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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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Journal ArticleDOI
TL;DR: In this article, a new algorithm for computing the eigenvalues of singular Sturm-Liouville problems with separated self-adjoint boundary conditions for both the limit-circle nonoscillatory and oscillatory cases is described.
Abstract: We describe a new algorithm to compute the eigenvalues of singular Sturm-Liouville problems with separated self-adjoint boundary conditions for both the limit-circle nonoscillatory and oscillatory cases. Also described is a numerical code implementing this algorithm and how it compares with SLEIGN. The latter is the only effective general purpose software available for the computation of the eigenvalues of singular Sturm-Liouville problems.

93 citations

Journal ArticleDOI
TL;DR: In this article, the eigenvalues and eigenfunctions of a two-level system interacting with a one-mode quantum field are calculated numerically using the operator method.
Abstract: Accurate eigenvalues and eigenfunctions of a two-level system interacting with a one-mode quantum field are calculated numerically. A special iteration procedure based on the operator method permits one to consider the solution within a wide range of the Hamiltonian parameters and to find the uniformly approximating analytical formula for the eigenvalues. Characteristic features of the model are considered, such as the level intersections, the population of the field states and the chaotization in the system through the doubling of the frequencies.

93 citations

Journal ArticleDOI
TL;DR: In this paper, the authors consider the case when a Hermitian linear operator A is slightly perturbed, and give a lower bound S > 0 for the gap that separates the cluster from all other eigenvalues.
Abstract: When a Hermitian linear operator A is slightly perturbed, by how much can its invariant subspaces change? Given some approximations to a cluster of neighboring eigenvalues and to the corresponding eigenvectors of a real symmetric matrix, and given a lower bound S > 0 for the gap that separates the cluster from all other eigenvalues, how much can the subspace spanned by the eigenvectors differ from that spanned by our approximations? These questions are closely related; both are investigated here. First the difference between the two subspaces is characterized in terms of certain angles through which one subspace must be rotated in order most directly to reach the other. The angles constitute the spectrum of a Hermitian operator ©, with which is associated a commuting skew-Hermitian operator J=—J; the unitary operator that differs least from the identity and rotates one subspace into the other turns out to be exp(/@). These operators unify the treatment of natural geometric, operatortheoretic and error-analytic questions concerning those subspaces. Given the gap ô, and given bounds upon either the perturbation (1st question) or a computable residual (2nd question), we obtain sharp bounds upon unitary-invariant norms of trigonometric functions of ©. (A norm is unitary-invariant whenever | | i | | = | | C /LFI | for all unitary U and V. Examples are the bound-norm | |L | | I = supj|Lx||/||x|| and the square-norm | | i | | , f f= (trace L*L) .) In this note we consider a finite-dimensional unitary space 3C in which the scalar product is denoted by y*x, and ||x|| = (#*#). Proofs of the following statements will appear elsewhere, together with extensions to infinite-dimensional Hubert spaces and to noncompact or unbounded operators [2]. Tha t article discusses the relation of 'our results to earlier work on the subject, such as [ l ] , [3], [4].

91 citations

Journal ArticleDOI
TL;DR: In this paper, the existence of large isolated (non-unit) eigenvalues of the Perron-Frobenius operator for expanding interval maps is discussed, and a systematic means of constructing maps which possess such isolated eigenfunctions is presented.
Abstract: We discuss the existence of large isolated (non-unit) eigenvalues of the Perron-Frobenius operator for expanding interval maps. Corresponding to these eigenvalues (or `resonances') are distributions which approach the invariant density (or equilibrium distribution) at a rate slower than that prescribed by the minimal expansion rate. We consider the transitional behaviour of the eigenfunctions as the eigenvalues cross this `minimal expansion rate' threshold, and suggest dynamical implications of the existence and form of these eigenfunctions. A systematic means of constructing maps which possess such isolated eigenvalues is presented.

91 citations

Journal ArticleDOI
TL;DR: In this article, the authors present a collection of relative perturbation results which have emerged during the past ten years and show that the derivation of many relative bounds can be based on absolute bounds.
Abstract: It used to be good enough to bound absolute of matrix eigenvalues and singular values. Not any more. Now it is fashionable to bound relative errors. We present a collection of relative perturbation results which have emerged during the past ten years.No need to throw away all those absolute error bound, though. Deep down, the derivation of many relative bounds can be based on absolute bounds. This means that relative bounds are not always better. They may just be better sometimes – and exactly when depends on the perturbation.

91 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20238
20229
20202
20193
20187
201731