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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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36.6%
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
Relevância na Pesquisa
36.68%
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...