scispace - formally typeset
Search or ask a question
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.


Papers
More filters
Journal ArticleDOI
TL;DR: In this article, it was shown that the eigenvalues of a Hermitian matrix H with matrix elements Hij = ΣkAkijak, where Akij are known numbers and Akijak is a set of parameters, can be exactly expanded as E i = ∂E i ∂a k )a k.

2 citations

Proceedings ArticleDOI
03 Oct 2005
TL;DR: In this paper, a general theory of coupling of eigenvalues of complex matrices of arbitrary dimension smoothly depending on real parameters is presented, and the cases of weak and strong coupling are distinguished and their geometric interpretation in two and three-dimensional spaces is given.
Abstract: The paper presents a general theory of coupling of eigenvalues of complex matrices of arbitrary dimension smoothly depending on real parameters. The cases of weak and strong coupling are distinguished and their geometric interpretation in two and three-dimensional spaces is given. General asymptotic formulae for eigenvalue surfaces near diabolic and exceptional points are presented demonstrating crossing and avoided crossing scenarios. Two numerical exam- ples from crystal optics illustrate effectiveness and accuracy of the presented theory. I. INTRODUCTION Behavior of eigenvalues of matrices dependent on param- eters is a problem of general interest having many important applications in natural and engineering sciences. In modern physics, e.g. quantum mechanics, crystal optics, physical chemistry, acoustics and mechanics, multiple eigenvalues in matrix spectra associated with specific effects attract great interest of researchers since the papers (1), (2). In recent papers, see e.g. (3)-(6), two important cases are distinguished: the diabolic points (DPs) and the exceptional points (EPs). From mathematical point of view DP is a point where the eigenvalues coalesce, while corresponding eigenvectors remain different; and EP is a point where both eigenvalues and eigenvectors merge forming a Jordan block. Both the DP and EP cases are interesting in applications and were observed in experiments (6), (7). In this paper we present a general theory of coupling of eigenvalues of complex matrices of arbitrary dimen- sion smoothly depending on multiple real parameters. Two essential cases of weak and strong coupling based on a Jordan form of the system matrix are distinguished. These two cases correspond to diabolic and exceptional points, respectively. We derive general formulae describing coupling and decoupling of eigenvalues, crossing and avoided crossing of eigenvalue surfaces. It is emphasized that the presented theory of coupling of eigenvalues of complex matrices gives not only qualitative, but also quantitative results on behavior of eigenvalues based only on the information taken at the singular points. The paper is based on the author's previous research on interaction of eigenvalues for matrices and differential operators depending on multiple parameters (8)- (11); for more references see the recent book (12).

2 citations

Proceedings ArticleDOI
07 Dec 1999
TL;DR: In this article, the authors study operator Lyapunov equations in which the infinitesimal generator is not necessarily stable, but it satisfies a spectrum decomposition assumption and it has at most finitely many unstable eigenvalues.
Abstract: We study operator Lyapunov equations in which the infinitesimal generator is not necessarily stable, but it satisfies a spectrum decomposition assumption and it has at most finitely many unstable eigenvalues. Under mild conditions, these have unique self-adjoint solutions. We give conditions under which the number of negative eigenvalues of this solution equals the number of unstable eigenvalues of the generator. An application to the bounded real lemma is treated.

2 citations

Journal ArticleDOI
TL;DR: In this paper, the root locus method is used to calculate the eigenvalues step by step from low order systems to high order systems, and the property of Schwartz matrix is also used to determine the factors of the characteristic equation by the searching method.
Abstract: Two methods for calculating eigenvalues of linear systems are proposed. One is to use the root locus method to calculate the eigenvalues step by step from low order systems to high order systems. The other is to use the property of Schwartz matrix to determine the factors of the characteristic equation by the searching method. A method for determining the number of eigenvalues in each half complex plane is also presented.

2 citations

Journal Article
TL;DR: In this paper, a criterion for the existence of an infinite sequence of eigenvalues is stated for insertion-type fillings, and the eigen values embedded in the continuous spectrum are shown to disappear under a small real perturbation of the filling.
Abstract: We consider eigenvalues embedded in the continuous spectrum of the eigenvalue problem for a filled waveguide. A criterion for the existence of an infinite sequence of eigenvalues is stated for insertion-type fillings. The eigenvalues embedded in the continuous spectrum are shown to disappear under a small real perturbation of the filling.

2 citations


Network Information
Related Topics (5)
Eigenvalues and eigenvectors
51.7K papers, 1.1M citations
81% related
Bounded function
77.2K papers, 1.3M citations
80% related
Linear system
59.5K papers, 1.4M citations
80% related
Differential equation
88K papers, 2M citations
80% related
Matrix (mathematics)
105.5K papers, 1.9M citations
79% related
Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20238
20229
20202
20193
20187
201731