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 standard method for finding eigenvalues of a single differential equation may be applied in generalized form to the coupled eigenvalue equations to yield the eigenfunctions themselves rather than the matrix elements.
Abstract: Previous study of the eigenvalue spectrum of time-dependent Hartree-Fock theory has involved the examination of the poles of linear response functions obtained by solving coupled perturbation equations. It is shown how the standard method for finding eigenvalues of a single differential equation may be applied in generalized form to the coupled eigenvalue equations to yield the eigenfunctions themselves rather than the matrix elements. A calculation is given for He(1s2p)1P.
2 citations
TL;DR: In this paper, the trees with exactly three Q-main eigenvalues, where one of them is zero, were identified, and the same authors also identified trees with one zero eigenvalue.
Abstract: For a simple graph G, the Q-eigenvalues are the eigenvalues of the signless Laplacian matrix Q of G. A Q-eigenvalue is said to be a Q-main eigenvalue if it admits a corresponding eigenvector non orthogonal to the all-one vector, or alternatively if the sum of its component entries is non-zero. In the literature the trees, unicyclic, bicyclic and tricyclic graphs with exactly two Q-main eigenvalues have been recently identied. In this paper we continue these investigations by identifying the trees with exactly three Q-main eigenvalues, where one of them is zero.
2 citations
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TL;DR: In this paper, a quantum particle moving in the one dimensional lattice Z and interacting with a indefinite sign external field v was considered, and it was shown that the associated discrete Schroedinger operator H can have one or two eigenvalues, situated as below the bottom of the essential spectrum, as well as above its top.
Abstract: We consider a quantum particle moving in the one dimensional lattice Z and interacting with a indefinite sign external field v. We prove that the associated discrete Schroedinger operator H can have one or two eigenvalues, situated as below the bottom of the essential spectrum, as well as above its top. Moreover, we show that the operator H can have two eigenvalues outside of the essential spectrum such that one of them is situated below the bottom of the essential spectrum, and other one above its top.
2 citations
TL;DR: In this paper, a Gershgorin's type result on the localisation of the spectrum of a matrix is presented, which relies upon the method of Schur complements and outperforms the one based on the Cassini ovals of Ostrovski and Brauer.
Abstract: We present a Gershgorin's type result on the localisation of the spectrum of a matrix. Our method is elementary and relies upon the method of Schur complements, furthermore it outperforms the one based on the Cassini ovals of Ostrovski and Brauer. Furthermore, it yields estimates that hold without major differences in the cases of both scalar and operator matrices. Several refinements of known results are obtained.
2 citations
TL;DR: In this paper, the authors give necessary and sufficient conditions for the occurrence of Q-spectral integral variation only in two places, as the first case never occurs, while the second case always occurs.
2 citations