Topic
Spectrum of a matrix
About: Spectrum of a matrix is a research topic. Over the lifetime, 1064 publications have been published within this topic receiving 19841 citations. The topic is also known as: matrix spectrum.
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28 citations
07 Aug 1992
TL;DR: In this paper, it was shown that a matrix has this property if and only if its associated bipartite graph is acyclic, and the same algorithm can compute the singular values of such a matrix to high relative accuracy.
Abstract: : It is known that small relative perturbations in the entries of a bidiagonal matrix only cause small relative perturbations in its singular values, independent of the values of the matrix entries. In this paper we show that a matrix has this property if and only if its associated bipartite graph is acyclic. We also show how to compute the singular values of such a matrix to high relative accuracy. The same algorithm can compute eigenvalues of symmetric acyclic matrices with tiny component-wise relative backward error. This class includes tridragonal matfices, arrow matrices, and exponentially many others.
28 citations
TL;DR: In this article, a geometric method, based on a decomposition of the space of complex matrices, is employed to study the variation of the spectrum of a matrix, which leads to some classical inequalities as well as some new ones.
Abstract: A geometric method, based on a decomposition of the space of complex matrices, is employed to study the variation of the spectrum of a matrix. When adapted to special cases, this leads to some classical inequalities as well as some new ones. As an example of the latter, we show that if U, V are unitary matrices and K is a skew-Hermitian matrix such that UV~' = exp K, then for every unitary-invariant norm the distance between the eigenvalues of V and those of Vis bounded by \\K\\. This generalises two earlier results which used particular unitary-invariant norms.
28 citations
TL;DR: In this article, the authors examined the selfadjoint operator of the spectrum of the self-adjoint function and showed that the potential tends to be asymptotic in the spectrum.
Abstract: We examine the selfadjoint operator in . We assume that the potential tends to as . Under these conditions the spectrum of is discrete. In the paper the well-known asymptotic formula (*)for the distribution function of the eigenvalues is justified under very weak assumptions on , namely the following conditions: 1) , where . 2) almost everywhere when . 3) There exist a continuous function , , , and an index such that for any , , .Bibliography: 12 items.
28 citations
TL;DR: The method is based on polynomial transformations of the Wilson–Dirac operator, leading to considerable improvements of the computation of eigenvalues, and can be applied to operators with a symmetric and bounded eigenspectrum.
27 citations