waterloo Publicador: University of Waterloo
Tipo: Artigo de Revista Científica
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We show that the number of acyclic directed graphs with n labeled vertices is equal to the number of n × n (0, 1)-matrices whose eigenvalues are positive real numbers.

Multi-point quasi-rational approximants for the energy eigenvalues of two-power potentials

Martin,P.; Castro,E.; Paz,J.L.
Fonte: Sociedad Mexicana de Física Publicador: Sociedad Mexicana de Física
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/08/2012 EN
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Analytic approximants for the eigenvalues of the one-dimensional Schrödinger equation with potentials of the form V(x) = xª + λx b are found using a multi-point quasi-rational approximation technique. This technique is based on the use of the power series and asymptotic expansion of the eigenvalues in λ, as well as the expansion at intermediate points. These expansions are found through a system of differential equations. The approximants found are valid and accurate for any values of λ > 0 (with b > a). As an example, the technique is applied to the quartic anharmonic oscillator.

Energy eigenvalues for free and confined triple-well potentials

Aquino,N.; Garza,J.; Campoy,G.; Vela,A
Fonte: Sociedad Mexicana de Física Publicador: Sociedad Mexicana de Física
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/02/2011 EN
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Some confined and unconfined (free) one-dimensional triple-well potentials are analyzed with two different numerical approaches. Confinement is achieved by enclosing the potential between two impenetrable walls. The unconfined (free) system is recovered as the positions of the walls move to infinity. The numerical solutions of the Schrodinger equation for the symmetric and asymmetric potentials without confinement, are comparable in precision with those obtained anaylitically. For the symmetric triple-well potentials, V (x) = αx2 - βx4 + x6, it is found that there are sets of two or three quasi-degenerate eigenvalues depending on the parameters a and ¡3. A heuristic analysis shows that if the conditions α= (β2 /4) ± 1 (with α > 0 and β > 0) are satisfied, then there are sets of three eigenvalues with similar energy. An interesting behavior is found when one impenetrable wall is fixed and the other is moved to different positions. In summary, the number of local minima that the potential has in the confined region determines a two- or three-fold degeneracy.