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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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TL;DR: In this article, the existence of point eigenvalues in gaps in the essential spectrum was studied and the problem of counting the number of such eigen values in a gap was studied.
Abstract: In this paper we consider a Schrodinger eigenvalue problem with a potential consisting of a periodic part together with a compactly supported defect potential. Such problems arise as models in condensed matter to describe color in crystals as well as in engineering to describe optical photonic structures. We are interested in studying the existence of point eigenvalues in gaps in the essential spectrum, and in particular in counting the number of such eigenvalues. We use a homotopy argument in the width of the potential to count the eigenvalues as they are created. As a consequence of this we prove the following significant generalization of Zheludev's theorem: the number of point eigenvalues in a gap in the essential spectrum is exactly one for sufficiently large gap number unless a certain Diophantine approximation problem has solutions, in which case there exists a subsequence of gaps containing 0,1 or 2 eigenvalues. We state some conditions under which the solvability of the Diophantine approximation problem can be established.

1 citations

Journal ArticleDOI
TL;DR: In this paper, the effect of a compact linear feedback control on the eigenvalues of a Hilbert space oscillator was considered and conditions were derived under which a sequence of complex numbers can be obtained as eigen values using such a feedback control.
Abstract: Consider the effect of a compact linear feedback control on the eigenvalues of a Hilbert space oscillator. It is shown that the kth eigenvalue can be perturbed a distance of only if a sequence is summable. Conditions are also derived under which a sequence of complex numbers can be obtained as eigenvalues using such a feedback control. The analysis gives an explicit form for the control in terms of the desired eigenvalues. A simple application to the stabilization problem for water waves in a finite tank is given.

1 citations

Journal ArticleDOI
TL;DR: The method of calculating the shifts in the eigenvalues of a perturbed matrix is given and the perturbation matrices of same types of perturbations are derived and these equations form the basis of an efficient computer programme that is used in various network esign problems.
Abstract: If a perturbed network or system is described by the sum of the matrix describing the original network and a perturbation matrix, then the shifts in the natural frequencies can be conveniently calculated and studied. In this paper the method of calculating the shifts in the eigenvalues of a perturbed matrix is given and the perturbation matrices of same types of perturbations are derived. The method is then used to formulate a set of simultaneous equations relating the eigenvalue shifts to the variations in the network elements. These equations form the basis of an efficient computer programme that is used in various network esign problems.

1 citations

Journal ArticleDOI

1 citations

Journal ArticleDOI
TL;DR: This paper presents a highly efficient algorithm, named EvArnoldi, for solving the large-scale eigenvalues problem, and in its basic formulation, is mathematically equivalent to ARPACK.
Abstract: Eigenvalues and eigenvectors are an essential theme in numerical linear algebra. Their study is mainly motivated by their high importance in a wide range of applications. Knowledge of eigenvalues is essential in quantum molecular science. Solutions of the Schrodinger equation for the electrons composing the molecule are the basis of electronic structure theory. Electronic eigenvalues compose the potential energy surfaces for nuclear motion. The eigenvectors allow calculation of diople transition matrix elements, the core of spectroscopy. The vibrational dynamics molecule also requires knowledge of the eigenvalues of the vibrational Hamiltonian. Typically in these problems, the dimension of Hilbert space is huge. Practically, only a small subset of eigenvalues is required. In this paper, we present a highly efficient algorithm, named EvArnoldi, for solving the large-scale eigenvalues problem. The algorithm, in its basic formulation, is mathematically equivalent to ARPACK (Sorensen, D. C. Implicitly Restart...

1 citations


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