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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TL;DR: In this paper, the authors considered the Sturm-Liouville problem with the eigenvalue parameter rationally and showed that the spectrum of the problem consists of a continuous component (the range of the function u ), discrete eigenvalues, and possibly a finite number of embedded eigen values.
22 citations
TL;DR: In this article, upper bounds for the solution of the Lyapunov matrix equation are presented for single eigenvalues, summations of eigen values including the trace, and products of Eigenvalues including the determinant.
Abstract: Eigenvalue upper bounds for the solution of the Lyapunov matrix equation are presented for single eigenvalues, summations of eigenvalues including the trace, and products of eigenvalues including the determinant. These bounds are compared to those for the trace and the extreme eigenvalues. >
22 citations
01 Jan 2006
TL;DR: In this paper, the eigenvalue problem for the mattix of a generalized linear operator is considered, and the problem is reduced to the analysis of an idempotent analogue of the charactetistic polynomial of the matix.
Abstract: The eigenvalue problem for the mattix of a generalized linear operator is considered. In the case of irreducible mattices, the problem is reduced to the analysis of an idempotent analogue of the charactetistic polynomial of the mattix. The eigenvectors are obtained as solutions to a homogeneous equation. The results are then extended to cover the case of an arbitrary mattix. It is shown how to build a basis of the eigensubspace of a mattix. In conclusion, an inequality for matlix powers and eigenvalues is presented, and some extremal properties of eigenvalues are considered.
22 citations
TL;DR: In this article, the authors considered the Neumann boundary-value problem of finding the small-parameter asymptotics of the eigenvalues and eigenfunctions for the Laplace operator in a singularly perturbed domain consisting of two bounded domains joined by a thin "handle".
Abstract: We consider the Neumann boundary-value problem of finding the small-parameter asymptotics of the eigenvalues and eigenfunctions for the Laplace operator in a singularly perturbed domain consisting of two bounded domains joined by a thin "handle". The small parameter is the diameter of the cross-section of the handle. We show that as the small parameter tends to zero these eigenvalues converge either to the eigenvalues corresponding to the domains joined or to the eigenvalues of the Dirichlet problem for the Sturm-Liouville operator on the segment to which the thin handle contracts. The main results of this paper are the complete power small-parameter asymptotics of the eigenvalues and the corresponding eigenfunctions and explicit formulae for the first terms of the asymptotics. We consider critical cases generated by the choice of the place where the thin "handle" is joined to the domains, as well as by the multiplicity of the eigenvalues corresponding to the domains joined.
22 citations
TL;DR: This work proposes, analyze, and test new algorithms for computing first and higher order derivatives of eigenvalues and eigenvectors that are valid much more generally and confirm the effectiveness of the methods for tightly clustered eigen values.
Abstract: We consider quadratic eigenvalue problems in which the coefficient matrices, and hence the eigenvalues and eigenvectors, are functions of a real parameter. Our interest is in cases in which these functions remain differentiable when eigenvalues coincide. Many papers have been devoted to numerical methods for computing derivatives of eigenvalues and eigenvectors, but most require the eigenvalues to be well separated. The few that consider close or repeated eigenvalues place severe restrictions on the eigenvalue derivatives. We propose, analyze, and test new algorithms for computing first and higher order derivatives of eigenvalues and eigenvectors that are valid much more generally. Numerical results confirm the effectiveness of our methods for tightly clustered eigenvalues.
22 citations